\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 52, pp. 1--51.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/52\hfil Dynamics of the characteristic curves]
{On the dynamics of the characteristic curves for the LSW model}
\author[J. J. L. Vel\'{a}zquez\hfil EJDE-2006/52\hfilneg]
{Juan J. L. Vel\'{a}zquez}

\address{Juan J. L. Vel\'{a}zquez \newline
 Departamento de Matem\'{a}tica Aplicada,
 Facultad de Matem\'{a}ticas, Universidad Complutense,
 Madrid 28040, Spain}
 \email{velazque@mat.ucm.es}
\date{}
\thanks{Submitted December 5, 2005. Published April 24, 2006.}
\thanks{Partially supported by grant MTM2004-05634 from  DGES}
\subjclass[2000]{35L40, 35A34, 34K60} 
\keywords{Oswald ripening; asymptotic behavior; Schwartzian derivative;
 \hfill\break\indent integral equations}

\begin{abstract}
 This paper describes in a rigorous manner how the dynamics
 of the characteristic curves for the Lifshitz-Slyozov-Wagner
 (LSW) model of coarsening  transforms a class of noncompactly
 supported initial data  in functions that behave in a
 self-similar manner for long times.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

The purpose of this paper is to obtain in a rigorous way some
properties for the dynamics of the characteristic curves
associated with the Lifshitz-Slyozov-Wagner (LSW) model
\begin{gather}
\frac{\partial f(R,t)}{\partial t}+\frac{\partial}{\partial R}
\Big(\big( -\frac{1}{R^{2}}+\frac{\Delta(t)}{R}\big)
f(R,t)\Big)  =0,\quad t>0\,,\; R>0\,,\label{S1E1}\\
f(R,0) =f_0(R)\geq0\,,\quad R>0\,,\label{S1E2}\\
\Delta(t) =\frac{\int_0^{\infty}f(R,t)\,\,dR}{\int_0^{\infty}f(R,t)R\,dR}\,.
\label{S1E3}
\end{gather}
This  system was introduced  to study the coarsening stage of the
so-called Oswald ripening (cf. \cite{LS,W}). The choice of
$\Delta(t)$ above ensures that the volume density of the particles
$\int_0^{\infty}f(R,t)R^{3}\,dR $ is preserved during their
evolution. Rigorous derivations of this system using
homogenization techniques, that take as starting point the
Mullins-Sekerka free boundary problem, have been obtained in
different scaling limits in \cite{N,NO,NV}.

The system \eqref{S1E1}--\eqref{S1E3} has a family of explicit self-similar
solutions with the form (cf. \cite{LS}):
\begin{equation}
f(R,t)=\frac{1}{t^{4/3}}\Phi(\rho),\quad \rho =\frac{R}{t^{1/3}}
\label{S1E4}
\end{equation}
The solutions of \eqref{S1E1}--\eqref{S1E3} with the form (\ref{S1E4})
having finite volume fraction
\[
\phi\equiv\int_0^{\infty}f(R,t)R^{3}\,dR<\infty
\]
are compactly supported on the space $\rho$.\ For each given value of the
volume fraction filled by the particles $\phi$ there exists a one-parameter
family of self-similar solutions of \eqref{S1E1}--\eqref{S1E3}.

It was rigorously proved in \cite{NP1} that the long time asymptotics of the
compactly supported solutions of \eqref{S1E1}--\eqref{S1E3} depends very
sensitively on the asymptotics of the initial data near the maximum radius.
Additional results concerning the long time asymptotics of the LSW system
\eqref{S1E1}--\eqref{S1E3} for compactly supported initial data can be found in
\cite{NV1}, \cite{NV2}.


The long time asymptotics of the solutions of
\eqref{S1E1}--\eqref{S1E3} for noncompactly supported initial data
has been studied in \cite{V2}. In that paper was shown that there
exist noncompactly supported initial data $f_0(R)$ yielding
nonselfsimilar behaviour as $t\to \infty$. Moreover, in
that paper were derived approximations of the noncompactly
supported solutions of the LSW model for long times by means of
asymptotic arguments. The resulting approximate LSW equations are
a set of integrodifferential equations with two different time
scales as $t\to \infty$.



The main goal of this paper is to provide some rigorous basis for some results
that were just obtained in a formal manner in \cite{V2}.

It was already implicit in the arguments of the seminal paper
\cite{LS} that the key problem in order to understand the dynamics
of the LSW model is to describe the behaviour of the
characteristic curves associated to \eqref{S1E1}--\eqref{S1E3} near
the so-called critical radius. In an informal manner we can say
that the critical radius is the value of the radius for which the
velocity of the characteristics in the space of radii, written in
self-similar variables is close to zero. Due to this fact the
characteristic curves ``leak'' very slowly from the region of
supercritical radii to the region of subcritical radii. To compute
in detail the rate of ``leaking'' is crucial in order to describe
asymptotically the long time behaviour of noncompactly supported
solutions of \eqref{S1E1}--\eqref{S1E3} behaving in a self-similar
manner as $t\to\infty$. In particular, the analysis of the
``leaking'' phenomenon yields an asymptotics for the rate of
change of the particles in the form
\begin{equation}
\frac{1}{t^{2}}+\frac{1}{t^{2}(\log(t))^{2}
}+\frac{1}{t^{2}(\log(t))^{2}(\log( \log( t))) ^{2}}+\dots
\quad\text{as }t\to \infty\label{logasymp}
\end{equation}
This type of formula were already obtained in \cite{LS}. A more detailed study
of these asymptotics was made in \cite{V}. On the other hand, the analysis in
\cite{V2} provides a simpler and more general explanation for the onset of the
asymptotics (\ref{logasymp}).

The main contribution of this paper is to develop mathematical
methods that allow to analyze the behaviour of the characteristic
curves near the critical radius in a fully rigorous manner. Only a
small fraction of the formal arguments in \cite{V2} are proved
rigorously in this paper. Nevertheless, the mathematical results
in this paper show that some of the main ideas in \cite{V2} can
indeed be made mathematically rigorous. Note that the results of
this paper are not obtained for the full LSW model, but for a
simplified problem that takes into account only the transition of
the characteristic curves near the critical line. Such transition
is the most relevant part in the transformation of general,
noncompactly supported initial data in self-similar solutions as
$t\to\infty$.

The plan of the paper is the following:
In Section 2 we introduce some basic notation and formulate the main result
proved in the paper.
In Section 3 we compute in a formal manner the stretching
generated by the dynamics of the characteristic curves. In
particular we will\ formally obtain in this Section that the
transition of the characteristic curves through the critical
region can be approximated by means of the problem (\ref{S5E1}
)-(\ref{S5E4}). This problem can be solved in an explicit manner,
as it was noticed in \cite{V2}. This fact plays an essential role
in many of the arguments in the rest of the paper. For this reason
we will recall the solution of \eqref{S5E1}--\eqref{S5E4} in
Section 4.

The rest of the paper consists studying the evolution of the characteristic
curves associated to \eqref{S1E1}, \eqref{S1E2} by means of a perturbative
argument. More precisely, it will be seen in Section 5 that the transition of
the characteristic curves of \eqref{S1E1}, \eqref{S1E2} near the critical
region can be approximated in some sense by means of the solution of the
problem \eqref{S5E1}--\eqref{S5E4}. To make precise the ``closedness''
of the solutions of these two problems it is convenient to reformulate the
solutions of the problem described in Section 5 by means of an integral
equations whose local solvability in time is obtained in Section 6. The local
solvability of this integral equation is given in Section 7, as well as the
long time asymptotics of the resulting solutions.

Some  technical properties of some auxiliary functions $f$ are given in
Appendix A at the end of the paper.

\section{Statement of the main result}

It is convenient, in order to simplify some computations, to replace the
density in the space of radii by the density in the space of volumes. We
define $\bar{f}(v,t)$ by means of:
\begin{gather*}
v    =R^{3}\\
f(R,t)\,dR    =\bar{f}(v,t)dv\\
\bar{t}    =3t
\end{gather*}
whence $\bar{f}(v,\bar{t})=\frac{1}{3v^{2/3}}f( v^{1/3},t)$. Using
this change of variables, equations \eqref{S1E1}-\eqref{S1E3}
become
\begin{gather}
\frac{\partial\bar{f}(v,\bar{t})}{\partial\bar{t}}
+\frac{\partial}{\partial v}\Big((-1+\Delta(\bar{t}) v^{1/3})
\bar{f}(v,\bar{t})\Big)=0\,,\quad \bar{t}>0\,,\;v>0\label{S2E1}\\
\bar{f}(v,0) =\frac{1}{3v^{2/3}}f_0( v^{1/3})
\equiv\bar{f}_0(v)\,,\quad v>0\label{S2E2}\\
\Delta(\bar{t})  =\frac{\int_0^{\infty}\bar{f}(
v,\bar{t})dv}{\int_0^{\infty}v^{1/3}\bar{f}( v,\bar{t}) dv}
\label{S2E3}
\end{gather}
The analysis of these equations becomes simpler if the density
$\bar{f}( v,\bar{t})$ is replaced by the distribution function
\begin{equation}
F(v,\bar{t})\equiv\int_{v}^{\infty}\bar{f}(\xi,\bar {t})d\xi \,.
\label{S2E4}
\end{equation}
Using this new variable the system (\ref{S2E1})-(\ref{S2E3})
becomes:
\begin{gather}
\frac{\partial F(v,\bar{t})}{\partial\bar{t}}+\Big(
-1+\Delta(\bar{t})v^{1/3}\Big)\frac{\partial F(
v,\bar{t})}{\partial v}    =0\,,\quad \bar{t}>0\,,\;v>0\label{S2E5}\\
F(v,0)  =F_0(v)\equiv\int_{v}^{\infty}
\bar{f}_0(v)d\xi\,,\quad v>0\label{S2E6}\\
\Delta(\bar{t})  =\frac{3F(0^{+},\bar{t})
}{\int_0^{\infty}v^{-2/3}F(v,\bar{t})dv}\,. \label{S2E7}
\end{gather}

Global well posedness for this problem has been studied in
\cite{NP2} for compactly supported initial data and in \cite{NP3}
for noncompactly supported initial data with fast enough decay. We
can assume that $\bar{f}( v,\bar{t})$ is a (nonnegative) measure
satisfying $\int v\bar {f}dv<\infty$, whence $F( v,\bar{t}) $ is
in general a decreasing bounded function. Note that the volume
preservation satisfied by the solutions of (\ref{S2E5}) becomes in
these variables,
\[
\frac{\partial}{\partial\bar{t}}(\int_0^{\infty}v\bar{f}(
v,\bar{t})dv)=\frac{\partial}{\partial\bar{t}}(\int
_0^{\infty}F(v,\bar{t})dv)=0\,.
\]
We introduce self-similar variables by means of:
\begin{equation}
F(v,t)=\frac{G(W,\tau)}{\bar{t}},\quad W=\frac{2v}{t},\quad
\tau=\log(\bar{t})\label{S3E1}
\end{equation}

In these set of variables (\ref{S2E5})-(\ref{S2E7}) become
\begin{gather}
\frac{\partial G}{\partial\tau}+(-2+3\lambda(\tau)
W^{1/3}-W)\frac{\partial G}{\partial W}    =G\label{S3E2}\\
\lambda(\tau) =\frac{2G(0^{+},\tau)}{\int
_0^{\infty}W^{-2/3}G(W,\tau)dW}\label{S3E3}\\
G(W,0) =G_0(W)\label{S3E4}
\end{gather}

The self-similar solution of the LSW system having maximal support
becomes, in this set of variables,
\begin{gather}
G_{s}(W)  =\exp\Big(-\int_0^{W}\frac{d\xi}{2-3\xi
^{1/3}+\xi}\Big), \\
  0\leq W<1\,,\;G_{s}(W) =0\quad \text{if }W>0.\label{S3E5}\\
\lambda(\tau)  =1 \label{S3E6}
\end{gather}
where we have used the normalization $G_{s}(0)=1$.

Let us define $\beta(\tau)=\lambda(\tau)-1$. Since
$\int_0^{\infty}W^{-2/3}G(W,\tau)dW=2$, it follows that
\begin{equation}
\beta(\tau)=\frac{\int_0^{\infty}W^{-2/3}(G( W,\tau)
-G(0^{+},\tau)G_{s}(W)) dW}{\int_0^{\infty}W^{-2/3}G(W,\tau)dW}
\label{S3E7}
\end{equation}
and (\ref{S3E2}) might be written as
\begin{equation}
\frac{\partial G}{\partial\tau}+\Big(-2+3W^{1/3}-W+3\beta(
\tau)W^{1/3}\Big)\frac{\partial G}{\partial W}=G \label{S3E8}
\end{equation}

Suppose that $G(W,\tau)$ behaves in a self-similar manner as
$t\to\infty$. Then (\ref{S3E7}) implies that
\begin{equation}
\lim_{\tau\to\infty}\beta(\tau)=0 \label{S3E8a}
\end{equation}
Therefore, the speed of the characteristics of (\ref{S3E8}) might
be approximated by means of the function $(-2+3W^{1/3}-W)$ that
has a double zero at $W=1$. The line $W=1$ will be termed from now
on as ``critical line''. Due to (\ref{S3E8a}) the characteristic
curves associated to (\ref{S3E8}) remain trapped near the critical
line for long times. For long times the only characteristic curves
that contribute to the asymptotics of $G(W,\tau)$ are those
starting at points $W=W_0$ with $W_0\to\infty$. Since
$G_0(W_0)\to0$ as $W_0\to\infty$, and the values of $G$ increase
exponentially on $\tau$ along characteristics, it follows that in
order to obtain $G(W,\tau)$ of order one for $W\in( 0,1)$ as $\tau
\to\infty$ the characteristics starting at $W=W_0$ with
$W_0\to\infty$ must remain trapped near the critical line $W=1$ a
very precisely tuned amount of time. A detailed computation of the
trapping time can be found in Section \ref{four} below.



The purpose of this paper is to obtain detailed information in a
rigorous manner on the transformations $G_0(W)\to
G(W,\infty)$ induced by the transition of the characteristic
curves along the critical line $W=1$. The main result obtained in
this paper is the following:

\begin{theorem}\label{thm1}
Suppose that $G_0(W)$ satisfies:
\begin{equation}
\lim_{W_0\to\infty}\sup_{0\leq\zeta\leq1}\Big| \frac{G_0(
W_0e^{\Lambda(W_0)\zeta})}{G_0( W_0)}-e^{-\zeta}\Big| =0,
\label{S4E26}
\end{equation}
where
\begin{equation}
\frac{d^{k}\Lambda}{dW_0^{k}}(W_0)\sim\frac{d^{k}}
{dW_0^{k}}(\frac{C}{(W_0)^{\alpha}}) \quad\text{as }
W_0\to\infty\label{S4E26a}
\end{equation}
for some $\alpha>0$ and $C>0$, $k=0,1,2$. Then there exists a
family of functions $\beta(\tau)\in L^{\infty}[0,\infty)$
satisfying
\begin{equation}
\lim_{\tau\to\infty}\beta(\tau)=0 \label{S4E26b}
\end{equation}
such that the corresponding solution of (\ref{S3E2}) with initial
data (\ref{S3E4}) satisfies
\begin{equation}
\lim_{W\to\infty}\frac{G(W,\tau)}{G(
0^{+},\tau)}=G_{s}(W)\label{S4E27}
\end{equation}
uniformly on $W\in\mathbb{R}^{+}$.
\end{theorem}

Condition (\ref{S4E26}) is reminiscent of some similar conditions
on the initial data that might be found in \cite{NP1}, \cite{NV1},
\cite{NV2}. In those papers it was proved that the initial data
$G_0(W)$ must satisfy rather strong conditions in order to yield
self-similar behaviours for long times. Roughly speaking, the
conditions on those papers, as well as (\ref{S4E26}), impose that
a suitable functional transformation of $G_0( W)$ behaves
asymptotically in a self-similar manner as $W\to\infty$.

The main shortcoming of Theorem \ref{thm1} concerning the LSW
theory is that the function $\beta(\tau)$ there is not necessarily
given by (\ref{S3E7}). Theorem \ref{thm1} shows only that it is
possible to transform initial data satisfying (\ref{S4E26}) into a
self-similar behaviour by means of the evolution equation
(\ref{S3E8}). In other words, the transformation induced by the
trapping of the characteristics associated to (\ref{S3E8}) near
the critical line is able to bring initial data satisfying
(\ref{S4E26}) into self-similar behaviour. In particular, the
choice of the function $\beta( \tau)$ derived in Theorem
\ref{thm1} does not yield preservation of the total volume of the
particles. To obtain an unique function $\beta(\tau)$
preserving the total volume of the particles it would be needed to
take into account also the dynamics of the characteristics in the
regions $0<W<1$ and $W>1$ as it was made at the formal level in
\cite{V2}. On the other hand, as it was seen in \cite{V2}, the
feasibility of the transformation described in Theorem \ref{thm1}
is the key feature that must be required on $G_0$ in order to
obtain also volume conservation.

Assumption (\ref{S4E26a}) looks very restrictive and certainly is
not the most general one possible. The main reason for this choice
of $\Lambda( W_0)$ is because under this assumption it will be
possible to handle easily many of the formula later. Nevertheless
(\ref{S4E26a}) could be weakened much. In any case (\ref{S4E26}),
(\ref{S4E26a}) covers several interesting initial data $G_0$. For
instance\ if $G_0$ satisfies
\begin{equation}
G_0(W)\sim CW^{B}(\log(W))^{D}e^{-W^{A}}\quad \text{as
}W\to\infty\,,\;A>0,\;B,D\in \mathbb{R} \label{S4E28}
\end{equation}
then formulae (\ref{S4E26}), (\ref{S4E26a}) hold with
\[
\Lambda(W)\sim\frac{1}{A}\frac{1}{(W)^{A}}
\]
as $W\to\infty$.

\section{A heuristic computation of the stretching generated
 by the evolution of the characteristics} \label{four}

In this Section we describe in a heuristic manner the key argument in this
paper. This argument has been described in a different, but essentially
equivalent manner in \cite{V2}. We recall it here for further reference.

The basic problem is to estimate the transformation induced by the
characteristics of the equation (\ref{S3E8}) for small functions
$\beta( \tau)$. Let us formulate the problem in a more precise
manner. We denote as $W(\tau;W_0)$ the solution of the
differential equation:
\begin{gather}
W_{\tau}   =-2+3W^{1/3}-W+3\beta(\tau)W^{1/3}\label{S4E1}\\
W(0;W_0) =W_0 \,.\label{S4E2}
\end{gather}

Our goal is to obtain approximations for the solutions of this
equation for small $\beta(\tau)$ uniformly valid in time. Since
$\beta(\tau)$is small it follows that the evolution of
$W(\tau;W_0)$ takes place in a very different manner in the
regions $W-1\gg \sqrt{|\beta(\tau)|}$, $1-W\gg
\sqrt{|\beta(\tau)|}$ and $|W-1| \approx\sqrt{|\beta(\tau)|}$.
Indeed, in the first two regions, away from the region $W=1$, we
can approximate the equation (\ref{S4E1}) as:
\begin{equation}
W_{\tau}=-2+3W^{1/3}-W\quad \text{for }|W-1|\gg \sqrt{|
\beta(\tau)|} \label{S4E3}
\end{equation}

This approximation cannot be used in the region $|W-1|
\approx\sqrt{|\beta(\tau)|}$. From now on we will call critical
line the line $W=1$. Using Taylor's expansion and keeping just the
leading terms we would approximate (\ref{S4E1}) in the region
$|W-1|\ll1$ as:
\begin{equation}
W_{\tau}=-\frac{1}{3}(W-1)^{2}+3\beta(\tau)
\label{S4E4}
\end{equation}
This equation shows that $W(\tau;W_0)$ is very sensitive in this
region to the precise form of $\beta(\tau)$. It follows from
(\ref{S4E3}), (\ref{S4E4}) that the evolution of the
characteristics away from the critical line does not depend much
on $\beta(\tau)$, but this evolution is extremely sensitive to the
values of $\beta(\tau)$ in the region $|W-1|
\approx\sqrt{|\beta(\tau)|}$.

To obtain a class of initial data that are transformed by
means of (\ref{S3E8}) in functions close to self-similar solutions
the key problem is to study the evolution of characteristics
$W(\tau;W_0)$ with $W_0$ large. Therefore, we will assume from now
on that $W_0>1$, i.e. that the characteristic curves cross the
critical line during their evolution. Using (\ref{S4E3}) it
follows that, as long as $1-W\gg \sqrt{|\beta(\tau) |}$, we
can approximate $W(\tau ;W_0)$ as:

\begin{equation}
\log\Big(\frac{W_0}{W(\tau;W_0)}\Big) +F_{\rm ext}(W(\tau;W_0))
-F_{\rm ext}(W_0)=\tau , \label{S4E5}
\end{equation}
where
\begin{equation}
F_{\rm ext}(W)\equiv\int_{W}^{\infty}\big[
\frac{1}{2-3\eta^{1/3}+\eta}-\frac{1}{\eta}\big]
d\eta=\int_{W}^{\infty }\frac{(3\eta^{1/3}-2)}{(
2-3\eta^{1/3}+\eta) \eta}d\eta \,.\label{S4E6}
\end{equation}
The asymptotic behaviour of $F(W)$ as $W\to1^{+}$ is
\begin{equation}
F_{\rm ext}(W)=\frac{3}{(W-1)} -\frac{5}{3}\log(W-1)+b_{\rm
ext}+o( 1) \label{S4E7}
\end{equation}
where $b_{\rm ext}\in\mathbb{R}$. We can rewrite (\ref{S4E5}) as
\begin{equation}
W(\tau;W_0)=w_{\rm ext}( \tau-\log( W_0) +F_{\rm ext}(W_0) -b_{\rm
ext})\label{S4E8}
\end{equation}
where
\begin{equation}
-\log(w_{\rm ext}(s)) +F_{\rm ext}(w_{\rm ext}(s) )=s+b_{\rm
ext}\,. \label{S4E9}
\end{equation}
Using (\ref{S4E7}) we obtain the asymptotics
\begin{equation}
w_{\rm ext}(s)\sim1+\frac{3}{s}-\frac{5\log(
\frac{3}{s})}{s^{2}}+o(\frac{1}{s^{2}})\quad \text{as }
s\to\infty\,. \label{S4E10}
\end{equation}
Formulae (\ref{S4E8}), (\ref{S4E10}) suggest to define the arrival
time to the critical line $W=1$ for $W(\tau;W_0)$ as:
\begin{equation}
T_{\rm arr}=\log(W_0)-F(W_0) +b_{\rm ext} \label{S4E11}
\end{equation}
Note that the transition of the characteristic $W(\tau;W_0)$ by
the region $|W-1|\approx\sqrt{|\beta(\tau)|}$ takes place in a
long time if $\beta(\tau)$ is small. Therefore, due to the
asymptotics (\ref{S4E10}) it makes sense to approximate
$W(\tau;W_0)$ near the critical line by means of the solution of
the problem
\begin{gather}
W_{\rm trans,\tau}   =-\frac{1}{3}(W_{\rm trans
}-1)^{2}+3\beta(\tau)\label{S4E12}\\
W_{\rm trans}((T_{\rm arr}) ^{+};W_0)
=+\infty\,.\label{S4E13}
\end{gather}
Indeed, note that the effect in $W_{\rm trans}$ of
the term $\beta(\tau)$ during the range of times in which
$W_{\rm trans}-1\gg \sqrt{|\beta( \tau) |}$ is
small.

On the other hand, after the transition near the critical line has
taken place, we can use again (\ref{S4E3}) to approximate
$W(\tau;W_0)$. Suppose that $W( \bar{\tau};W_0) =\bar{W}$. Arguing
as in the region $W-1\gg \sqrt{|\beta(\tau) |} $ we obtain the
approximation:
\begin{equation}
F_{\rm int}(W(\tau;W_0))
=F_{\rm int}(\bar{W})+(\bar{\tau} -\tau),\label{S4E14}
\end{equation}
where
\begin{equation}
F_{\rm int}(W)=\int_0^{W}\frac{d\eta}
{2-3\eta^{1/3}+\eta}\,.\label{S4E15}
\end{equation}
We have the  asymptotics
\begin{equation}
F_{\rm int}(W)\sim\frac{3}{(1-W)
}+\frac{5}{3}\log(1-W)+b_{\rm int}+o(
1)\quad \text{as }W\to1^{-}\,. \label{S4E16}
\end{equation}
We then write
\begin{equation}
W(\tau;W_0)=w_{\rm int}( \tau-\bar{\tau
}-F_{\rm int}(\bar{W})+b_{\rm int}),\label{S4E17}
\end{equation}
where
\begin{equation}
F_{\rm int}(w_{\rm int}(s)
)=b_{\rm int}-s\,. \label{S4E18}
\end{equation}
Therefore,
\begin{equation}
w_{\rm int}(s)
\sim1+\frac{3}{s}-\frac{5\log(-\frac{3}{s})}{s^{2}}+o\big(
\frac{1}{s^{2}}\big)
\quad\text{as }s\to-\infty\label{S4E19}
\end{equation}
Equations (\ref{S4E17}), (\ref{S4E19}) suggest to define the
``exit time'' from the transition region for $W(\tau;W_0) $ as
\begin{equation}
T_{\rm exit}=\bar{\tau}+F_{\rm int}(\bar {W})-b_{\rm int}\,. \label{S4E19a}
\end{equation}
Using again the smallness of $\beta(\tau)$ it would be natural to
assume that, to the leading order, the function
$W_{\rm trans}$ that solves (\ref{S4E12}),
(\ref{S4E13}) satisfies
\begin{equation}
W_{\rm trans}((T_{\rm exit})
^{-};W_0)=-\infty\,. \label{S4E20}
\end{equation}

Let us summarize. The previous argument indicates that for
$\beta(\tau)$ small we can approximate $W( \tau;W_0)$ using (\ref{S4E8}),
(\ref{S4E9}) if $W-1\gg \sqrt{|\beta(\tau) |}$,
(\ref{S4E17}), (\ref{S4E18}) if $1-W\gg \sqrt{|
\beta(\tau)|}$ and the function
$W_{\rm trans}(\tau;W_0)$ that solves (\ref{S4E12}),
(\ref{S4E13}), (\ref{S4E20}) for $| W-1|
\approx\sqrt{|\beta(\tau)|}$. Note that $W(\tau;W_0)$ is, to
the leading order, independent on $\beta(\tau)$ for $|W-1|
\gg \sqrt{|\beta(\tau)|}$. Therefore, the stretching of the
characteristic curves $W(\tau;W_0)$, is contained, to the leading
order, in the dynamics of the function
$W_{\rm trans}(\tau;W_0)$. Note that for each
$T_{\rm arr}$ and each function $\beta(\tau)$ there is
a unique $T_{\rm exit}$. Let us write the functional
relation between these quantities as
\begin{equation}
T_{\rm arr}=S(T_{\rm exit}),\label{S4E20a}
\end{equation}
where $S(\cdot)$ is an increasing function such that $S(x) <x$.

We can then examine the precise manner in which we should choose
$T_{\rm arr}$, $T_{\rm exit}$ in order to
transform the initial data $G_0(W)$ into a self-similar solution.
Let us denote as $\bar{W}_0$ the starting value of the
characteristic $W(\tau;\bar{W}_0)$ arriving at $W=0$ at the time
$\tau=\bar{\tau}$. By assumption
\begin{equation}
\frac{G(\bar{W},\bar{\tau})}{G( 0^{+},\bar{\tau})}\approx
G_{s}(\bar{W})=e^{-F_{\rm int}(\bar{W})}\,,\quad
0\leq\bar{W}<1\,. \label{S4E21}
\end{equation}
Then, since along the characteristic curves we have
$\frac{dG}{d\tau}=G$, it follows that
\begin{equation}
\frac{G(\bar{W},\bar{\tau})}{G( 0^{+},\bar{\tau})
}=\frac{G_0(W_0)}{G_0(\bar{W}_0)}\,. \label{S4E22}
\end{equation}
Combining (\ref{S4E11}), (\ref{S4E19a}), (\ref{S4E20a}) we obtain
\begin{gather}
\log(W_0)-F(W_0)+b_{\rm ext}
=S(\bar{\tau}+F_{\rm int}(\bar{W})
-b_{\rm int}),\label{S4E22a}\\
\log(\bar{W}_0)-F(\bar{W}_0) +b_{\rm ext}
=S(\bar{\tau}-b_{\rm int} )\,.\label{S4E22b}
\end{gather}
Subtracting these equations we obtain the following relation
between $W_0$ and $\bar{W}_0$,
\begin{equation}
\frac{W_0}{\bar{W}_0}e^{-F(W_0)+F(\bar{W} _0)}=e^{[
S(\bar{\tau}+F_{\rm int}(\bar{W})
-b_{\rm int})-S(\bar{\tau }-b_{\rm int})]}
\label{S4E23}
\end{equation}
We are interested in finding the asymptotic behaviours of $G_0(
W_0)$ that might be transformed into self-similar behaviours as
$\bar{\tau}\to\infty$. Using (\ref{S4E21})-(\ref{S4E23})
as well as the fact that $\lim_{W_0\to\infty}F(W_0)=0$, we
deduce that such functions $G_0(W_0)$ must satisfy:
\begin{equation}
\frac{G_0\Big(\bar{W}_0e^{[S(\bar{\tau}
+F_{\rm int}(\bar{W})-b_{\rm int} )-S(\bar{\tau}-b_{\rm int})] }\Big)}{G_0(\bar{W}_0)}\sim
e^{-F_{\rm int} (\bar{W})}\quad\text{as }\bar{W}_0\to
\infty\label{S4E24}
\end{equation}
We then obtain that the initial data $G_0(W_0)$ that might be
transformed in the self-similar solution (\ref{S3E5}) must satisfy
(\ref{S4E24}). There are many asymptotic behaviour of the
function $G_0(W_0)$ for which the transformation (\ref{S4E24}) can
be achieved by means of functions $S(\cdot)$ satisfying $|
S'(x)|\ll1$ as $x\to\infty$. In that case, (\ref{S4E24}) can be
written as
\[
\frac{G_0\Big(\bar{W}_0e^{S'(\bar{\tau})
F_{\rm int}(\bar{W})}\Big)}{G_0( \bar{W}_0) }\sim
e^{-F_{\rm int}( \bar{W})}\quad \text{as }\bar{W}_0\to\infty\,.
\]
Therefore, there exists a function $\lambda(W_0)>0$ such that
\begin{equation}
\frac{G_0(W_0e^{\Lambda(W_0)\zeta})} {G_0(W_0)}\sim
e^{-\zeta}\quad\text{as }W_0 \to\infty\label{S4E25}
\end{equation}
uniformly on compact sets of $\zeta$.

On the other hand, if (\ref{S4E25}) holds it is possible to find
an approximation of the function $S(\cdot)$ that must relate
$T_{\rm arr}$, $T_{\rm exit}$. Since
$S'(\bar{\tau})\approx\Lambda( \bar{W}_0)$ and
$\log(\bar{W}_0)-F( \bar{W}_0) +b_{\rm ext}\approx
S(\bar{\tau}-b_{\rm int} )$ we would obtain, to the
leading order
\begin{equation}
S'(\bar{\tau})\approx\Lambda \exp\big(-b_{\rm ext}+S(\bar{\tau})\big)\,.
\label{S4E25a}
\end{equation}
The solution of this differential equation would provide an
approximation for the function $S(\cdot)$ in (\ref{S4E20a}).
Note that for $\lambda(\cdot)$ satisfying (\ref{S4E26a}),
equation (\ref{S4E25a}) would imply a rough asymptotics of the
form
\begin{equation}
S(\bar{\tau})\approx\frac{1}{\alpha}\log(\bar{\tau
})\quad\text{as }\bar{\tau}\to\infty\,. \label{S4E29}
\end{equation}

The rest of this paper is devoted to a rigorous proof of the
previous argument. The key difficulty consists in proving the
existence of functions $\beta(\tau)$ satisfying (\ref{S4E26b}) and
solving (\ref{S4E1}), (\ref{S4E13}), (\ref{S4E20}) with
$T_{\rm arr}$, $T_{\rm exit}$ related by
means of\ (\ref{S4E20a}), where $S$ solves (\ref{S4E25a}). It is
not hard to obtain an explicit solution of this problem if
(\ref{S4E1}) is replaced by the approximated equation
(\ref{S4E12}). This solution will be given in the next Section.
Nevertheless, the analysis of the neglected error terms is
technically more involved and it will be made in the remaining
Sections of the paper.

\section{Analysis of the simplified transition problem for the
 characteristic curves}\label{five}

In this Section we solve the transition problem (\ref{S4E12}),
(\ref{S4E13}), (\ref{S4E20}), (\ref{S4E20a}) that we reformulate
here for convenience. Since deriving this problem we have
neglected higher order terms in the Taylor's series we will term
it as ``Simplified Transition Problem''. The solution of this
problem has been found in \cite{V2}, but we recall it here for
convenience. Let us write $W_{\rm trans}=1+X$. Our goal
is, given a strictly monotonic increasing function $S(\cdot) $
satisfying $S(x)<x$, for $x>0$, to find functions $\beta(\tau)$
such that for any $T_{\rm arr}>T_0$ the unique solution
of the problem
\begin{gather}
X_{\tau}    =-\frac{1}{3}X^{2}+3\beta(\tau),\text{
}T_{\rm arr}<\tau<T_{\rm exit}\label{S5E1}\\
X((T_{\rm arr})^{+})=+\infty\label{S5E2}
\end{gather}
satisfies
\begin{equation}
X((T_{\rm exit})^{-})=-\infty
\label{S5E3}
\end{equation}
with
\begin{equation}
S(T_{\rm exit})=T_{\rm arr}\,. \label{S5E4}
\end{equation}
There are several ways of solving problem
(\ref{S5E1} )-(\ref{S5E4}). The method used in this Section is
convenient because it can be adapted in a perturbative manner to
the transition problem one obtains if (\ref{S4E12}) is replaced by
(\ref{S4E1}). We compute the solution of \eqref{S5E1}--\eqref{S5E4}
in a basically explicit manner. In particular, we will assume that
$S$ is a smooth as required in the forthcoming computations.


We then proceed to solve \eqref{S5E1}--\eqref{S5E4}. Let us define
a function $f(\zeta)\geq T_0$ strictly monotonically increasing in
$\zeta$, defined in $\zeta\geq\zeta_0$ satisfying
\begin{gather}
S(f(\zeta+1))  =f(\zeta)
\label{S5E5}\\
f(\zeta_0)  =0, \label{S5E6}
\end{gather}
where $S(\cdot)$ is as in (\ref{S5E4}).

Our assumptions imply the existence of  the inverse function
$S^{-1}( \cdot)$. Therefore, given an arbitrary function
$f(\zeta)$ in $[\zeta_0,\zeta_0+1)$ satisfying (\ref{S5E6}) we
obtain $f$ defined in $[\zeta_0,\infty)$ iterating the formula
\begin{equation}
f(\zeta+1)=S^{-1}(f(\zeta)). \label{S5E6a}
\end{equation}
In further computations we will need $f$ to be differentiable
three times. This regularity would hold if the function $f(\zeta)$
defined in $[\zeta_0,\zeta_0+1)$ belongs to $C^{3}[\zeta
_0,\zeta_0+1]$ and satisfies the following compatibility
conditions
\begin{gather}
f(\zeta_0)  =f(\zeta_0+1)\label{S5E7}\\
f'(\zeta_0)  =S'(f(\zeta _0+1))f'(\zeta_0+1)\label{S5E8}\\
f''(\zeta_0) =S''( f(
\zeta_0+1))(f'(\zeta _0+1)) ^{2}+S'(f(\zeta_0+1)
)f''(\zeta_0+1)\label{S5E9}\\
\begin{aligned}
f'''(\zeta_0)& =S'''(f(\zeta_0+1))(f'( \zeta_0+1)) ^{3}\\
 &\quad +3S''(f(\zeta _0+1)) f'(\zeta_0+1)f''(\zeta_0+1)
 + S'(f(\zeta_0+1))f'''(\zeta_0+1)\,.
\end{aligned}\label{S5E11}
\end{gather}
Therefore, given $f\in C^{3}[\zeta_0,\zeta_0+1]$ strictly
monotonically increasing in $[ \zeta_0,\zeta_0+1]$ satisfying
(\ref{S5E7})-(\ref{S5E11}) we obtain, iterating (\ref{S5E6a}), an
increasing function $f\in C^{3}[\zeta_0,\infty)$. Moreover, since
$S(x)<x$ for any $x$, it follows that
\[
\lim_{\zeta\to\infty}f(\zeta)=\infty\,.
\]
Given such a function $f(\cdot)$ we define a new function
$\varphi(Y,\zeta)$ as
\begin{equation}
\varphi(Y,\zeta)=\frac{3Y}{f'(\zeta)
}+\frac{3f''(\zeta)}{2(f'( \zeta))^{2}}\,. \label{S5E12}
\end{equation}
Let us introduce the change of variables
\begin{gather}
\tau   =f(\zeta)\label{S5E13}\\
X    =\varphi(Y,\zeta)\label{S5E14}
\end{gather}
Suppose that the function $\beta(\tau)$ satisfies
\begin{equation}
\beta(f(\zeta))=\frac{1}{(f'( \zeta))
^{2}}\big(\frac{1}{2}\left\{
f,\zeta\right\}  -\pi^{2}\big), \label{S5E15}
\end{equation}
where $\{f,\zeta\}$ is the Schwartzian derivative (cf.
\cite{Hille}),
\begin{equation}
\left\{  f,\zeta\right\} =\frac{f'''(\zeta)}{f'(
\zeta) }-\frac{3}{2}(\frac{f''( \zeta)
}{f'(\zeta)})^{2}\,. \label{S5E16}
\end{equation}
Under this assumption the change of variables (\ref{S5E13}), (\ref{S5E14})
transforms the original transition problem \eqref{S5E1}--\eqref{S5E4} into
\begin{gather}
\frac{dY}{d\zeta}+Y^{2}+\pi^{2}
=0\,,\quad \zeta>\zeta_{\rm arr}\label{S5E17}\\
Y((\zeta_{\rm arr})^{+}) =+\infty\label{S5E18}\\
Y((\zeta_{\rm arr}+1)^{-})=-\infty\label{S5E19}
\end{gather}
where $\tau_{\rm arr}=f(\zeta_{\rm arr})$,
and we have used (\ref{S5E4}), (\ref{S5E5}) to obtain
(\ref{S5E19}).

Problem (\ref{S5E17})-(\ref{S5E19}) can be easily solved, since the
solution of (\ref{S5E17}), (\ref{S5E18}) is given by
\begin{equation}
Y=Y(\zeta,\zeta_0)=\pi\coth(\pi(\zeta
-\zeta_{\rm arr}))\label{S5E20}
\end{equation}
and it can be inmediately checked that $Y(\zeta,\zeta_0) $
satisfies (\ref{S5E19}).

The previous computations provide a method of finding functions
$\beta( \tau)$ solving \eqref{S5E1}--\eqref{S5E4}. Indeed, given
any strictly increasing function $f\in C^{3}[ \zeta_0,\zeta_0+1]$
satisfying (\ref{S5E7})-(\ref{S5E11}) we can obtain $f\in C^{3}[
\zeta_0,\infty)$ iterating (\ref{S5E6a}) with the properties
stated above. We then define $\beta(\tau)$ by means of
(\ref{S5E15}). Then, the function $\beta(\tau)$ solves
(\ref{S5E1})-(\ref{S5E4}).

\begin{remark} \label{rmk2} \rm
It has been proved in \cite{V2} that for functions $S$ satisfying
(\ref{S4E29}), (i.e. $\lambda$ as in (\ref{S4E26a})), the
asymptotics of $\beta(\tau)$ is
\[
\beta(\tau)\sim-\frac{1}{4}\Big[\frac{1}{( \tau)
^{2}}+\frac{1}{(\tau)^{2}(\log(\tau)) ^{2}}+\frac{1}{(\tau)
^{2}(\log(\tau))^{2}(\log(\log(\tau)))^{2}}+\dots \Big]
\]
as $\tau\to\infty$. This asymptotics has been previously
derived in \cite{LS}, \cite{V} using different methods.
\end{remark}

The main contribution of the rest of the paper we obtain rigorous
results analogous to the ones of this Section replacing
(\ref{S5E1}) by the complete equation (\ref{S4E1}) and
(\ref{S5E2}), (\ref{S5E3}) by a precise formulation of the
matching conditions (\ref{S4E8}), (\ref{S4E10}), (\ref{S4E17}),
(\ref{S4E19}). To this end we treat this problem as a perturbation
of the problem \eqref{S5E1}--\eqref{S5E4}. Nevertheless the
rigorous implementation of this perturbative argument is
technically involved. The main reason for this is that, as
indicated in \cite{V2} there exist infinitely many different ways
of finding functions $\beta(\tau)$ that solve
(\ref{S5E1})-(\ref{S5E4}). In particular, the effect of the
corrective terms (\ref{S4E1}) as well as the matching condition
generate some corrections on $\beta( \tau)$ whose asymptotics is
``beyond all the orders'' if the asymptotics of $\beta( \tau)$ is
computed that could yield an important effect on the behaviour of
the function $S(\cdot)$ and for this reason must be studied
carefully.

\section{Rigorous formulation of the transition problem}\label{six}

\subsection{Restricting the class of functions $\beta(\tau)$}

In this Section we formulate in a precise manner the transition
problem that must be solved by characteristics to understand the
transformation experienced by the solution of (\ref{S3E8}) as the
characteristics cross the critical region $W\approx1$. In
particular we give in this Section a precise meaning to the formal
matching conditions (\ref{S4E13}), (\ref{S4E20}).

 From now on, we will suppose that the assumptions in Theorem \ref{thm1}
hold. Let us define $S(\cdot)$ by means of
\begin{equation}
\int_{S_0}^{S(\tau)}\frac{d\eta}{\Lambda( e^{-b_{\rm
ext}+\eta})}=\tau\,, \label{S6E11}
\end{equation}
where $S_0$ is a large number that will be precised later. Note
that $S(\tau)$ solves the ODE (cf. (\ref{S4E25a})):
\begin{equation}
S'(\tau)=\Lambda(e^{-b_{\rm ext}+S(\tau)})\,.\label{S6Eminus1}
\end{equation}
It is convenient to impose some constraints in the class of
functions $\beta(\tau)$ that will be taken into account in Theorem
\ref{thm1}. By assumption we restrict ourselves to the class of
functions $\lambda(W)$ in Theorem \ref{thm1}.\thinspace Therefore
$S(\cdot)$ behaves roughly like in (\ref{S4E29}). For these
functions $S( \cdot)$ the function $\beta(\cdot)$ defined by means
of (\ref{S5E15}) satisfies (cf. (\ref{F11}) in the Appendix)
\begin{equation}
\log(\beta(\tau))\sim-2\log(\tau)
\quad\text{as }\tau\to\infty \,.\label{S6E3}
\end{equation}
Therefore, it is natural to assume that the functions
$\beta(\cdot)$ belong to the class of functions verifying
\begin{equation}
|\beta(\tau)|\leq\frac{A}{( \tau) ^{2-\delta}+1}\quad
\text{for }\tau\geq0 \label{S6E4}
\end{equation}
for some $\delta>0$, $A>0$.

\subsection{Approximating the evolution of the characteristics away
from the critical line}

As a first step, we obtain a rigorous version of the approximation
of $W(\tau;W_0)$ given in (\ref{S4E5}).

The following two Lemmas show that the effect of  $\beta(
\tau)W^{1/3}$ in (\ref{S4E1}) yields a negligible contribution to
the dynamics of the characteristic curves $W( \tau;W_0)$ for $W_0$
large if $|W-1|$ is not too small.

\begin{lemma} \label{L1}
Suppose that $W(\tau;W_0)$ is the solution of
(\ref{S4E1}), (\ref{S4E2}), that $w_{\rm ext}( s) $ is as in
(\ref{S4E9}) with $F_{\rm ext}( \cdot)$ defined in (\ref{S4E6}).
Let us assume also (\ref{S6E4}). There exists a function
$\varepsilon(W_0)$ satisfying $\lim_{W_0\to\infty}$
$\varepsilon(W_0)=0$ such that, for any $W_0$ large enough there
holds
\begin{equation}
|W(\tau;W_0)-w_{\rm ext}(
\tau-T_{\rm arr})|\leq\frac{\varepsilon( W_0)(w_{\rm
ext}(\tau-T_{\rm arr}))^{1/3}}{1+(\tau-T_{\rm arr}) _{+}^{2}} \label{S6E5n}
\end{equation}
for $0\leq\tau\leq T_{\rm arr}+L(T_{\rm arr})$ where
$T_{\rm arr}$ is as in (\ref{S4E11}) and $(x)_{+}=\max\{  x,0\}  $
and where $L(\cdot)$ is a
smooth, increasing function, satisfying
\begin{gather}
C_{1}(\zeta+1)^{3}\leq L(f(\zeta))
\leq C_{2}(\zeta+1)^{3} \label{Lhyp}\\
L'(\tau)\ll \frac{1}{\tau} \label{Lhypder}
\end{gather}
for some positive constants $C_{1}$, $C_{2}$, with $f(\zeta) $
defined by means of (\ref{S5E5}), (\ref{S5E6}).
\end{lemma}

\begin{remark} \label{rmk4} \rm
Due to (\ref{S5E5}), the function $f(\zeta)$ can be thought as the function
$\exp(\exp(\exp(\dots \exp(\zeta))))$ for large $\zeta$. Then,
assumption (\ref{Lhyp}) means, roughly, that $L(\tau)$ is like
$\log(\log( \dots \log( \tau)))$. The estimate (\ref{S6E5n}) means
that $W(\tau;W_0)$ might be approximated by $w_{\rm
ext}(\tau-T_{\rm arr})$ in a ``matching region'' having
the width $\log(\log( \dots \log( \tau)))$. Due to this slow
growth of the function $L$ it is not hard to see that it is
possible to choose it satisfying (\ref{Lhypder}).
\end{remark}


\begin{proof}
The proof of Lemma \ref{L1} is basically to a ``Gronwall-like''
argument for the difference $W(\tau;W_0) -w_{\rm
ext}(\tau-T_{\rm arr})\equiv Z(\tau;W_0)$. Since
$W_{\rm ext}\equiv w_{\rm ext}(\tau-T_{\rm arr})$
solves (\ref{S4E1}) with $\beta(\cdot)\equiv0$, and $W_{\rm
ext}>1$, it follows that
\begin{gather}
Z_{\tau}    =(W_{\rm ext})^{-2/3}Z-Z+O( ( W_{\rm
ext})^{-5/3}Z^{2}+|\beta(\tau)|W_{\rm ext}^{1/3})\label{S6E6}\\
Z(0;W_0) =W_0-w_{\rm ext}( -T_{\rm arr})\,.\label{S6E7}
\end{gather}
Using (\ref{S4E9}) and the definition of $T_{\rm arr}(W_0)$,
it follows that
\[
w_{\rm ext}(-T_{\rm arr})=W_0 \exp(F_{\rm ext}(w_{\rm
ext}( -T_{\rm arr})) -F_{\rm ext}(W_0) )\,,
\]
whence
\begin{equation}
\lim_{W_0\to\infty}Z(0;W_0)=0\,. \label{S6E8}
\end{equation}
Solving (\ref{S6E6}) we arrive at
\begin{equation}
\begin{aligned}
|\frac{Z(\tau;W_0)}{Z_{h}( \tau;W_0)}|
& \leq| Z(0;W_0)|+C\int_0^{\tau }
\frac{(W_{\rm ext}( s;W_0)) ^{-5/3}(Z(s;W_0)
)^{2}}{Z_{h}(s;W_0)}ds\\
&\quad+ C\int_0^{\tau}\frac{|\beta(s)|( W_{\rm
ext}(s;W_0))^{1/3}}{Z_{h}( s;W_0)}ds,
\end{aligned} \label{S6E9}
\end{equation}
where
\[
Z_{h}(\tau;W_0)\equiv\exp\Big(\int_0^{\tau}[ (W_{\rm
ext}(s;W_0))^{-2/3} -1]ds\Big)
\]
and $C>0$ is a generic constant that might change from line to line.
The asymptotics (\ref{S4E10}) implies
\[
0<C_{1}\leq\frac{(1+(\tau-T_{\rm arr}) _{+}^{2}) |
Z_{h}(\tau;W_0)|}{e^{(
-(\tau-T_{\rm arr})_{-}-T_{\operatorname{arr} })}}\leq
C_{2},\;0\leq\tau\leq T_{\rm arr}+L(
T_{\rm arr}),
\]
where $(x)_{-}=\min\{  x,0\}  $.

Therefore, since $w_{\rm ext}\geq1$, the first term on the
right-hand side of (\ref{S6E9}) combined with (\ref{S6E8}) might
be estimated as the right-hand side of (\ref{S6E5n}). On the other
hand, using (\ref{S6E4}) as well as the asymptotics of $w_{\rm
ext}$, we can estimate the last term in (\ref{S6E9}) by means of
the right-hand side of (\ref{S6E5n}) uniformly in the region
$0\leq\tau\leq T_{\rm arr}+L( T_{\rm arr})$ as $W_0\to\infty$.
Note that estimating this term we are using the fact that
$(L(\tau) ) ^{n}\ll\tau^{2-\delta}$ as $\tau\to\infty$ for any
$n\in \mathbb{R}$. Finally, the term (\ref{S6E9}) might be
estimated by means of a classical continuity argument. Indeed,
(\ref{S6E5n}) holds for small values of $\tau$. Therefore, we can
use (\ref{S6E5n}) to estimate the second term on the right-hand
side of (\ref{S6E9}) and since the resulting contribution is
smaller than the right-hand side of (\ref{S6E5n}) as
$W_0\to\infty$, the result follows.
\end{proof}



On the other hand, we can approximate the characteristics
$W(\tau ;W_0)$ after leaving the critical region in an analogous manner.

\begin{lemma} \label{L2}
Let $W(\tau;W_0)$ be a solution of (\ref{S4E1}) with
initial value (\ref{S4E2}). Suppose that at some time
$\tau=\bar{\tau}$ we have $W(\bar{\tau};W_0)=\bar{W}$ where
$0\leq\bar{W} \leq1-\delta$. Let us define
$w_{\rm int}(s)$ as in (\ref{S4E18}) with
$F_{\rm int}(W)$ as in (\ref{S4E15}). Then, for any
$\delta>0$ there exists $\varepsilon_{\delta }(W_0)$ satisfying
$\lim_{W_0\to\infty}$ $\varepsilon_{\delta}( W_0)=0$ such
that, for any $W_0$ large enough and $\tau\geq0$ there holds
\begin{equation}
\begin{gathered}
|W(\tau;W_0)-w_{\rm int}(
\tau-T_{\rm exit})|\leq\frac{\varepsilon_{\delta
}(W_0)}{1+( T_{\rm exit}-\tau)
_{+}^{2}}\,,\\
T_{\rm exit}-L(T_{\rm exit})\leq\tau\leq\bar{\tau},
\end{gathered} \label{S6E10bis}
\end{equation}
where $T_{\rm exit}$ is as in (\ref{S4E19a}) and
$L(\tau)$ satisfies (\ref{Lhyp}).
\end{lemma}

Since the proof of this result is basically analogous to the one
of Lemma \ref{L1} we omit it.

\subsection{Relating the evolution of the characteristics $W(\tau
,W_0)$ with the long time asymptotics of $G(W,\tau)$}

As a next step we formulate in a rigorous manner the main heuristic
result in Section \ref{four}.

\begin{lemma} \label{L3}
Suppose that $G_0(\cdot)$ satisfies the assumptions of
in Theorem \ref{thm1}. Let us assume that $T_{\rm arr}$
defined in (\ref{S4E11}) as $T_{\rm exit}$ defined in
(\ref{S4E19a}) are related by means of (\ref{S5E4}) where
$S(\cdot)$ is as in (\ref{S6E11}). Suppose that $G(W,\tau)$ solves
(\ref{S3E8}) with initial data $G_0(\cdot)$. Then
\begin{equation}
\lim_{\tau\to\infty}\frac{G(W,\tau)}{G(0^{+},\tau)}=G_{s}(W)\label{S6E12}
\end{equation}
uniformly on compact sets of $W\in[0,1)$, where $G_{s}( W)$ is as
in (\ref{S3E5}).
\end{lemma}

\begin{proof}
Suppose that $W(\tau;W_0)$ solves (\ref{S4E1}), (\ref{S4E2}).
Integration by characteristics yields
\[
G(W(\tau;W_0),\tau)=G_0(W_0)\,.
\]
By assumption $W(\bar{\tau};W_0)=\bar{W}$. Let us define
$\bar{W}_0$ as the starting point of the characteristic vanishing
at time $\tau=\bar{\tau}$, i.e. $W( \bar{\tau};\bar{W}_0)=0$.
Therefore (\ref{S4E22}) holds and the definitions of
$T_{\rm arr}$, $T_{\rm exit}$ in
(\ref{S4E11}), (\ref{S4E19a}) imply, arguing as in Section
\ref{four} the formulae (\ref{S4E22a}), (\ref{S4E22b}),
(\ref{S4E23}). Henceforth,
\begin{equation}
\frac{G(\bar{W},\bar{\tau})}{G( 0^{+},\bar{\tau})
}=\frac{G_0(\bar{W}_0e^{[S(\bar{\tau}
+F_{\rm int}(\bar{W})-b_{\rm int} )-S(\bar{\tau}-b_{\rm int})]+[ F_{\rm
ext}(W_0)-F_{\rm ext}(\bar{W}_0)]})}{G_0(\bar{W} _0)}
\label{S6E13}
\end{equation}
Since $S(\cdot)$ solves (\ref{S6Eminus1}), the asymptotics
(\ref{S4E26a}) implies
\[
S(\bar{\tau}+F_{\rm int}(\bar{W})
-b_{\rm int})-S(\bar{\tau})=o( 1)\quad\text{as }\bar{\tau}\to\infty
\]
uniformly on compacts sets of $\bar{W}\in[0,1)$. Therefore, using
again (\ref{S6Eminus1}) and (\ref{S4E26a}) we obtain
\begin{equation}
\begin{aligned}
&S(\bar{\tau}+F_{\rm int}(\bar{W})
-b_{\rm int})-S(\bar{\tau}-b_{\rm int} )\\
&=\int_{\bar{\tau}-b_{\rm int}}^{\bar{\tau
}+F_{\rm int}(\bar{W})-b_{\rm int}
}\lambda(e^{-b_{\rm ext}+S(s)})ds\\
&=\frac{CF_{\rm int}(\bar{W})(1+o(1)) }{(e^{-b_{\rm
ext}+S(\bar{\tau })})^{\alpha}}  \\
&=\lambda(e^{-b_{\rm ext}+S(\bar{\tau})})F_{\rm int}( \bar {W}) (1+o(1)
)\quad \text{as }\bar{\tau }\to\infty
\end{aligned}\label{S6E14}
\end{equation}
Using (\ref{S4E22b}) and (\ref{S6Eminus1}) as well as the fact
that $\lim_{W_0\to\infty}F_{\rm ext}(W_0) =0$, we deduce that
\[
\bar{W}_0=e^{-b_{\rm ext}+F(\bar{W}_0)+S(
\bar{\tau}-b_{\rm int})}=e^{-b_{\rm ext}+S(\bar{\tau})}(1+o(1))
\]
as $\bar{\tau}\to\infty$.
Therefore, (\ref{S4E26a}) and (\ref{S6E14}) yield
\[
S(\bar{\tau}+F_{\rm int}(\bar{W})
-b_{\rm int})-S(\bar{\tau}-b_{\rm int} )=\lambda(\bar{W}_0)F_{\rm int}(
\bar{W})(1+o(1))\quad\text{as } \bar{\tau}\to\infty\,.
\]
Plugging this formula in (\ref{S6E13}) and using the asymptotics
of $F_{\rm ext}$ as $W_0\to\infty$ we obtain
\[
\frac{G(\bar{W},\bar{\tau})}{G( 0^{+},\bar{\tau})
}=\frac{G_0\big(\bar{W}_0e^{\lambda(\bar{W}_0)
F_{\rm int}(\bar{W})(1+o(1))
+O(\frac{W_0-\bar{W}_0}{\bar{W}_0^{5/3}})}\big)
}{G_0(\bar{W}_0)}\quad \text{as }\bar{\tau }\to\infty
\]
Using (\ref{S4E26}) it then follows that
\[
\frac{G(\bar{W},\bar{\tau})}{G( 0^{+},\bar{\tau})}\sim
e^{-F_{\rm int}(\bar{W})}( 1+o( 1))
=G_{s}(\bar{W})(1+o(1))\quad\text{as }\bar{\tau}\to\infty
\]
uniformly on compact sets of $\bar{W}\in[0,1)$, whence
(\ref{S6E12}) follows.
\end{proof}

\subsection{Dynamics of the characteristic curves near the critical line
$W=1$}

We have then obtained that in order to transform by means of
(\ref{S3E8}) the initial data $G_0(\cdot)$ in a function
$G(W,\tau)$ behaving in a self-similar manner as
$\tau\to\infty$, we need to find functions $\beta( \cdot)$
such that (\ref{S5E4}) holds, with $T_{\rm arr}$,
$T_{\rm exit}$ as in (\ref{S4E11}), (\ref{S4E19a}). As
a next step we transform this question in a problem analogous to
the Simplified Transition Problem (\ref{S5E1} )-(\ref{S5E4}) but
with a slightly perturbed equation (\ref{S5E1}).
More precisely, we rewrite (\ref{S4E1}) as
\begin{equation}
W_{\tau}=-\frac{1}{3}(W-1)^{2}+\bar{h}_{1}(W) +3\beta(\tau)
(1+\bar{h}_{2}(W)),
\label{S6E15}
\end{equation}
where
\begin{gather}
\frac{|\bar{h}_{1}(W)|}{|W-1|^{3}}+\frac{| \bar{h}_{2}(W)|
}{|W-1|}   \leq C\;\;\text{for \ }|W-1|\leq\frac{1}{2},\label{S6E16}\\
\frac{|\bar{h}_{1}'(W)|}{|W-1|^{2}}+| \bar{h}_{2}'(W)|
 \leq C\quad \text{for \ }|W-1|\leq\frac{1}{2}\,. \label{S6E16a}
\end{gather}
We define smooth functions $h_{1}(W,\tau),\;h_{2}( W,\tau) $ as
\begin{gather*}
h_{1}(W,\tau) \equiv h_{1}(W)\xi(
(W-1)L(\tau))\\
h_{2}(W,\tau)  \equiv h_{2}(W)\xi(( W-1)L(\tau))
\end{gather*}
where $\xi(\cdot)$ is a smooth function satisfying
\[
\xi(s)  =\begin{cases}
1 & \text{for }|s|\leq 1,\\
0 & \text{for }|s|\geq 2\,.
\end{cases}
\]
Note that (\ref{S6E16}), (\ref{S6E16a}) imply
\begin{gather}
\begin{gathered}
h_{1}(W,\tau)=\bar{h}_{1}(W),\quad \text{if }
|W-1|\leq\frac{6}{L(\tau)},\\
|h_{1}(W,\tau)|\leq\frac{K}{(L( \tau))
^{3}}, \quad \text{if }|W-1|\geq\frac {6}{L(\tau)},
\end{gathered} \label{S6E17}
\\
\begin{gathered}
h_{2}(W,\tau)=\bar{h}_{2}(W),\quad \text{if }
|W-1|\leq\frac{6}{L(\tau)},\\
|h_{2}(W,\tau)|\leq\frac{K}{L( \tau) },\quad \text{if }|W-1|
\geq\frac{6}{L( \tau)},
\end{gathered} \label{S6E18}
\end{gather}
where $K>0$ is a fixed numerical constant.
Suppose that
\begin{equation}
|W(\tau;W_0)-1|\leq\frac{1}{2}\frac
{1}{L(\tau)}\;\;\;\text{for}\;\;\tau\in[
T_{\rm arr}+L(T_{\rm arr})
,T_{\rm exit}-L(T_{\rm exit})].
\label{S6E19}
\end{equation}
Then Lemmas \ref{L1}, \ref{L2} as well as the asymptotics
(\ref{S4E10}), (\ref{S4E19}) imply that this inequality holds for
$\tau=T_{\rm arr}+L(T_{\rm arr})$,
$\tau=T_{\rm exit}-L(T_{\rm exit})$. Under the assumption (\ref{S6E19})
would be possible to replace (\ref{S6E15}) by
\begin{equation}
\tilde{W}_{\tau}=-\frac{1}{3}(\tilde{W}-1)^{2}+h_{1}(
\tilde{W},\tau)+3\beta(\tau)(1+h_{2}(
\tilde{W},\tau))\,.\label{S6E20}
\end{equation}
for $\tau\in[T_{\rm arr}+L(T_{\rm arr}),T_{\rm exit}-L(T_{\rm exit}) ]$.
 Moreover,
if (\ref{S6E19}) holds for the solutions of (\ref{S6E20}), a
similar inequality would also be satisfied for the solutions of
(\ref{S6E15}).

The advantage of (\ref{S6E20}) with respect to (\ref{S6E15}) is
that the functions $h_{1}(W,\tau)$, $h_{2}( W,\tau)$ are globally
bounded in $\mathbb{R}$. Therefore, it is possible to define for
it a transition problem with singular boundary conditions
analogous to (\ref{S5E2}), (\ref{S5E3}). It turns out that such a
singular boundary conditions will be shown to be convenient in
order to deal with the corresponding transition problem by means
of perturbations of (\ref{S5E1} )-(\ref{S5E4}).

Note that since $\beta(\tau)\to0$ as $\tau
\to\infty$ (cf. (\ref{S6E4})) and due to (\ref{S6E17}),
(\ref{S6E18}) the solution of (\ref{S6E20}) satisfying
$\tilde{W}(T_{\rm arr}+L( T_{\rm arr})
)=W(T_{\rm arr}+L( T_{\rm arr}) ;W_0)$
becomes singular for some $\tau<T_{\operatorname{arr}
}+L(T_{\rm arr})$ for $T_{\rm arr}$ large
enough. In a similar manner, the solution of (\ref{S6E20})
satisfying
$\tilde{W}(T_{\rm exit}-L(T_{\rm exit}))=W(T_{\rm exit}-L( T_{\rm exit}) ;W_0)$
blows up for some $\tau>T_{\rm exit}-L(
T_{\rm exit})$. It would be then possible to define a
transition problem analogous to (\ref{S5E4}) between those blow-up
times. Moreover, due to the fact that the function $L(\tau) $
varies very slowly we can approximate $h_{1}(\tilde{W},\tau)$ as
$h_{1}(\tilde{W} ,T_{\rm arr})$, $h_{1}(\tilde{W}
,T_{\rm exit})$in the regions $\tau\approx
T_{\rm arr}$, $\tau\approx T_{\rm exit}$
respectively. On the other hand, for long times, and since
$\beta(\tau)\to0$ as $\tau\to\infty$ it would be
natural to approximate (\ref{S6E20}), in the computations of the
singularity times, by means of the simpler equations
\begin{gather}
\tilde{W}_{\tau}    =-\frac{1}{3}(\tilde{W}-1)^{2}
+h_{1}(\tilde{W},T_{\rm arr}),\label{S6E21}\\
\tilde{W}_{\tau}    =-\frac{1}{3}(\tilde{W}-1)^{2}
+h_{1}(\tilde{W},T_{\rm exit})\,. \label{S6E21a}
\end{gather}

We then define new singular times as follows. We define
$T_{\rm arr, mod}$, $T_{\rm exit,mod}$
by means of
\begin{gather}
\int_{w_{\rm ext}(L( T_{\rm arr})
)}^{\infty}\frac{d\eta}{\frac{(\eta-1)^{2}}{3}
-h_{1}(\eta,T_{\rm arr})}    =(
T_{\rm arr}+L(T_{\rm arr})
-T_{\rm arr, mod}),\label{S6E22}\\
\int_{-\infty}^{w_{\rm int}(-L(
T_{\rm exit}))}\frac{d\eta}{\frac{( \eta-1)
^{2}}{3}-h_{1}( \eta,T_{\rm exit})}
 =(T_{\rm exit,mod}-T_{\rm exit}+L(T_{\rm exit}))\,.\label{S6E23}
\end{gather}
Using the asymptotics of $w_{\rm ext}$, $w_{\rm int}$
in (\ref{S4E10}), (\ref{S4E19}) as well as the bounds for $h_{1}$
in (\ref{S6E17})) it follows that the differences
$|T_{\rm arr}-T_{\rm arr,mod}|$,
$|T_{\rm exit, mod}-T_{\rm exit}|$ are smaller than
$L(T_{\rm arr})$, $L(T_{\rm exit})$ respectively.

Given a function $S(\cdot)$ defined by means of (\ref{S6E11}) we
define a new function $\tilde{S}( \cdot)$ by means of:
\begin{equation}
\tilde{S}(T_{\rm exit,mod})
=T_{\rm arr, mod}, \label{S6E24}
\end{equation}
where
$T_{\rm arr, mod}$, $T_{\rm exit,mod}$ are as in
(\ref{S6E22}), (\ref{S6E23}) and
$T_{\rm exit}$, $T_{\rm arr}$ are related by
means of (\ref{S5E4}).

We now show in a rigorous manner that the effect of replacing
(\ref{S6E20}) by (\ref{S6E21}) is small as $W_0\to\infty$.

\begin{lemma}\label{L4}
Let us denote as $\tilde{W}(\tau;W_0)$ the unique
solution of (\ref{S6E20}) that verifies
$\tilde{W}( ( T_{\rm arr, mod})^{+};W_0) =+\infty$,
with $T_{\rm arr, mod}$ as in
(\ref{S6E22}) and $T_{\rm arr}$ as in (\ref{S4E11}).
There exists $\varepsilon(W_0)$, satisfying
$\lim_{W_0\to\infty }\varepsilon(W_0)=0$ such that
\begin{equation}
|\tilde{W}(T_{\rm arr}+L(T_{\rm arr});W_0)
-w_{\rm ext}(L( T_{\rm arr}) )|
\leq\frac{\varepsilon(W_0)}{(L( T_{\rm arr}))^{2}}\,.
\label{S6E25}
\end{equation}
Moreover, if $\tilde{W}(\tau;W_0)$ satisfies $\tilde
{W}((T_{\rm exit,mod}) ^{-};W_0)
=-\infty$ with $T_{\rm exit,mod}$ as in
(\ref{S6E23}) and $T_{\rm exit}$ as in (\ref{S4E19a})
there exists $\varepsilon(W_0)$ such that
\begin{equation}
|\tilde{W}(T_{\rm exit}-L(
T_{\rm exit});W_0)
-w_{\rm int}(-L(T_{\rm exit})) |\leq
\frac{\varepsilon(W_0)}{(L( T_{\rm exit}))^{2}}\,.
\label{S6E26}
\end{equation}
\end{lemma}

\begin{proof}
Let $\tilde{W}_{\rm app }(\tau;W_0) $ be the
solution of (\ref{S6E21}) satisfying
$\tilde{W}_{\rm app }((T_{\rm arr, mod})^{+};W_0)=+\infty$.
Since $\tilde{W}_{\rm app }( \tau;W_0)$ might be
computed explicitly it follows from (\ref{S6E22}) that
$\tilde{W}_{\rm app }( \tau;W_0)=w_{\rm ext}(L(
T_{\rm arr}))$. Subtracting (\ref{S6E20}) and
(\ref{S6E21}) we obtain
\begin{equation}
\begin{aligned}
(\tilde{W}-\tilde{W}_{\rm app })_{\tau}
&=-\frac{2}{3}(\tilde{W}_{\rm app }-1)(
\tilde{W}-\tilde{W}_{\rm app })-\frac{1}{3}( \tilde
{W}-\tilde{W}_{\rm app })^{2}\\
&\quad  +\big(h_{1}(\tilde{W},\tau)-h_{1}(\tilde
{W}_{\rm app },T_{\rm arr})\big)
+3\beta(\tau)(1+h_{2}(\tilde{W},\tau))\,,
\end{aligned}\label{S6E26a}
\end{equation}
where, unless it is explicitly needed we will drop the dependence
of the functions on $W_0$ for simplicity.

A local analysis of the asymptotics of
$\tilde{W}(\tau)$, $\tilde{W}_{\rm app }(\tau)$ as
$\tau \to(T_{\rm arr, mod})^{+}$ imply that
$|\tilde{W}(\tau)-\tilde{W} _{\rm app }(\tau)|
=O\Big((\tau-( T_{\rm arr, mod})^{+})\Big)$ as
$\tau\to(T_{\rm arr, mod})^{+}$.
Therefore, integrating the equation for
$\tilde{W}-\tilde {W}_{\rm app }$ we obtain
\begin{align*}
&\tilde{W}(\tau)-\tilde{W}_{\rm app }(\tau)\\
&  =-\frac{1}{3}\int_{T_{\rm arr, mod}}^{\tau}
ds\exp\Big(-\frac{2}{3}\int_{s}^{\tau}(\tilde{W}_{\rm app
}(\xi)-1)d\xi\Big)\\
&\quad \times \Big[(\tilde{W}(s)-\tilde{W}_{\rm app
}(s))^{2}+\Big(h_{1}(\tilde{W}(s),s)
-h_{1}(\tilde{W}_{\rm app }(s),T_{\rm arr})\Big)\\
&\quad +3\beta(s)\big( 1+h_{2}(\tilde{W}(s) ,s)\big)\Big]\,.
\end{align*}
Using (\ref{Lhypder}), (\ref{S6E16}), (\ref{S6E16a}) we obtain
\begin{align*}
&  |\tilde{W}(\tau)-\tilde{W}_{\rm app
}(\tau)|\\
&  \leq C\int_{T_{\rm arr, mod}}^{\tau}\frac
{\Phi(\tau-T_{\rm arr, mod})}
{\Phi(s-T_{\rm arr, mod})}\\
&\times  \Big[(\tilde{W}(s)-\tilde{W}_{\rm app
}(s))^{2}+\frac{|\tilde{W}(s)
-\tilde{W}_{\rm app }(s)|}{L^{2}}+\frac{1}
{s}+|\beta(s)|\Big]ds,
\end{align*}
where
\[
\Phi(\tau-T_{\rm arr, mod})\equiv
\exp\Big(-\frac{2}{3}\int_{T_{\rm arr, mod}
-1}^{\tau}(\tilde{W}_{\rm app }(\xi)-1) d\xi\Big)
\]
satisfies
\[
0<C_{1}\leq\tau^{2}\Phi(\tau)\leq C_{2}<+\infty,\quad \tau>0;
\]
whence,
\begin{equation}\label{S6E27}
\begin{aligned}
&  |\tilde{W}(\tau)-\tilde{W}_{\rm app}(\tau)|\\
&  \leq C\int_{T_{\rm arr, mod}}^{\tau}\Big(
\frac{s-T_{\rm arr, mod}}{\tau-T_{\operatorname{arr}
,\operatorname{mod}}}\Big)^{2}\\
&\times  \Big[(\tilde{W}(s)-\tilde{W}_{\rm app
}(s))^{2}+\frac{|\tilde{W}(s)
-\tilde{W}_{\rm app }(s)|}{( L( \tau) )
^{2}}+\frac{1}{s}+|\beta( s)|\Big]ds\,.
\end{aligned}
\end{equation}
Due to (\ref{S6E4}), (\ref{Lhyp}) the last two terms in
(\ref{S6E27}) can be estimated as
$\varepsilon(W_0)\min\{ (\tau-T_{\rm arr, mod}),\frac
{1}{(\tau-T_{\rm arr, mod})^{2} }\}$
with $\lim_{W_0\to\infty}\varepsilon( W_0)=0$ for
$T_{\rm arr, mod}\leq\tau\leq
T_{\rm arr}+L(T_{\rm arr})$. Then, a
Gronwall-like estimate yields the bound
\[
|\tilde{W}(\tau) -\tilde{W}_{\rm app }(\tau)| \leq
C\varepsilon(W_0)\min\big\{ (
\tau-T_{\rm arr, mod}),\frac{1}{(\tau-T_{\rm arr, mod})^{2} }\big\}
\]
that implies (\ref{S6E25}). Estimate (\ref{S6E26}) can be proved
in a similar manner.
\end{proof}

Combining Lemmas \ref{L1}, \ref{L2}, \ref{L4}, we obtain the following
result.

\begin{proposition}\label{L5}
Suppose that $W(\tau;W_0)$ is the unique solution of
(\ref{S4E1}), (\ref{S4E2}). Let us assume also that $W(\bar{\tau
};W_0)=\bar{W}$ where $0\leq\bar{W}\leq1-\delta$ for some $\bar
{\tau}>0$. Suppose that $\tilde{W}(\tau;W_0)$ is as in Lemma
\ref{L4} and that $T_{\rm arr}$, and
$T_{\rm exit}$ are as in (\ref{S4E11}), (\ref{S4E19a})
respectively and are related by means of (\ref{S5E4}). Then there
exists $\varepsilon( W_0)$ satisfying
$\lim_{W_0\to\infty}\varepsilon(W_0)=0$, such that
\begin{gather}
|\tilde{W}(T_{\rm arr}+L(T_{\rm arr});W_0)-W(T_{\rm arr}
+L( T_{\rm arr});W_0)|
\leq\frac {\varepsilon(W_0)}{(L(T_{\rm arr}))^{2}} ,\label{S6E28}\\
|\tilde{W}(T_{\rm exit}-L(
T_{\rm exit});W_0)-W(T_{\rm exit}-L(T_{\rm exit}) ;W_0) |
\leq\frac{\varepsilon( W_0) }{(L(T_{\rm exit}))^{2}}\,.
\label{S6E29}
\end{gather}
\end{proposition}

As indicated above, if (\ref{S6E19}) holds we would have that $W(
\tau;W_0)$ solves (\ref{S6E20}) for $\tau\in[
T_{\rm arr}+L(T_{\rm arr})
,T_{\rm exit}-L(T_{\rm exit})]$. However,
due to the error terms in (\ref{S6E28}), (\ref{S6E29}) we cannot
ensure that $W(\tau;W_0)=\tilde{W}( \tau;W_0)$. Nevertheless, the
following result holds.



\begin{lemma}\label{L6}
Suppose that $W(\tau;W_0)$, $\tilde{W}( \tau;W_0)$ are
as in Proposition \ref{L5}. Let us suppose also that (\ref{S6E19})
holds in the interval $[T_{\rm arr}+L(T_{\rm arr})
,T_{\rm exit}-L(T_{\rm exit})]$.
Then there exists a function $\mu(W_0)$ such that for $W_0$
large enough,
\begin{equation}
W(\tau;W_0+\mu(W_0))=\tilde{W}( \tau;W_0) \label{S6E30}
\end{equation}
for $\tau\in[T_{\rm arr}+L(T_{\rm arr}),T_{\rm exit}-L(
T_{\rm exit})]$,
where
\begin{equation}
\frac{\mu(W_0)}{W_0}\to0\quad\text{as } W_0\to\infty\,.\label{S6E31}
\end{equation}
Moreover, suppose that $T_{\rm exit}$ is defined as in
(\ref{S4E19a}). Let us denote as $\bar{W}=W(\bar{\tau};W_0)$ Then
\begin{equation}
|W(\bar{\tau};W_0+\mu(W_0))-\bar {W}|\to0\quad\text{as }
W_0\to\infty\label{S6E32}
\end{equation}
uniformly on compact sets of $0\leq\bar{W}<1$.
\end{lemma}

\begin{proof}
The basic idea consists in estimating the derivative
$\frac{\partial W( \tau;W_0)}{\partial W_0}$ for
$\tau=T_{\operatorname{arr} }+L(T_{\rm arr})$. To this
end, notice that, differentiating (\ref{S4E1}), (\ref{S4E2}) we
obtain
\begin{gather}
(\frac{\partial W}{\partial W_0})_{\tau}
  =(W^{-2/3}-1+\beta(\tau)W^{-2/3})\frac{\partial
W}{\partial W_0},\label{S6E33}\\
\frac{\partial W}{\partial W_0}(0;W_0)  =1 \,.\label{S6E33a}
\end{gather}
Let us recall that $W_{\rm ext}(\tau,W_0) \equiv w_{\rm ext}(
\tau-T_{\rm arr}( W_0) )$. Using (\ref{S4E8}),
(\ref{S4E9}) and (\ref{S4E11}) we obtain
\begin{gather}
(\frac{\partial W_{\rm ext}}{\partial W_0}) _{\tau}
 =(W_{\rm ext}^{-2/3}-1)\frac{\partial W_{\rm ext}}{\partial W_0},
 \label{S6E33new}\\
\frac{\partial W_{\rm ext}}{\partial W_0}(0;W_0)  =1\,.
\label{S6E33newb}
\end{gather}
Subtracting (\ref{S6E33}), (\ref{S6E33new}) we obtain, after some
computations
\begin{equation}
\begin{aligned}
(\frac{\partial W}{\partial W_0}-\frac{\partial
W_{\rm ext}}{\partial W_0})_{\tau}
&=(W_{\rm ext}
^{-2/3}-1)(\frac{\partial W}{\partial W_0}-\frac{\partial
W_{\rm ext}}{\partial W_0})\\
&\quad +(W^{-2/3} -W_{\rm ext}^{-2/3})
\frac{\partial  W}{\partial W_0}
+\beta(\tau)W^{-2/3}\frac{\partial W}{\partial W_0}\,.
\end{aligned}\label{S5E33dif}
\end{equation}
We now claim that
\begin{equation}
|\frac{\partial W}{\partial W_0}-\frac{\partial
W_{\rm ext}}{\partial W_0}|\leq\varepsilon(W_0)
\frac{\partial W_{\rm ext}}{\partial W_0} \label{S5E33est}
\end{equation}
for $0\leq\tau\leq T_{\rm arr}+L(T_{\operatorname{arr}
})$, where $\lim_{W_0\to\infty}\varepsilon( W_0) =0$.
Estimate (\ref{S5E33est}) follows by means of a continuation
argument. Indeed, suppose that the following estimate holds
\begin{equation}
\frac{\partial W}{\partial W_0}\leq2\frac{\partial W_{\rm ext}
}{\partial W_0}\,. \label{S5E33cont}
\end{equation}
This estimate is certainly valid for $\tau=0$,
and it might be shown to be extended to arbitrary values of
$\tau\leq T_{\operatorname{arr} }+L(T_{\rm arr})$,
assuming that it is valid for previous times. On the other hand it
is possible to obtain an improvement of (\ref{S6E5n}) as follows.
The term $|\beta(\tau)|W_{\rm ext}^{1/3}$ in (\ref{S6E6})\ plays
the role of ''source'' in that differential equation. Due to the
exponential decay of $W_{\rm ext}$ it follows that this term is
smaller than $CW_0^{1/3}e^{-\frac{T_{\rm arr}}{2}}$ for
$0\leq\tau\leq \frac{T_{\rm arr}}{2}$, and due to
(\ref{S6E4}) it can be estimated as $\frac{CW_{\rm
ext}^{1/3}}{(T_{\rm arr})^{2-\delta}}$ for
$\frac{T_{\rm arr}}{2}\leq\tau\leq T_{\rm arr}+L(T_{\rm arr})$.
Arguing as in the Proof of Lemma \ref{L1} it follows that similar
 estimates might be obtained for the difference $| W-W_{\rm ext}|$.
Therefore,
the term $( W^{-2/3}-W_{\rm ext}^{-2/3})$ in (\ref{S5E33dif}) can
be estimated as $\frac{C}{W_{\rm ext}}$ for
$0\leq\tau\leq\frac{T_{\rm arr}}{2}$ and as $\frac{C}{(
T_{\rm arr})^{2-\delta}}$ for
$\frac{T_{\operatorname{arr} }}{2}\leq\tau\leq
T_{\rm arr}+L(T_{\operatorname{arr} })$. A Gronwall's
like argument as the one in the proof of Lemma \ref{L1} combined
with the fact that $L(T_{\rm arr})
\ll T_{\rm arr}\ll W_{\rm ext}(\frac
{T_{\rm arr}}{2},W_0)\sim C\sqrt{W_0}$ yields the
estimate (\ref{S5E33est}), that for the particular value $\tau
=T_{\rm arr}+L(T_{\rm arr})$ becomes
\[
\frac{\partial W}{\partial W_0}(T_{\rm arr}+L(
T_{\rm arr});W_0)=-\frac{\partial
T_{\rm arr}}{\partial W_0}w_{\rm ext}'(L(
T_{\rm arr}))(1+o(1))\quad\text{as }W_0\to\infty\,.
\]
Since $w_{\rm ext}'(L(T_{\operatorname{arr} }))=-\frac{3}{(L(
T_{\rm arr}))^{2}}$, and $T_{\rm arr}'( W_0)
\sim\frac{1}{W_0}$ as $W_0\to\infty$ (cf. (\ref{S4E11})), we have
\begin{equation}
\frac{\partial W}{\partial W_0}(T_{\rm arr}+L(
T_{\rm arr});W_0)
=\frac{3}{W_0}\frac{1}{(L(T_{\rm arr})) ^{2}}(1+o(
1))\quad\text{as }W_0\to\infty\,.\label{S6E33b}
\end{equation}
Then (\ref{S6E28}) implies the existence of $\delta W_0$ as
indicated in Lemma \ref{L6} satisfying $|\frac{3\delta
W_0}{W_0}|\leq2\varepsilon(W_0)$, whence (\ref{S6E31}) follows.

To obtain (\ref{S6E32}) we write
\[
W(\tau;W_0)=V(\tau;\bar{W})\,.
\]
Note that by assumption $V(\bar{\tau};\bar{W})=\bar{W}$.
Differentiating (\ref{S4E1}) we obtain
\begin{gather}
(\frac{\partial V}{\partial\bar{W}})_{\tau}
 =\big(
V^{-2/3}-1+\beta(\tau)V^{-2/3}\big)\frac{\partial
V}{\partial\bar{W}},\label{S6E34}\\
\frac{\partial V}{\partial\bar{W}}(\bar{\tau};\bar{W}) =1\,.
\label{S6E35}
\end{gather}
Using (\ref{S6E33}) and (\ref{S6E10bis}) we obtain arguing as in the
derivation of (\ref{S6E33b}),
\[
\frac{\partial V}{\partial\bar{W}}(T_{\rm exit}-L(
T_{\rm exit});\bar{W})=-\frac{\partial
T_{\rm exit}}{\partial\bar{W}}w_{\rm int}'(-L(T_{\rm exit}))
(1+o(1))\quad\text{as }W_0\to\infty\,.
\]
Taking into account that $\frac{\partial
T_{\rm exit}} {\partial\bar{W}}=F_{\rm int}'(\bar{W}) $, and
$w_{\rm int}'(-L(T_{\rm exit}))=-\frac{3}{(L( T_{\rm exit}))^{2}}$
is bounded in compact sets of $\bar{W}\in[0,1)$ we obtain
from (\ref{S6E29}) that
$\tilde{W}( \tau;W_0) =V(\tau;\bar{W}+\delta\bar{W})$ with
$\delta\bar{W} \to0$ as $W_0\to\infty$ whence
Lemma \ref{L6}.
\end{proof}

It is possible to obtain some regularity for  $\mu(W_0)$ defined
in Lemma \ref{L6}.

\begin{lemma}
\label{L6b}Suppose that $\mu(W_0)$ is as in Lemma \ref{L6}. Then
\[
\lim_{W_0\to\infty}|\frac{\partial\mu(W_0) }{\partial W_0}|
=0\,.
\]
\end{lemma}

\begin{proof}
Given $W_0$ large enough, let us write $\hat{\tau}\equiv
T_{\rm arr}+L(T_{\rm arr})$. It then follows
from (\ref{S6E30}) that
\[
W(\hat{\tau};W_0+\mu(W_0))=\tilde{W}( \hat{\tau};W_0)\,.
\]
Differentiating this formula with respect to $W_0$, we obtain
\begin{align*}
\frac{\partial\mu(W_0)}{\partial W_0}
&  =\Big( \frac{\partial
W(\hat{\tau};W_0+\mu(W_0))}{\partial W_0}\Big)^{-1}\\
&\quad \times \Big[\frac{\partial\hat{\tau}}{\partial W_0}
\Big(\frac {\partial\tilde{W}(\hat{\tau};W_0)}{\partial\tau}
 -\frac{\partial W(\hat{\tau};W_0+\mu(W_0))
}{\partial\tau}\Big) \\
&\quad  + \frac{\partial\tilde{W}(\hat{\tau};W_0) }{\partial
W_0}-\frac{\partial W( \hat{\tau};W_0+\mu( W_0) ) }{\partial W_0}\Big].
\end{align*}
Using (\ref{S6E33b}) we obtain
\begin{equation}
|\frac{\partial\mu(W_0)}{\partial W_0}|\leq
C(L(T_{\rm arr}))^{2}W_0[ K+J_{1}+J_{2}] \label{S6E35b}
\end{equation}
for some $C>0$ independent on $L,\;W_0$, where
\begin{gather*}
K    \equiv\frac{\partial\hat{\tau}}{\partial W_0}
\Big(\frac
{\partial\tilde{W}(\hat{\tau};W_0)}{\partial\tau} -\frac{\partial
W(\hat{\tau};W_0+\mu(W_0))
}{\partial\tau}\Big),\\
J_{1}    \equiv\big|\frac{\partial(\tilde{W}(\hat{\tau };W_0)-w_{\rm
ext}(\hat{\tau}
-T_{\rm arr}))}{\partial W_0}\big|,\\
J_{2}    \equiv\big|\frac{\partial(W(\hat{\tau};W_0 +\mu(W_0))-w_{\rm
ext}( \hat{\tau }-T_{\rm arr}))}{\partial W_0}\big|\,.
\end{gather*}
To estimate $J_{1}$, we use the auxiliary function
$\tilde {W}_{\rm app }(\hat{\tau};W_0)$ defined in the
proof of Lemma \ref{L4}. Then
$J_{1}\leq J_{1,1}+J_{1,2}$,
where
\begin{gather*}
J_{1,1}   \equiv\big|\frac{\partial(\tilde{W}( \hat{\tau
};W_0)-\tilde{W}_{\rm app }(\hat{\tau}
;W_0))}{\partial W_0}\big|,\\
J_{1,2}   \equiv\big| \frac{\partial(\tilde{W}_{\rm app
}(\hat{\tau};W_0)-w_{\rm ext}( \hat{\tau
}-T_{\rm arr}))}{\partial W_0}\big|\,.
\end{gather*}
To estimate $J_{1,1}$ we recall that $\tilde{W}(\tau
;W_0,L)-\tilde{W}_{\rm app }( \tau;W_0,L)$ satisfies
(\ref{S6E26a}) with $\tilde{W}-\tilde{W}_{\rm app
}=O((\tau-(T_{\rm arr, mod})^{+}))$ as
$\tau\to( T_{\rm arr, mod})^{+}$.
Differentiating (\ref{S6E26a}) with respect to
$T_{\rm arr, mod}$ we obtain
\begin{align*}
&\Big(\frac{\partial( \tilde{W}-\tilde{W}_{\rm app })
}{\partial T_{\rm arr, mod}}\Big) _{\tau} \\
 & =-\frac{2}{3}( \tilde{W}_{\rm app }-1)\frac {\partial(
\tilde{W}-\tilde{W}_{\rm app })}{\partial
T_{\rm arr, mod}}-\frac{2}{3}(\frac
{\partial\tilde{W}_{\rm app }}{\partial T_{\rm arr
,mod}})(\tilde{W}-\tilde{W}_{\rm app })\\
&\quad  -\frac{2}{3}(\tilde{W}-\tilde{W}_{\rm app })
\frac{\partial( \tilde{W}-\tilde{W}_{\rm app })
}{\partial
T_{\rm arr, mod}}+\frac{\partial(
h_{1}(\tilde{W},W_0)-h_{1}(\tilde{W}
_{\rm app },W_0))}{\partial T_{\rm arr, mod}}\\
&\quad+  3\beta(\tau)\frac{\partial(h_{2}(\tilde {W},W_0))}{\partial
T_{\rm arr, mod}},
\end{align*}
\[
\frac{\partial(\tilde{W}-\tilde{W}_{\rm app })
}{\partial T_{\rm arr, mod}}    =O( 1)
\quad\text{as }\tau\to(T_{\rm arr , mod})^{+}\,.
\]
Note that the term $\frac{2}{3}(\frac{\partial\tilde{W}
_{\rm app }}{\partial W_0})(\tilde{W}-\tilde
{W}_{\rm app })$ might be bounded as $\frac{C}{(\tau-(
T_{\rm arr, mod})^{+})}$. The asymptotics
of the linear term $-\frac{2}{3}(\tilde
{W}_{\rm app }-1)\frac{\partial(\tilde{W}-\tilde
{W}_{\rm app })}{\partial T_{\rm arr , mod}}$ is exactly
the same as in (\ref{S6E26a}).
Using then Lemma \ref{L4} to estimate
$(\tilde{W}-\tilde{W}_{\rm app  })$ as well as
Gronwall-like arguments analogous to the ones used in the proof of
Lemma \ref{L4}, as well as the fact that
$\frac{dT_{\rm arr, mod}}{dW_0}\approx\frac{C}{W_0}$
we obtain
\begin{equation}
\lim_{W_0\to\infty}W_0J_{1,1}=0 \,.\label{S6E35c}
\end{equation}
We now proceed to estimate the term $J_{1,2}$. Note that
$\tilde{W}_{\rm app }(\tau;W_0)=w_{\rm app
}(\tau-T_{\rm arr, mod},W_0)$, with
\begin{gather*}
w_{\rm app ,s}   =-\frac{1}{3}(w_{\rm app
}-1)^{2}+h_{1}(w_{\rm app },W_0)\,,\quad s>0\,,\\
w_{\rm app }(0^{+})  =+\infty\,.
\end{gather*}
It then follows from (\ref{S6E22}) that $w_{\rm app }(
L(T_{\rm arr}),W_0)=w_{\rm ext}(L(T_{\rm arr}))$, whence since
the ODE satisfied for
$w_{\rm app },\;w_{\rm ext}$ is the same for $s>L$ it
follows that
\begin{equation}
J_{1,2}=0\,. \label{S6E35c1}
\end{equation}
Finally we estimate $J_{2}$. To this end, we argue as in the proof
of Lemma \ref{L1}. Note that $\frac{\partial(W(\hat{\tau};W_0
+\mu(W_0)))}{\partial W_0}-\frac {\partial(w_{\rm ext}(\hat{\tau}
-T_{\rm arr}))}{\partial W_0}$ solves (\ref{S5E33dif})
with zero initial data at $\tau=0,W$ since $\frac{\partial
W}{\partial W_0}(0,W_0) =\frac{\partial(w_{\rm ext}(
-T_{\rm arr}))}{\partial W_0}=1$. We can then argue as
in the proof of (\ref{S5E33est}) to obtain
\[
J_{2}\leq\frac{\varepsilon(W_0)}{W_0(L( \tau))^{2}};
\]
whence
\begin{equation}
W_0L^{2}J_{2}\to0\quad\text{as }W_0\to\infty\,.
\label{S6E35d}
\end{equation}
Finally we estimate $K$ in (\ref{S6E35b}). Using (\ref{S6E15}),
(\ref{S6E17}), (\ref{S6E20}), (\ref{S6E30})\ as well as the fact
that for $\tau=\hat{\tau}$, $|W-1|
\leq\frac{6}{L(T_{\rm arr})}$ (cf. (\ref{S6E19})) it
follows that $K=0$. Combining this with (\ref{S6E35c}),
(\ref{S6E35c1}) and (\ref{S6E35d}) the result follows.
\end{proof}

We can now prove the main result of this Section.

\begin{theorem}
\label{T2}Suppose that $G_0(\cdot)$ satisfies the assumptions in
Theorem \ref{thm1}. Let us define $S$ as in (\ref{S6E11}).
Suppose that $\beta(\tau)$ in (\ref{S6E20}) is chosen in such a
way that the function $\tilde{W}( \tau;W_0)$ defined by means of
(\ref{S6E20}) and
\begin{equation}
\tilde{W}\big((T_{\rm arr, mod})
^{+};W_0\big)=+\infty. \label{S6E36}
\end{equation}
Suppose that
\begin{equation}
\tilde{W}\big(( T_{\rm exit,mod})
^{-};W_0\big)=-\infty\,,\label{S6E37}
\end{equation}
where $T_{\rm arr, mod}$,
$T_{\rm exit ,mod }$ are related by means
of (\ref{S6E24}). Then, $G( W,\tau)$ solution of (\ref{S3E8}) with
initial data $G_0(\cdot)$ satisfies
\begin{equation}
\lim_{\tau\to\infty}\frac{G(W,\tau)}{G(
0^{+},\tau)}=G_{s}(W)\label{S6E38}
\end{equation}
uniformly on compact sets of $W\in[0,1)$.
\end{theorem}

\begin{remark} \label{rmk12}\rm
Theorem \ref{T2} reduces the Proof of Theorem \ref{thm1} to the
problem of finding $\beta(\tau)$ solving the Transition Problem
(\ref{S6E20}), (\ref{S6E36}), (\ref{S6E37}).
\end{remark}

\begin{proof}
Due to Proposition \ref{L5} and Lemma \ref{L6} we have that the
solutions of (\ref{S4E1}) starting at $W_0+\mu(W_0)$ at time
$\tau=0$ arrives to a point $\bar{W}+\nu( \bar{W})$ at time $\tau
=\bar{\tau}$, with $\frac{\mu(W_0)}{W_0}\ll1$, $\nu( \bar{W})\ll1$.
Let us denote as $\bar{W}_0+\mu(\bar{W} _0)$ the starting point
for the characteristic vanishing at time $\tau=\bar{\tau}$, i.e.
the characteristic for which $\bar{W}+\nu( \bar{W})=0$ at time
$\tau=\bar{\tau}$. Arguing as in Lemma \ref{L3} we have
\[
\frac{G(\bar{W}+\nu(\bar{W}),\bar{\tau})}{G(
0^{+},\bar{\tau})}=\frac{G_0(W_0+\mu(
W_0))}{G_0(\bar{W}_0+\mu(\bar{W} _0))}\,.
\]
Since $W_0$, $\bar{W}_0$ are related as in Lemma \ref{L3} we can
argue as in the Proof of that result to obtain
\begin{equation}
\frac{G(\bar{W},\bar{\tau})}{G( 0^{+},\bar{\tau})
}=\frac{G_0\Big(\bar{W}_0e^{\lambda(\bar{W}_0)
F_{\rm int}(\bar{W})(1+o(1)
)+O(\frac{W_0-\bar{W}_0}{\bar{W}_0^{5/3}})}
+\mu(W_0)\Big)}{G_0\big(\bar{W}_0+\mu( \bar{W}_0)\big) }
\label{S6G}
\end{equation}
 as $\bar{\tau}\to \infty$.
Using Lemma \ref{L6b} it follows that
$|\mu(W_0)-\mu(\bar{W}_0)|=o(|W_0-\bar{W} _0|)$ as
$\bar{\tau}\to\infty$. On the other hand, to the leading order
$\frac{W_0-\bar{W}_0}{\bar{W}_0}\sim\lambda(\bar{W}_0)F_{\rm int}(\bar{W})$
(cf. (\ref{S6E14})).
Therefore,
\begin{align*}
&  \bar{W}_0e^{\lambda(\bar{W}_0)F_{\rm int} (\bar{W})(1+o(1))+O( \frac{W_0-\bar{W}_0}{\bar{W}_0^{5/3}})}+\mu(
W_0)
\\
&  =(\bar{W}_0+\mu(\bar{W}_0))e^{\lambda ( \bar{W}_0)
F_{\rm int}(\bar{W})
(1+o(1))}\\
&\quad +  (\mu(\bar{W}_0)-\mu(W_0))+O( \lambda(\bar{W}_0)
F_{\rm int}(
\bar{W}))\mu(\bar{W}_0)\\
&  =(\bar{W}_0+\mu(\bar{W}_0))e^{\lambda ( \bar{W}_0)
F_{\rm int}(\bar{W}) (1+o(1))};
\end{align*}
whence (\ref{S6G}) becomes
\[
\frac{G(\bar{W},\bar{\tau})}{G( 0^{+},\bar{\tau})
}=\frac{G_0((\bar{W}_0+\mu(\bar{W}_0))
e^{\lambda(\bar{W}_0)F_{\rm int}( \bar{W})
(1+o(1))})}{G_0( \bar{W}_0+\mu(\bar{W}_0))}\;\text{\ as\ \ }
\bar{\tau}\to\infty\,.
\]
Arguing then as in the Proof of Lemma \ref{L3} we obtain (\ref{S6E38}).
\end{proof}


\section{Analysis of the transition problem: Local well posedness}

\subsection{The Transition Problem}

The main result that we have obtained in the previous Section is
that the problem of transforming an initial data $G_0(W)$ in a
self-similar solution of the form (\ref{S3E5}) by means of the
evolution (\ref{S3E8}) for a suitable choice of $\beta( \cdot)$
might be reduced to the Transition Problem (\ref{S6E20}),
(\ref{S6E36}), (\ref{S6E37}). We rewrite this problem here for
convenience:

Let us fix a function $S(\cdot)$ by means of (\ref{S6E11}). To
find $\beta(\tau)$ such that $\tilde{W}(\tau ;W_0)$, solution of
\begin{gather}
\tilde{W}_{\tau}   =-\frac{1}{3}(\tilde{W}-1)^{2}
+h_{1}(\tilde{W},\tau)+3\beta(\tau)(
1+h_{2}(\tilde{W},\tau)), \label{S7E1}\\
\tilde{W}((T_{\rm arr, mod}) ^{+};W_0)
=+\infty\label{S7E2}
\end{gather}
satisfies
\begin{equation}
\tilde{W}(( T_{\rm exit,mod})^{-};W_0)=-\infty\label{S7E3}
\end{equation}
for any $W_0$ large enough, where
\begin{equation}
\tilde{S}(T_{\rm exit,mod})
=T_{\rm arr, mod} \label{S7E4}
\end{equation}
and where the function $\tilde{S}$ is defined by means of the
relation $S(T_{\rm exit}) =T_{\rm arr}$ as
well as (\ref{S6E22}), (\ref{S6E23}).

The key idea to solve the Transition Problem (\ref{S7E1})-(\ref{S7E4}) might
be considered, is to treat it as a perturbation of the Simplified Transition
Problem \eqref{S5E1}--\eqref{S5E4} that was solved in an explicit manner in
Section \ref{five}. We then proceed to reformulate (\ref{S7E1})-(\ref{S7E4})
in a more convenient way.



\subsection{Reformulating the Transition Problem as a perturbation of the
Simplified Transition Problem}

Arguing as in Section \ref{five} we define a function
$f(\zeta)\geq T_0$ strictly monotonically increasing in $\zeta$,
defined in $\zeta\geq\zeta_0$ satisfying
\begin{gather}
\tilde{S}(f(\zeta+1))  =f(\zeta), \label{S7E5}\\
f(\zeta_0)  =T_0\,. \label{S7E6}
\end{gather}
Such a function $f$ is uniquely defined by its values in the
interval $\zeta\in[\zeta_0,\zeta_0+1]$. Moreover, arguing as in
Section \ref{five} we would have that $f\in C^{3}[\zeta_0
,\infty)$ if $f\in C^{3}[ \zeta_0,\zeta_0+1]$ and satisfies the
compatibility conditions (\ref{S5E7})-(\ref{S5E11}).

We define a new set of variables $(\zeta,Y)$ by means of (cf.
(\ref{S5E13}), (\ref{S5E14}))
\begin{gather}
\tau  =f(\zeta), \label{S7E7}\\
\tilde{W}-1  =\psi(Y,\zeta),\label{S7E8}
\end{gather}
where $\psi(Y,\zeta)$ is as in (\ref{S5E12}).
We define $\bar{\beta}(\cdot)$ as (cf. (\ref{S5E15}))
\begin{equation}
\bar{\beta}(f(\zeta))=\frac{1}{( f'( \zeta))
^{2}}(\frac{1}{2}\{ f,\zeta\}  -\pi^{2})\,.\label{S7E9}
\end{equation}
We will look for solutions of (\ref{S7E1})-(\ref{S7E4}) in the form
\begin{equation}
\beta(f(\zeta))=\bar{\beta}(f(\zeta))+\mu(\zeta)\,. \label{S7E10}
\end{equation}
We have
\begin{equation}
\frac{1}{f'(\zeta)}\big(-(Y^{2}+\pi ^{2}) \frac{\partial\psi}{\partial
Y}+\frac{\partial\psi}{\partial
\zeta}\big)+\frac{\psi^{2}}{3}-3\bar{\beta}(f( \zeta) )=0\,.
\label{S7E11}
\end{equation}
On the other hand, the change of variables (\ref{S7E7}), (\ref{S7E8})
transforms (\ref{S7E1}) into
\begin{equation}
\frac{1}{f'(\zeta)}( Y_{\zeta}\frac{\partial\psi }{\partial
Y}+\frac{\partial\psi}{\partial\zeta})+\frac{\psi^{2}}
{3}-h_{1}(\psi,f(\zeta))-3\beta(f( \zeta)) (1+h_{2}(\psi,f(\zeta)
))=0 \,. \label{S7E12}
\end{equation}
Subtracting (\ref{S7E11}) from (\ref{S7E12}) and using
$\frac{\partial\psi }{\partial Y}=\frac{1}{f'(\zeta) }$ we obtain
\begin{equation}
\frac{1}{(f'(\zeta))^{2}}( Y_{\zeta}+Y^{2}+\pi^{2})
-h_{1}(\psi,f(\zeta))-3\mu(\zeta)-3\beta(f(\zeta)
)h_{2}(\psi,f(\zeta))=0\,. \label{S7E13}
\end{equation}
Let us define
\begin{gather}
\lambda(\zeta)  =3\mu(\zeta)(f'(\zeta))^{2}, \label{S7E14}\\
\bar{R}(Y,\zeta)  =(f'(\zeta))^{2}[
h_{1}(\psi(Y,\zeta),f(\zeta))+3\beta(f(\zeta))
h_{2}(\psi(Y,\zeta),f( \zeta)) ]\,. \label{S7E15}
\end{gather}
Then (\ref{S7E13}) might be rewritten as
\begin{equation}
Y_{\zeta}+Y^{2}+\pi^{2}=\lambda(\zeta)+\bar{R}(Y,\zeta)\,.\label{S7E16}
\end{equation}
Using (\ref{S7E4}), (\ref{S7E5}) the boundary conditions (\ref{S7E2}),
(\ref{S7E3}) become
\begin{gather}
Y(\bar{\zeta})  =+\infty,\label{S7E17}\\
Y(\bar{\zeta}+1)  =-\infty\label{S7E18}
\end{gather}
for any $\bar{\zeta}\geq\zeta_0$. Note that the new formulation of
the Transition Problem (\ref{S7E16})-(\ref{S7E18}) is a
perturbation of (\ref{S5E17})-(\ref{S5E19}). By technical reasons
it is convenient to transform (\ref{S7E16})-(\ref{S7E18}) into a
new problem without singular boundary conditions. Let us denote as
$Y(\zeta,\bar{\zeta})$ the solution of
(\ref{S7E16})-(\ref{S7E18}). We define a new function $Z(
\zeta,\bar{\zeta})$ as
\begin{equation}
Y(\zeta,\bar{\zeta})=\pi\cot(\pi(\zeta-\bar
{\zeta}-Z(\zeta,\bar{\zeta})))\,. \label{S7E19}
\end{equation}
Using this new function, (\ref{S7E16})-(\ref{S7E18}) becomes
\begin{gather}
Z_{\zeta}    =\frac{\sin^{2}(\pi(\zeta-\bar{\zeta}-Z(
\zeta,\bar{\zeta})))}{\pi^{2}}[\lambda(
\zeta)+R(Z,\zeta,\bar{\zeta})],\label{S7E20}\\
Z(\bar{\zeta}^{+},\bar{\zeta})  =0,\label{S7E21}\\
Z((\bar{\zeta}+1)^{-},\bar{\zeta})  =0,
\label{S7E22}
\end{gather}
where
\begin{equation}
R(Z,\zeta,\bar{\zeta})\equiv\bar{R}(\psi(\pi \cot(
\pi(\zeta-\bar{\zeta}-Z(\zeta,\bar{\zeta})
))),\zeta)\,. \label{S7E22a}
\end{equation}
Note that the function $R(Z,\zeta,\bar{\zeta})$ is a smooth
function, bounded in compact sets of $\zeta,\bar{\zeta}$.

Let us summarize. We have transformed the Transition Problem
(\ref{S7E1} )-(\ref{S7E4}) into the problem
\eqref{S7E20}-\eqref{S7E22}. Therefore, our goal is to find
functions $\lambda(\zeta)$ ,\ $\zeta\geq \zeta_0$ such that the
unique solution of \eqref{S7E20}, \eqref{S7E21} satisfies
\eqref{S7E22}. We will solve this problem transforming it in an
integral equation coupled with a differential equation.

It turns out that it is possible obtain functions $\lambda(
\zeta)$ solving \eqref{S7E20}-\eqref{S7E22} for any function
$\lambda(\zeta)$ defined in $[\zeta_0,\zeta _0+1]$ satisfying
suitable compatibility conditions. To explain in an
intuitive manner the meaning of these compatibility conditions we
proceed to solve \eqref{S7E20}-\eqref{S7E22} in the particular
case $R\equiv0$ with $\lambda(\cdot)$ small.

\subsection{Solving \eqref{S7E20}-\eqref{S7E22} with $R\equiv0$ and
$\lambda(\cdot)$ small}\label{subsectsim}

It is illuminating to solve \eqref{S7E20}-\eqref{S7E22} under the assumptions
stated in the heading of this Subsection, because this can be made in an
explicit manner and it provides some intuition about the main arguments used later.

In the case $R\equiv0$ the equation \eqref{S7E20} becomes
\[
Z_{\zeta}=\frac{\sin^{2}(\pi(\zeta-\bar{\zeta}-Z(
\zeta,\bar{\zeta})))}{\pi^{2}}\lambda( \zeta)\,.
\]
On the other hand, if $\lambda(\cdot)$ is small $Z(
\zeta,\bar{\zeta})$ would be small too. Then the problem
\eqref{S7E20}-\eqref{S7E22} might be approximated as
follows:

Find functions $\lambda(\cdot)\in L^{\infty}[\zeta _0,\infty)$
such that for any $\bar{\zeta}\geq\zeta_0$ the function
$Z(\zeta,\bar{\zeta})$ solution of
\begin{equation}
Z_{\zeta}=\frac{\sin^{2}(\pi(\zeta-\bar{\zeta}))
}{\pi^{2}}\lambda(\zeta)\label{S7E23}
\end{equation}
with initial condition \eqref{S7E21} satisfies \eqref{S7E22}. Since, in the
case $R\equiv0$ \eqref{S7E20}-\eqref{S7E22} is just a reformulation of the
Simplified Transition Problem solved in Section \ref{five}, the problem
\eqref{S7E21}-(\ref{S7E23}) is a linearized version of the problem studied
there, and it could be studied using similar methods. However, the solution
obtained here can be easily adapted to solve the whole Transition Problem
\eqref{S7E20}-\eqref{S7E22}.

The solution of \eqref{S7E21}, (\ref{S7E23}) is given by
\[
Z(\zeta,\bar{\zeta})=\frac{1}{\pi^{2}}\int_{\bar{\zeta}}
^{\zeta}\sin^{2}(\pi(\xi-\bar{\zeta}))\lambda( \xi)d\xi\,.
\]
Using \eqref{S7E22}, we obtain the following integral equation, for
$\lambda(\cdot)$,
\begin{equation}
\Phi(\bar{\zeta})\equiv\int_{\bar{\zeta}}^{\bar{\zeta}+1}
\sin^{2}(\pi(\xi-\bar{\zeta}))\lambda(
\xi)d\xi=0\,. \label{S7E24}
\end{equation}
To solve \eqref{S7E24} we differentiate $\Phi$ three times with
respect to $\bar{\zeta}$. We have
\begin{gather*}
\Phi'(\bar{\zeta})  =\pi\int_{\bar{\zeta}}
^{\bar{\zeta}+1}\sin(2\pi(\xi-\bar{\zeta}))
\lambda(\xi)d\xi, \\
\Phi''(\bar{\zeta})  =2\pi^{2}\int_{\bar{\zeta}
}^{\bar{\zeta}+1}\cos(2\pi(\xi-\bar{\zeta}))
\lambda(\xi)d\xi, \\
\Phi'''(\bar{\zeta})  =2\pi^{2}[
\lambda(\bar{\zeta}+1)-\lambda(\bar{\zeta})]
-4\pi^{3}\int_{\bar{\zeta}}^{\bar{\zeta}+1}\sin(2\pi(
\xi-\bar{\zeta}))\lambda(\xi)d\xi\,.
\end{gather*}
Therefore,
\begin{equation}
\Phi'''(\bar{\zeta})+4\pi^{2}\Phi'(\bar{\zeta})=2\pi^{2}[\lambda(\bar{\zeta
}+1)-\lambda(\bar{\zeta})]\,.\label{S7E25}
\end{equation}
Suppose that $\lambda(\cdot)$ solves \eqref{S7E24}. Since
$\Phi(\bar{\zeta})=0$ for $\bar{\zeta}\geq\zeta_0$, it follows
from (\ref{S7E25}) that
\begin{equation}
\lambda(\bar{\zeta}+1)=\lambda(\bar{\zeta})
\,,\quad \bar{\zeta}\geq\zeta_0 \,. \label{S7E26}
\end{equation}
Moreover, if $\lambda(\cdot)$ solves \eqref{S7E24} we have
$\Phi(\zeta_0)=\Phi'(\zeta_0) =\Phi''(\zeta_0)=0$,
i.e. $\lambda( \cdot)$ satisfies
\begin{equation}
\begin{aligned}
\int_{\zeta_0}^{\zeta_0+1}\sin^{2}(\pi( \xi-\zeta_0)
)\lambda(\xi)d\xi &  =\int_{\zeta_0}^{\zeta_0
+1}\sin(2\pi(\xi-\zeta_0))\lambda(\xi)d\xi \\
&=\int_{\zeta_0}^{\zeta_0+1}\cos(2\pi( \xi-\zeta_0)
)\lambda(\xi)d\xi   =0
\end{aligned}\label{S7E27}
\end{equation}
If, on the contrary $\lambda(\zeta)$ is any function that
satisfies \eqref{S7E27} for $\zeta\in( \zeta_0,\zeta_0+1)$ and we
extend $\lambda( \cdot)$ to $\bar{\zeta}\geq\zeta_0$ using
\eqref{S7E26} it would follow that the resulting $\lambda(\cdot)$
would solve \eqref{S7E24}, because under these assumptions
(\ref{S7E25}) implies
\begin{equation}
\Phi'''(\bar{\zeta}) +4\pi^{2}\Phi'(\bar{\zeta})=0\,,\quad
\bar{\zeta}>\zeta_0\,. \label{S7E27a}
\end{equation}
Also \eqref{S7E27} implies
\[
\Phi(\zeta_0)=\Phi'(\zeta_0) =\Phi''(\zeta_0)=0,
\]
whence the fact that $\Phi(\bar{\zeta})=0$ for $\bar{\zeta}
\geq\zeta_0$ just follows using standard uniqueness result for
ODEs. We summarize this result as follows.

\begin{proposition}
A function $\lambda(\cdot)$ solves the integral equation
\eqref{S7E24} if and only if $\lambda(\cdot)$ satisfies \eqref{S7E26},
\eqref{S7E27}.
\end{proposition}

The idea explained in this Subsection is the same that will be used to solve
the Transition Problem \eqref{S7E20}-\eqref{S7E22}. Note that for the
simplified version just considered we have obtained a set of compatibility
conditions \eqref{S7E27}. A similar set of compatibility conditions arises in
the study of \eqref{S7E20}-\eqref{S7E22}. We find them in next Subsection.



\subsection{Transition problem: Compatibility conditions}\label{compat}

To study  \eqref{S7E20}-\eqref{S7E22} we begin by finding a
set of compatibility conditions analogous to \eqref{S7E27}.
Note that \eqref{S7E20}-\eqref{S7E22} imply
\begin{equation}
\Phi(\bar{\zeta})\equiv\int_{\bar{\zeta}}^{\bar{\zeta}+1}
\frac{\sin^{2}(\pi(\zeta-\bar{\zeta}-Z(\zeta,\bar{\zeta }) )
)}{\pi^{2}}[\lambda(\zeta)+R( Z(\zeta,\bar{\zeta})
,\zeta,\bar{\zeta}) ]d\zeta=0\,,
\label{S7E28}
\end{equation}
where $\bar{\zeta}\geq\zeta_0$.
We then argue as in Subsection \ref{subsectsim}. Note that a
first compatibility condition for $\lambda(\zeta)$ in $\zeta
\in(\zeta_0,\zeta_0+1)$ is
\begin{equation}
\int_{\zeta_0}^{\zeta_0+1}\sin^{2}(\pi(\zeta-\zeta
_0-Z(\zeta,\zeta_0)))[\lambda(\zeta)
+R(Z,\zeta,\zeta_0)]d\zeta=0\; \label{S7E29}
\end{equation}
with $Z(\zeta,\zeta_0)$ defined by means of \eqref{S7E20},
\eqref{S7E21} with $\bar{\zeta}=\zeta_0$. Differentiating
$\Phi(\bar{\zeta})$ in (\ref{S7E28}), and choosing
$\bar{\zeta}=\zeta_0$ we obtain
\begin{equation}
\begin{aligned}
&  -\pi\int_{\zeta_0}^{\zeta_0+1}\sin(2\pi(\zeta-\zeta
_0-Z(\zeta,\zeta_0)))\big(1+\frac {\partial
Z(\zeta,\zeta_0)}{\partial\zeta_0}\big)[\lambda(\zeta)
+R(Z,\zeta,\zeta_0)]d\zeta\\
& +\int_{\zeta_0}^{\zeta_0+1}\sin^{2}(\pi(\zeta-\zeta
_0-Z(\zeta,\zeta_0)))[\frac{\partial R}{\partial\zeta_0}(
Z,\zeta,\zeta_0)+\frac{\partial R}{\partial Z}\frac{\partial
Z}{\partial\zeta_0}(\zeta,\zeta
_0)]d\zeta  =0
\end{aligned}\label{S7E30}
\end{equation}
We can simplify (\ref{S7E30}) after computing an equation for
$\frac{\partial Z(\zeta,\zeta_0)}{\partial\zeta_0}$.
Differentiating \eqref{S7E20}, with respect to $\bar{\zeta}$, we
obtain
\begin{equation}
\begin{aligned}
(\frac{\partial Z}{\partial\bar{\zeta}})_{\zeta}
&=-\frac{\sin(2\pi(\zeta-\bar{\zeta}-Z(\zeta,\bar{\zeta }) )
)}{\pi}[\lambda(\zeta)+R( Z,\zeta,\bar{\zeta})] (1+\frac{\partial
Z}{\partial\bar{\zeta}}) \\
&\quad  +\frac{\sin^{2}(\pi(\zeta-\bar{\zeta}-Z(\zeta
,\bar{\zeta})))}{\pi^{2}}g( \zeta,\bar{\zeta}),
\end{aligned}\label{S7E31}
\end{equation}
where
\begin{equation}
g(\zeta,\bar{\zeta})\equiv\frac{\partial R}{\partial\bar{\zeta
}}+\frac{\partial R}{\partial Z}\frac{\partial
Z}{\partial\bar{\zeta}}\,.
\label{S7E31a}
\end{equation}
On the other hand, differentiating \eqref{S7E21} we arrive at
\[
\frac{\partial Z}{\partial\zeta}(\bar{\zeta}^{+},\bar{\zeta})
+\frac{\partial Z}{\partial\bar{\zeta}}(
\bar{\zeta}^{+},\bar{\zeta })=0\,.
\]
Note that \eqref{S7E20} implies $\frac{\partial
Z}{\partial\zeta}( \bar{\zeta}^{+},\bar{\zeta})=0$. Then
\begin{equation}
\frac{\partial Z}{\partial\bar{\zeta}}(\bar{\zeta}^{+},\bar{\zeta
})=0 \,. \label{S7E32}
\end{equation}
Integrating (\ref{S7E31}), (\ref{S7E32}), we obtain
\begin{equation} \label{S7E33}
\begin{aligned}
&\frac{\partial Z}{\partial\bar{\zeta}}(\zeta,\bar{\zeta})\\
&=\int_{\bar{\zeta}}^{\zeta}e^{-\psi( \zeta,\bar{\zeta})
+\psi(\eta,\bar{\zeta})}\Big[-\frac{\sin( 2\pi(
\eta-\bar{\zeta}-Z(\eta,\bar{\zeta})))}{\pi }[
\lambda(\eta)+R(Z(\eta,\bar{\zeta
}),\eta,\bar{\zeta})] \\
& \quad   +\frac{\sin^{2}(\pi(\eta-\bar{\zeta}-Z(
\eta,\bar{\zeta})))}{\pi^{2}}g( \eta,\bar{\zeta })\Big] d\eta,
\end{aligned}
\end{equation}
where
\begin{equation}
\psi(\zeta,\bar{\zeta})\equiv\frac{1}{\pi}\int_{\bar{\zeta}
}^{\zeta}\sin(2\pi(\eta-\bar{\zeta}-Z(\eta,\bar{\zeta }) ) )
[\lambda(\eta)+R(Z( \eta,\bar{\zeta}) ,\eta,\bar{\zeta})]
d\eta \,. \label{S7E34}
\end{equation}
We can rewrite (\ref{S7E33}) as
\begin{equation}
\frac{\partial Z}{\partial\bar{\zeta}}(\zeta,\bar{\zeta})
+1=e^{-\psi(\zeta,\bar{\zeta})}
\Big(1+\int_{\bar{\zeta}
}^{\zeta}e^{\psi(\eta,\bar{\zeta})}\Big[\frac{\sin
^{2}(\pi(\eta-\bar{\zeta}-Z(\eta,\bar{\zeta})))
}{\pi^{2}}g(\eta,\bar{\zeta})\Big]d\eta\Big)\label{S7E35}
\end{equation}
substituting (\ref{S7E35}) with $\bar{\zeta}=\zeta_0$ in
(\ref{S7E30}) and using \eqref{S7E34}, we obtain
\begin{align*}
&  -\int_{\zeta_0}^{\zeta_0+1}\frac{\partial\psi(\zeta,\zeta
_0)}{\partial\zeta}e^{-\psi(\zeta,\zeta_0)}
d\zeta
 -\frac{1}{\pi^{2}}\int_{\zeta_0}^{\zeta_0+1}\frac{\partial\psi(
\zeta,\zeta_0)}{\partial\zeta}e^{-\psi( \zeta,\zeta _0)}\\
&\times \int_{\zeta_0}^{\zeta}e^{\psi( \eta,\zeta_0)
}\sin^{2}(\pi(\eta-\zeta_0-Z( \eta,\zeta_0)
))g(\eta,\zeta_0)\,d\eta \,d\zeta\\
& +\frac{1}{\pi^{2}}\int_{\zeta_0}^{\zeta_0+1}\sin^{2}(\pi(
\zeta-\zeta_0-Z(\zeta,\zeta_0)))g(
\zeta,\zeta_0)d\zeta
  =0\,.
\end{align*}
After some integrations by parts we arrive to the following
compatibility condition
\begin{equation}
(1-e^{\psi(\zeta_0+1,\zeta_0)})+\frac{1}
{\pi^{2}}\int_{\zeta_0}^{\zeta_0+1}e^{\psi(\zeta,\zeta_0)
}\sin^{2}(\pi(\zeta-\zeta_0-Z(\zeta,\zeta_0)
))g(\zeta,\zeta_0)d\zeta=0 \,. \label{S7E36}
\end{equation}
Actually \eqref{S7E36} holds for any $\bar{\zeta}\geq\zeta_0$ if
$\zeta_0 $ is replaced by $\bar{\zeta}$. We obtain an additional
compatibility condition that must be satisfied by $\lambda(\cdot)$
differentiating the resulting equation with respect to
$\bar{\zeta}$ and particularizing the value $\bar{\zeta}=\zeta_0$.
Equivalently we can just differentiate with respect to $\zeta_0$
in \eqref{S7E36} to obtain
\begin{equation}
\begin{aligned}
&  -e^{\psi(\zeta_0+1,\zeta_0)}\frac{d( \psi(
\zeta_0+1,\zeta_0))}{d\zeta_0}\\
&  +\frac{1}{\pi^{2}}\int_{\zeta_0}^{\zeta_0+1}e^{\psi(\zeta
,\zeta_0)}\frac{\partial\psi( \zeta,\zeta_0)
}{\partial\zeta_0}\sin^{2}(\pi(\zeta-\zeta_0-Z( \zeta,\zeta_0)))g(
\zeta,\zeta_0)
d\zeta\\
&  -\frac{1}{\pi}\int_{\zeta_0}^{\zeta_0+1}e^{\psi(\zeta,\zeta
_0)}\sin(2\pi(\zeta-\zeta_0-Z(\zeta,\zeta _0))) (1+\frac{\partial
Z(\zeta ,\zeta_0) }{\partial\zeta_0}) g(\zeta,\zeta
_0)d\zeta\\
&  +\frac{1}{\pi^{2}}\int_{\zeta_0}^{\zeta_0+1}e^{\psi(\zeta
,\zeta_0)}\sin^{2}(\pi(\zeta-\zeta_0-Z( \zeta,\zeta_0)
))\frac{\partial g( \zeta
,\zeta_0)}{\partial\zeta_0}d\zeta
  =0\,.
\end{aligned}\label{S7E37}
\end{equation}
Using \eqref{S7E34}, we obtain
\begin{align*}
&  \frac{d(\psi(\zeta_0+1,\zeta_0))}
{d\zeta_0}\\
&  =-2\int_{\zeta_0}^{\zeta_0+1}\cos(2\pi(\eta-\zeta
_0-Z(\eta,\zeta_0)))(1+\frac {\partial Z(\eta,\zeta_0)
}{\partial\zeta_0})\\
&\times\big[\lambda(\eta) +R(Z(\eta,\zeta_0)
,\eta,\zeta_0)\big]d\eta\\
&\quad  +\frac{1}{\pi}\int_{\zeta_0}^{\zeta_0+1}\sin(2\pi(
\eta-\zeta_0-Z(\eta,\zeta_0)))g( \eta,\zeta_0)d\eta\,.
\end{align*}
Therefore, we can use (\ref{S7E35}) to transform (\ref{S7E37}) into
\begin{equation} \label{S7E38}
\begin{aligned}
&  -2e^{\psi(\zeta_0+1,\zeta_0)}\int_{\zeta_0}^{\zeta
_0+1}e^{-\psi( \eta,\zeta_0)}\cos(2\pi(
\eta-\zeta_0-Z(\eta,\zeta_0)))\\
&\times \big[\lambda(\eta) +R(Z(\eta,\zeta_0)
,\eta,\zeta_0)\big]d\eta
  +K_{1}(\zeta_0)+K_{2}(\zeta_0)  =0,
\end{aligned}
\end{equation}
where
\begin{equation} \label{S7E39}
\begin{aligned}
&K_{1}(\zeta_0)\\
& \equiv-2e^{\psi(\zeta_0 +1,\zeta_0)
}\int_{\zeta_0}^{\zeta_0+1}\int_{\bar{\zeta}}^{\eta
}e^{\psi(\theta,\bar{\zeta})-\psi( \eta,\zeta_0)
}\cos(2\pi(\eta-\zeta_0-Z(\eta,\zeta_0)
)) \\
& \quad \times\big[\frac{\sin^{2}(\pi( \theta-\bar{\zeta}-Z(
\theta,\bar{\zeta})))}{\pi^{2}}g( \theta ,\bar{\zeta}) \big]
[\lambda(\eta)+R( Z(\eta,\zeta_0) ,\eta,\zeta_0) ] d\theta
d\eta\\
&\quad +\frac{e^{\psi(\zeta_0+1,\zeta_0)}}{\pi}\int_{\zeta_0
}^{\zeta_0+1}\sin(2\pi(\eta-\zeta_0-Z(\eta,\zeta _0) )
)g(\eta,\zeta_0)d\eta
\\
& \quad +\frac{1}{\pi^{2}}\int_{\zeta_0}^{\zeta_0+1}e^{\psi(\zeta
,\zeta_0)}\frac{\partial\psi( \zeta,\zeta_0)
}{\partial\zeta_0}\sin^{2}(\pi(\zeta-\zeta_0-Z( \zeta,\zeta_0)))g(
\zeta,\zeta_0)
d\zeta\\
& \quad  -\frac{1}{\pi}\int_{\zeta_0}^{\zeta_0+1}e^{\psi(\zeta,\zeta
_0)}\sin(2\pi(\zeta-\zeta_0-Z(\zeta,\zeta _0))) (1+\frac{\partial
Z(\zeta ,\zeta_0) }{\partial\zeta_0}) g(\zeta,\zeta
_0)d\zeta\,,
\end{aligned}
\end{equation}
\begin{equation}
K_{2}(\zeta_0)\equiv\frac{1}{\pi^{2}}\int_{\zeta_0}
^{\zeta_0+1}e^{\psi(\zeta,\zeta_0)}\sin^{2}( \pi(
\zeta-\zeta_0-Z(\zeta,\zeta_0))) \frac{\partial
g(\zeta,\zeta_0)}{\partial\zeta_0} d\zeta \,. \label{S7E40}
\end{equation}
In the definitions of $K_{1}(\zeta_0)$, $K_{2}( \zeta_0) $ we have
included in $K_{1}( \zeta_0) ,\;K_{2}(\zeta_0)$ all the terms
depending on $g( \zeta,\zeta_0)$ and in $\frac{\partial g(
\zeta,\zeta _0)}{\partial\zeta_0}$ respectively.

Equations \eqref{S7E29}, \eqref{S7E36} and \eqref{S7E38} are the
compatibility conditions that we will need to assume on
$\lambda(\cdot)$ in order to solve \eqref{S7E20}-\eqref{S7E22}. We
remark that these conditions reduce to \eqref{S7E27} if $R\equiv0$
and only linear terms on $\lambda(\cdot)$ are kept, as could be
expected.

The rest of this Section is devoted to obtaining local existence
and uniqueness results for \eqref{S7E20}-\eqref{S7E22} given a
function $\lambda(\cdot)$ defined in $[\zeta_0,\zeta _0+1]$
satisfying the compatibility conditions \eqref{S7E29},
\eqref{S7E36} and \eqref{S7E38}. The key idea is to adapt the
argument that was used in Subsection \ref{subsectsim} to transform
(\ref{S7E21} )-(\ref{S7E23}) into \eqref{S7E26}, (\ref{S7E27a}).

\subsection{Local existence and uniqueness Theorem}

The main result in this Subsection is as follows.

\begin{theorem}\label{Th2}
Given $\lambda(\cdot)$ in $C[\zeta_0 ,\zeta_0+1]$ we
define a function $Z(\zeta;\zeta_0),\;$for $\zeta\in[
\zeta_0,\zeta_0+1]$ as the unique solution of \eqref{S7E20},
\eqref{S7E21} with $\bar{\zeta}=\zeta_0$. Suppose that
$\lambda(\zeta)$, $Z(\zeta;\zeta_0)$ satisfy the compatibility
conditions \eqref{S7E29}, \eqref{S7E36}, \eqref{S7E38} with
$g(\zeta;\zeta_0)$ as in \eqref{S7E31a}, $\psi(\zeta;\zeta_0)$ is
defined by means of \eqref{S7E34} and $K_{1}(\zeta_0)$,
$K_{2}(\zeta_0)$ are as in \eqref{S7E39}, \eqref{S7E40}. Then, for
any $\zeta_0>0$ large enough, there exists $\delta>0$ depending
only on $\| \lambda(\cdot)\|_{L^{\infty}[\zeta_0,\zeta_0+1]}$
such that the problem
\eqref{S7E20}-\eqref{S7E22} has a unique solution $\lambda(\cdot)$
$\in C[\zeta_0,\zeta_0 +1+\delta]$.
\end{theorem}

\begin{proof}
We wish to transform the problem in a perturbed version of
\eqref{S7E26} where it is possible to apply classical fixed point
arguments. We remark that for an arbitrary function
$\lambda(\cdot)\in C[\zeta_0 ,\zeta_0+1+\delta]$ a function
$Z(\zeta;\bar{\zeta})$ solving \eqref{S7E20}, \eqref{S7E21} does
not satisfy in general \eqref{S7E22}. It is then convenient to
define, by technical reasons, two functions
$Z_{-}(\zeta;\bar{\zeta})$, $Z_{+}(\zeta ;\bar{\zeta})$ as
follows. For any $\bar{\zeta}\in[\zeta _0,\zeta_0+\delta]$ we
denote as $Z_{-}( \zeta;\bar{\zeta })$, $Z_{+}(
\zeta;\bar{\zeta})$ the unique solutions of \eqref{S7E20},
\eqref{S7E21} and \eqref{S7E20}, \eqref{S7E22} respectively. We
also define
\begin{equation}
Z(\zeta;\bar{\zeta})\equiv\begin{cases}
 Z_{-}(\zeta;\bar{\zeta})\,,&\zeta\in[\zeta_0,\zeta_0+\frac{1}{2})\\
Z_{+}(\zeta;\bar{\zeta})\,, &\zeta\in[\zeta_0+\frac{1}{2},\zeta_0+1]
\end{cases}
 \label{S7E41}
\end{equation}
Note that for an arbitrary function $\lambda(\cdot)$ $\in
C[\zeta_0,\zeta_0+1+\delta]$ the function $Z(\zeta;\bar{\zeta})$
is discontinuous at the point $\zeta=\zeta _0+\frac{1}{2}$.
Actually the functions $\lambda(\cdot)$ solving
\eqref{S7E20}-\eqref{S7E22} are precisely those functions for
which:
\[
Z_{-}(\zeta_0+\frac{1}{2};\bar{\zeta})=Z_{+}( \zeta
_0+\frac{1}{2};\bar{\zeta})\,.
\]
We define a function $\Phi(\bar{\zeta}) ,\;\bar{\zeta}\in[
\zeta_0,\zeta_0+\delta]$ by means of (\ref{S7E28}). Due to
(\ref{S7E41}) we have
\begin{equation} \label{S7E42}
\begin{aligned}
\Phi(\bar{\zeta})
&\equiv\int_{\bar{\zeta}}^{\zeta_0+\frac{1}{2}}\frac{\sin^{2}(
\pi(\zeta-\bar{\zeta}-Z_{-}(\zeta,\bar{\zeta}))) }{\pi^{2}}[
\lambda(\zeta)+R( Z_{-}( \zeta,\bar{\zeta})
,\zeta,\bar{\zeta})]d\zeta
\\
&\quad +\int_{\zeta_0+\frac{1}{2}}^{\bar{\zeta}+1}\frac{\sin^{2}(\pi(
\zeta-\bar{\zeta}-Z_{+}(\zeta,\bar{\zeta}))) }{\pi^{2}}[
\lambda(\zeta)+R(Z_{+}( \zeta,\bar{\zeta})
,\zeta,\bar{\zeta})]d\zeta
\\
&=0\,,\quad \bar{\zeta}\geq\zeta_0
\end{aligned}
\end{equation}
Arguing as in the proof of (\ref{S7E35}) we obtain
\begin{equation}
\begin{aligned}
&\frac{\partial Z_{-}}{\partial\bar{\zeta}}(\zeta,\bar{\zeta})
+1 \\
&=e^{-\psi_{-}(\zeta,\bar{\zeta})}\Big(1+\int_{\bar{\zeta}
}^{\zeta}e^{\psi_{-}(\eta,\bar{\zeta})}[\frac{\sin
^{2}(\pi(\eta-\bar{\zeta}-Z_{-}(\eta,\bar{\zeta})))
}{\pi^{2}}g(\eta,\bar{\zeta})]
d\eta\Big),
\end{aligned}\label{S7E43}
\end{equation}
\begin{equation}
\begin{aligned}
&\frac{\partial Z_{+}}{\partial\bar{\zeta}}(\zeta,\bar{\zeta})
+1\\
&=e^{-\psi_{+}(\zeta,\bar{\zeta})}\Big(1-\int_{\zeta}
^{\bar{\zeta}+1}e^{\psi_{+}(\eta,\bar{\zeta})}[
\frac{\sin^{2}(\pi(\eta-\bar{\zeta}-Z_{+}(\eta,\bar {\zeta})
))}{\pi^{2}}g(\eta,\bar{\zeta}) ]d\eta\Big),
\end{aligned}\label{S7E44}
\end{equation}
\begin{equation}
\psi_{-}(\zeta,\bar{\zeta}) \equiv\frac{1}{\pi}\int_{\bar{\zeta
}}^{\zeta}\sin( 2\pi( \eta-\bar{\zeta}-Z_{-}(\eta
,\bar{\zeta})))[\lambda(\eta)
+R(Z_{-}(\eta,\bar{\zeta}),\eta,\bar{\zeta})
]d\eta, \label{S7E45}
\end{equation}
\begin{equation}
\begin{aligned}
&\psi_{+}(\zeta,\bar{\zeta})\\
&\equiv-\frac{1}{\pi}\int_{\zeta
}^{\bar{\zeta}+1}\sin(2\pi(\eta-\bar{\zeta}-Z_{+}(
\eta,\bar{\zeta})))[\lambda( \eta)
+R(Z_{+}(\eta,\bar{\zeta}),\eta,\bar{\zeta}) ] d\eta,
\end{aligned}\label{S7E46}
\end{equation}
where $g(\eta,\bar{\zeta})$ is defined by means of \eqref{S7E31a}
with the values of $Z( \zeta;\bar{\zeta})$ required in each region
of integration.
We rewrite (\ref{S7E43}), (\ref{S7E44}) as
\begin{equation}
\frac{\partial Z}{\partial\bar{\zeta}}(\zeta,\bar{\zeta})
+1=H_0(\zeta,\bar{\zeta})+H_{1}(\zeta,\bar{\zeta }),\label{S7E44a}
\end{equation}
where
\begin{equation}
H_0(\zeta,\bar{\zeta})\equiv\begin{cases}
 e^{-\psi_{-}(\zeta,\bar{\zeta})}\,, &\zeta \in[\zeta_0,\zeta_0+\frac{1}{2})\\
e^{-\psi_{+}(\zeta,\bar{\zeta})}\,, &\zeta\in[\zeta_0+\frac{1}{2},
\zeta_0+1]\,.
\end{cases} \label{S7E44b}
\end{equation}
Note that
\begin{equation}
\begin{gathered}
H_0(\bar{\zeta},\bar{\zeta})=H_0(\bar{\zeta}
+1,\bar{\zeta})=1\,,\\
H_{1}(\bar{\zeta},\bar{\zeta})=H_{1}(\bar{\zeta}
+1,\bar{\zeta})=0\,.
\end{gathered} \label{S7E44c}
\end{equation}
Differentiating (\ref{S7E42}) and using (\ref{S7E43}), (\ref{S7E44}), we
obtain
\begin{equation}
\frac{d\Phi(\bar{\zeta})}{d\bar{\zeta}}=-e^{-\psi_{-}(
\zeta_0+\frac{1}{2},\bar{\zeta})}+e^{-\psi_{+}( \zeta
_0+\frac{1}{2},\bar{\zeta})}-G_{1}( \bar{\zeta},\zeta
_0),\label{S7E47}
\end{equation}
\begin{align*}
&G_{1}(\bar{\zeta},\zeta_0)\\
&\equiv\frac{e^{-\psi_{-}(
\zeta_0+\frac{1}{2},\bar{\zeta})}}{\pi^{2}}\int_{\bar{\zeta}}
^{\zeta_0+\frac{1}{2}}e^{\psi_{-}(\eta,\bar{\zeta})}\sin
^{2}(\pi(\eta-\bar{\zeta}-Z_{-}(\eta,\bar{\zeta})
))g(\eta,\bar{\zeta})d\eta\\
&\quad +\frac{e^{-\psi_{+}(\zeta_0+\frac{1}{2},\bar{\zeta})}}
{\pi^{2}}\int_{\zeta_0+\frac{1}{2}}^{\bar{\zeta}+1}e^{\psi_{+}(
\eta,\bar{\zeta})}\sin^{2}(\pi(\eta-\bar{\zeta}
-Z_{+}(\eta,\bar{\zeta})))g(\eta ,\bar{\zeta}) d\eta,
\end{align*}
\begin{equation}
\begin{aligned}
\frac{d^{2}\Phi(\bar{\zeta})}{d\bar{\zeta}^{2}}
&=e^{-\psi_{-}(\zeta_0+\frac{1}{2},\bar{\zeta})}\frac{\partial\psi
_{-}(\zeta_0+\frac{1}{2},\bar{\zeta})}{\partial\bar{\zeta}}\\
&\quad -e^{-\psi_{+}(\zeta_0+\frac{1}{2},\bar{\zeta})}
\frac{\partial\psi_{+}(\zeta_0+\frac{1}{2},\bar{\zeta})
}{\partial\bar{\zeta}}-\frac{\partial G_{1}( \bar{\zeta},\zeta
_0)}{\partial\bar{\zeta}}\,.
\end{aligned}\label{S7E48}
\end{equation}
Using (\ref{S7E47}) to eliminate $\exp\big(-\psi_{+}(\zeta_0+\frac{1}
{2},\bar{\zeta})\big)$ in (\ref{S7E48}), we obtain
\begin{equation}
\begin{aligned}
&\frac{d^{2}\Phi(\bar{\zeta})}{d\bar{\zeta}^{2}}+\frac
{\partial\psi_{+}(\zeta_0+\frac{1}{2},\bar{\zeta})}
{\partial\bar{\zeta}}\frac{d\Phi(\bar{\zeta})}{d\bar{\zeta}
}\\
&=e^{-\psi_{-}(\zeta_0+\frac{1}{2},\bar{\zeta})}\frac
{\partial(\psi_{-}(\zeta_0+\frac{1}{2},\bar{\zeta})
-\psi_{+}(\zeta_0+\frac{1}{2},\bar{\zeta}))}
{\partial\bar{\zeta}}\\
&\quad -\frac{\partial\psi_{+}(\zeta_0+\frac{1}{2},\bar{\zeta})
}{\partial\bar{\zeta}}G_{1}(\bar{\zeta},\zeta_0) -\frac{\partial
G_{1}(\bar{\zeta},\zeta_0)}{\partial \bar{\zeta}}\,.
\end{aligned}\label{S7E49}
\end{equation}
Differentiating (\ref{S7E49}) with respect to $\bar{\zeta}$, we obtain
\begin{equation} \label{S7E50}
\begin{gathered}
\frac{d}{d\bar{\zeta}}\Big(\frac{d^{2}\Phi(\bar{\zeta})
}{d\bar{\zeta}^{2}}+\frac{\partial\psi_{+}(\zeta_0+\frac{1}{2}
,\bar{\zeta})}{\partial\bar{\zeta}}\frac{d\Phi(\bar{\zeta
})}{d\bar{\zeta}}\Big)\\
=e^{-\psi_{-}(\zeta_0+\frac{1}{2},\bar{\zeta})}\frac
{\partial^{2}(\psi_{-}( \zeta_0+\frac{1}{2},\bar{\zeta})
-\psi_{+}(\zeta_0+\frac{1}{2},\bar{\zeta}))}
{\partial\bar{\zeta}^{2}}+U(\bar{\zeta},\zeta_0)\,,
\end{gathered}
\end{equation}
where
\begin{equation}
\begin{aligned}
U(\bar{\zeta},\zeta_0)
&\equiv-\frac{\partial\psi_{-}(
\zeta_0+\frac{1}{2},\bar{\zeta}) }{\partial\bar{\zeta}}\frac
{\partial(\psi_{-}( \zeta_0+\frac{1}{2},\bar{\zeta})
-\psi_{+}(\zeta_0+\frac{1}{2},\bar{\zeta}))}
{\partial\bar{\zeta}}e^{-\psi_{-}(\zeta_0+\frac{1}{2},\bar{\zeta
})}\\
&\quad -\frac{\partial}{\partial\bar{\zeta}}
\Big(\frac{\partial\psi_{+}(
\zeta_0+\frac{1}{2},\bar{\zeta}) }{\partial\bar{\zeta}}G_{1}(
\bar{\zeta},\zeta_0)+\frac{\partial G_{1}(\bar{\zeta}
,\zeta_0)}{\partial\bar{\zeta}}\Big)\,.
\end{aligned}\label{S7E50a}
\end{equation}
Equation (\ref{S7E50}) will play a role analogous to (\ref{S7E25})
in the analysis of the linearized problem considered in Subsection
\ref{subsectsim}. To formulate the analogous of the
problem \eqref{S7E26} we need to compute the term
$\frac{\partial^{2}(\psi_{-}(\zeta_0+\frac
{1}{2},\bar{\zeta})-\psi_{+}( \zeta_0+\frac{1}{2},\bar{\zeta
}))}{\partial\bar{\zeta}^{2}}$.
Using (\ref{S7E41}), (\ref{S7E45}), (\ref{S7E46}) we  have
\begin{equation} \label{S7E50b}
\begin{aligned}
&\psi_{-}(\zeta_0+\frac{1}{2},\bar{\zeta})-\psi_{+}(
\zeta_0+\frac{1}{2},\bar{\zeta}) \\
&=\frac{1}{\pi}\int_{\bar{\zeta}}^{\bar{\zeta}+1}\sin(2\pi(
\eta-\bar{\zeta}-Z(\eta,\bar{\zeta})))[\lambda(\eta)
+g(\eta,\bar{\zeta})]d\eta;
\end{aligned}
\end{equation}
whence, using (\ref{S7E44a}),
\begin{align*}
&\frac{\partial(\psi_{-}(\zeta_0+\frac{1}{2},\bar{\zeta })
-\psi_{+}(\zeta_0+\frac{1}{2},\bar{\zeta}))
}{\partial\bar{\zeta}}\\
&=-2\int_{\bar{\zeta}}^{\bar{\zeta}+1}\cos(2\pi( \eta-\bar{\zeta
}-Z(\eta,\bar{\zeta})))(H_0( \zeta,\bar{\zeta})
+H_{1}(\zeta,\bar{\zeta}))[\lambda(\eta) +g(\eta,\bar{\zeta})
]d\eta\\
&\quad+\frac{1}{\pi}\int_{\bar{\zeta}}^{\bar{\zeta}+1}\sin(2\pi(
\eta-\bar{\zeta}-Z(\eta,\bar{\zeta})))\frac{\partial
g(\eta,\bar{\zeta})}{\partial\bar{\zeta}}d\eta\,.
\end{align*}
Differentiating this formula and using (\ref{S7E44c}), we obtain
\begin{equation}
\frac{\partial^{2}(\psi_{-}( \zeta_0+\frac{1}{2},\bar{\zeta
})-\psi_{+}(\zeta_0+\frac{1}{2},\bar{\zeta}))
}{\partial\bar{\zeta}^{2}}
=-2[\lambda(\bar{\zeta}+1)-\lambda( \bar{\zeta
})]+V(\bar{\zeta},\zeta_0)\,,
\label{S7E51}
\end{equation}
\begin{equation}
\begin{aligned}
V(\bar{\zeta},\zeta_0)
&\equiv-4\pi\int_{\bar{\zeta}}
^{\bar{\zeta}+1}\sin(2\pi(\eta-\bar{\zeta}-Z(\eta
,\bar{\zeta})))(H_0( \zeta,\bar{\zeta })\\
&\quad +H_{1}(\zeta,\bar{\zeta}))^{2}[\lambda(\eta) +g(\eta,\bar{\zeta})]
d\eta
-2\int_{\bar{\zeta}}^{\bar{\zeta}+1}\cos(2\pi( \eta-\bar{\zeta
}-Z(\eta,\bar{\zeta}))) \\
&\quad\times \frac{\partial(
(H_0(\zeta,\bar{\zeta})+H_{1}(\zeta,\bar{\zeta }))[
\lambda(\eta)+g(\eta
,\bar{\zeta})])}{\partial\bar{\zeta}}d\eta\\
&\quad +\frac{1}{\pi}\int_{\bar{\zeta}}^{\bar{\zeta}+1}
\frac{\partial}{\partial
\bar{\zeta}}\Big(\sin(2\pi(\eta-\bar{\zeta}-Z( \eta,\bar{\zeta})
))\frac{\partial g(\eta
,\bar{\zeta})}{\partial\bar{\zeta}}\Big)d\eta\,.
\end{aligned}\label{S7E51a}
\end{equation}
Suppose that $\lambda(\bar{\zeta})$\ satisfies
\begin{equation}
[\lambda(\bar{\zeta}+1)-\lambda(\bar{\zeta })]
=\frac{V(\bar{\zeta},\zeta_0)}{2}
+\frac{e^{\psi_{-}(\zeta_0+\frac{1}{2},\bar{\zeta})}U(
\bar{\zeta},\zeta_0)}{2} \,. \label{S7E52}
\end{equation}
Then (\ref{S7E50})  implies
\begin{equation}
\frac{d}{d\bar{\zeta}}(\frac{d^{2}\Phi(\bar{\zeta})
}{d\bar{\zeta}^{2}}+\frac{\partial\psi_{+}(\zeta_0+\frac{1}{2}
,\bar{\zeta})}{\partial\bar{\zeta}}\frac{d\Phi(\bar{\zeta
})}{d\bar{\zeta}})=0. \label{S7E53}
\end{equation}

Note that the compatibility conditions \eqref{S7E29},
\eqref{S7E36}, \eqref{S7E38} yield
\[
\Phi(\zeta_0)=\Phi'(\zeta_0)=\Phi''( \zeta_0) =0\,.
\]
 Therefore, (\ref{S7E53}) and, under suitable regularity assumptions for
$\frac {\partial\psi_{+}(\zeta_0+\frac{1}{2},\bar{\zeta})}
{\partial\bar{\zeta}}$, classical uniqueness theory for ODEs would
imply $\Phi(\bar{\zeta})=0$ for the values of $\bar{\zeta}$ for
which (\ref{S7E52}) holds. Reciprocally, if $\lambda(\cdot) $
solves \eqref{S7E20}-\eqref{S7E22} in an interval
$(\zeta_0,\zeta_0 +\delta), $ $\Phi(\bar{\zeta})=0$ in such
interval, whence (\ref{S7E52}) would follow.

We have then reduced the problem of proving Theorem \ref{Th2} to
the solution of the equation (\ref{S7E52}), under suitable
regularity conditions for $\lambda(\cdot)$. Let us then precise
the suitable framework in which it is possible to solve
(\ref{S7E52}) as well as (\ref{S7E53}).
Let us choose an arbitrary function $\lambda(\zeta)$ for
$\zeta\in[\zeta_0,\zeta_0+1]$,
$\lambda( \cdot)\in C[\zeta_0,\zeta_0+1]$. Suppose that
$|\lambda(\zeta)|\leq\varepsilon_0$ for
$\zeta \in[ \zeta_0,\zeta_0+1]$.
We define a Banach space
$X_{\delta}=C[\zeta_0+1,\zeta_0+1+\delta]$
endowed with the $L^{\infty}$ norm
\[
\|  \lambda\|  _{X_{\delta}}\equiv\sup_{\zeta\in[ \zeta
_0+1,\zeta_0+1+\delta]}|\lambda(\zeta)|\,.
\]
We rewrite (\ref{S7E52}) as the fixed point problem:
\begin{equation}
\lambda(\bar{\zeta}+1)=T[\lambda]( \bar{\zeta})\,,
\quad \bar{\zeta}\in[\zeta_0,\zeta_0 +\delta], \label{S7E54}
\end{equation}
where
\begin{equation}
T[\lambda](\bar{\zeta})\equiv\lambda(
\bar{\zeta})+\frac{V(\bar{\zeta},\zeta_0)+e^{\psi
_{-}(\zeta_0+\frac{1}{2},\bar{\zeta})}U(\bar{\zeta },\zeta_0)}{2}
\label{S7E55}
\end{equation}
with $U(\bar{\zeta},\zeta_0)$, $V(\bar{\zeta} ,\zeta_0)$ as in
(\ref{S7E50a}), (\ref{S7E51a}) and $\psi_{-}( \zeta,\bar{\zeta})$
is as in (\ref{S7E45}). Note that
$\|V(\cdot,\zeta_0)\|_{L^{\infty}[\zeta_0
,\zeta_0+\delta]}\leq C(\varepsilon_0+\eta_{L}) $
where $\eta_{L}\to0$ as $L\to\infty$. A similar estimate
might be obtained for
$\|  U( \cdot,\zeta_0)\|_{L^{\infty}[\zeta_0,\zeta_0+\delta]}$.
To derive such estimate some care is needed with the term
$\frac{\partial}{\partial
\bar{\zeta}}(\frac{\partial\psi_{+}(\zeta_0+\frac{1}{2}
,\bar{\zeta})}{\partial\bar{\zeta}}G_{1}(\bar{\zeta},\zeta
_0)+\frac{\partial G_{1}( \bar{\zeta},\zeta_0)
}{\partial\bar{\zeta}})$ in (\ref{S7E50a}), and in particular the
term $\frac{\partial^{2}\psi_{+}( \zeta_0+\frac{1}{2},\bar{\zeta})
}{\partial\bar{\zeta}^{2}}G_{1}(\bar{\zeta},\zeta_0)$. The term
$\frac{\partial^{2}\psi_{+}( \zeta_0+\frac{1}{2},\bar{\zeta
})}{\partial\bar{\zeta}^{2}}$ contains a term proportional to
$\lambda(\bar{\zeta})$. However, $G_{1}(\bar{\zeta },\zeta_0)$
might be estimated by a small constant if $\delta>0$ is chosen
small enough. Therefore the operator $T$ transforms the ball
$\| \lambda(\cdot)\| _{X_{\delta}}\leq1$ in a ball
$\| T[\lambda](\cdot)\| _{X_{\delta}}\leq\nu$ where
$\nu$ is small if $L$ and $\zeta_0$ are large and $\delta>0$ is
small. Moreover, similar bounds show that
\[
\|  T[\lambda_{1}](\cdot)-T[\lambda_{2}] (\cdot)\|
_{X_{\delta}}\leq \theta\| \lambda_{1}(\cdot)-\lambda_{2}(
\cdot)\|  _{X_{\delta}}
\]
where $\theta$ is small if $L$ and $\zeta_0$ are large enough and
$\delta>0$ is small. A standard contractive fixed point argument
then shows that (\ref{S7E54}) (or equivalently (\ref{S7E52})) has
a unique solution for this range of values of $L,\zeta_0$,
$\delta$. Moreover, for these functions $\lambda\in X_{\delta}$,
$\frac{\partial^{2}\psi_{+}(\zeta_0+\frac
{1}{2},\bar{\zeta})}{\partial\bar{\zeta}^{2}}$ is a continuous
function, whence (\ref{S7E53}) implies that $\Phi( \bar{\zeta})=0$
for $\bar{\zeta}\in[\zeta_0,\zeta_0+\delta]$ and the Theorem
follows.
\end{proof}

\subsection{On the existence of functions $\lambda(\cdot)$
satisfying the compatibility conditions for $\zeta_0 $ large}

Note that a key assumption in Theorem \ref{Th2} is the existence
of a function $\lambda(\cdot)$ for which the compatibility
conditions \eqref{S7E29}, \eqref{S7E36}, \eqref{S7E38} hold for
$\zeta_0$ large. The existence of such functions $\lambda( \cdot)$
is not obvious at all. The purpose of this Section, is to show
that there is indeed a large class of functions $\lambda(\cdot)$
satisfying $|\lambda(\bar{\zeta}) |\leq\varepsilon_0$ as well as
\eqref{S7E29}, \eqref{S7E36}, \eqref{S7E38}.Several of the
formulae derived in this Subsection will be useful later proving
that the function $\lambda(\bar{\zeta})$ is globally defined for
$\bar{\zeta}\in[\zeta _0,\infty)$.

As a first step we rewrite the compatibility conditions obtained
in Subsection \ref{compat} in a more convenient manner.
Note that we can rewrite the compatibility condition
\eqref{S7E29} for $\lambda(\zeta)$ in $\zeta\in( \zeta_0,\zeta
_0+1)$ as
\begin{equation}
\int_{\zeta_0}^{\zeta_0+1}\big[\frac{\sin^{2}(\pi(
\zeta-\zeta_0-Z(\zeta,\zeta_0)))\lambda(\zeta)
}{\pi^{2}}+H(\zeta,\zeta_0) \big]d\zeta=0\; \label{S7E29N}
\end{equation}
with $Z(\zeta,\zeta_0)$ defined by means of \eqref{S7E20},
\eqref{S7E21} with $\bar{\zeta}=\zeta_0$, and where from now on:
\begin{equation}
H(\zeta,\zeta_0)\equiv\frac{\sin^{2}(\pi(
\zeta-\zeta_0-Z(\zeta,\zeta_0)))R( Z(\zeta,\zeta_0)
,\zeta,\zeta_0)}{\pi^{2}} \,.\label{S7E29aN}
\end{equation}
Differentiating $\Phi(\bar{\zeta})$ in (\ref{S7E28}), and choosing
$\bar{\zeta}=\zeta_0$ we obtain:
\begin{equation} \label{S7E30N}
\begin{aligned}
&  -\frac{1}{\pi}\int_{\zeta_0}^{\zeta_0+1}\sin(2\pi(
\zeta-\zeta_0-Z(\zeta,\zeta_0)))( 1+\frac{\partial
Z(\zeta,\zeta_0)}{\partial\zeta_0})
\lambda(\zeta)d\zeta\\
&  +\frac{d}{d\zeta_0}(\int_{\zeta_0}^{\zeta_0+1}H(
\zeta,\zeta_0)d\zeta)  =0
\end{aligned}
\end{equation}
We can simplify the above expression after computing
$\frac{\partial Z(\zeta,\zeta_0)}{\partial\zeta_0}$.
Differentiating \eqref{S7E20},
with respect to $\bar{\zeta}$, we obtain
\begin{equation}
(\frac{\partial Z}{\partial\bar{\zeta}})_{\zeta}
=-\frac{\sin(2\pi(\zeta-\bar{\zeta}-Z(\zeta,\bar{\zeta }) )
)\lambda(\zeta)}{\pi}(
1+\frac{\partial Z}{\partial\bar{\zeta}})
  +\frac{\partial H(\zeta,\bar{\zeta}) }{\partial\bar{\zeta}}\,.
\label{S7E31N}
\end{equation}
Integrating (\ref{S7E31N}), (\ref{S7E32}), we obtain
\begin{equation} \label{S7E33N}
\begin{aligned}
\frac{\partial Z}{\partial\bar{\zeta}}(\zeta,\bar{\zeta})
&=-\int_{\bar{\zeta}}^{\zeta}e^{-\bar{\psi}(
\zeta,\bar{\zeta})+\bar{\psi}(\eta,\bar{\zeta})
}\frac{\partial\bar{\psi}(
\eta,\bar{\zeta})}{\partial\zeta}d\eta\\
& \quad +\int_{\bar{\zeta}}^{\zeta}e^{-\bar{\psi}(\zeta,\bar{\zeta})
+\bar{\psi}(\eta,\bar{\zeta})}\frac{\partial H(\eta
,\bar{\zeta})}{\partial\bar{\zeta}}d\eta\\
&  =(e^{-\bar{\psi}(\zeta,\bar{\zeta})}-1)
+\int_{\bar{\zeta}}^{\zeta}e^{-\bar{\psi}(
\zeta,\bar{\zeta})+\bar{\psi}(\eta,\bar{\zeta}) }\frac{\partial
H(\eta ,\bar{\zeta})}{\partial\bar{\zeta}}d\eta,
\end{aligned}
\end{equation}
where
\begin{equation}
\bar{\psi}(\zeta,\bar{\zeta})\equiv\frac{1}{\pi}\int
_{\bar{\zeta}}^{\zeta}\sin(2\pi(\eta-\bar{\zeta}-Z(
\eta,\bar{\zeta})))\lambda( \eta)
d\eta\,. \label{S7E34N}
\end{equation}
We can rewrite (\ref{S7E33}) as
\begin{equation}
\frac{\partial Z}{\partial\bar{\zeta}}(\zeta,\bar{\zeta})
+1=e^{-\bar{\psi}(\zeta,\bar{\zeta})}\Big(1+\int_{\bar
{\zeta}}^{\zeta}e^{\bar{\psi}(\eta,\bar{\zeta}) }\frac{\partial
H(\eta,\bar{\zeta})}{\partial\bar{\zeta}}d\eta\Big)\,.
\label{S7E35N}
\end{equation}
Substituting (\ref{S7E35N}) with $\bar{\zeta}=\zeta_0$ into
(\ref{S7E30N}) and using also (\ref{S7E34N}), we obtain
\begin{align*}
&  -\int_{\zeta_0}^{\zeta_0+1}\frac{\partial\bar{\psi}(\zeta
,\zeta_0)}{\partial\zeta}e^{-\bar{\psi}( \zeta,\zeta
_0)}d\zeta \\
&  -\int_{\zeta_0}^{\zeta_0+1}\frac{\partial\bar{\psi}(\zeta
,\zeta_0)}{\partial\zeta}e^{-\bar{\psi}( \zeta,\zeta _0)
}\int_{\zeta_0}^{\zeta}e^{\bar{\psi}( \eta,\bar{\zeta })
}\frac{\partial H( \eta,\zeta_0)}{\partial\bar
{\zeta}}d\eta d\zeta
  +\frac{d}{d\zeta_0}\Big(\int_{\zeta_0}^{\zeta_0+1}H(
\zeta,\zeta_0)d\zeta\Big)\\
&  =0\,.
\end{align*}
After some integrations by parts we arrive to the following compatibility
condition
\begin{equation}
(e^{\bar{\psi}(\zeta_0+1,\zeta_0)}-1)
+\int_{\zeta_0}^{\zeta_0+1}e^{\bar{\psi}(
\zeta,\zeta_0)}\frac{\partial H( \zeta,\zeta_0)
}{\partial\zeta_0}d\zeta=0\,. \label{S7E36N}
\end{equation}

Actually (\ref{S7E36N}) holds for any $\bar{\zeta}\geq\zeta_0$ if
$\zeta _0$ is replaced by $\bar{\zeta}$. We obtain an additional
compatibility condition that must be satisfied by $\lambda(\cdot)$
differentiating the resulting equation with respect to
$\bar{\zeta}$ and particularizing the value $\bar{\zeta}=\zeta_0$.
Equivalently we can just differentiate with respect to $\zeta_0$
in (\ref{S7E36N}) to obtain
\begin{equation}
e^{\bar{\psi}(\zeta_0+1,\zeta_0)
}\frac{d\bar{\psi}(\zeta_0+1,\zeta_0)
}{d\zeta_0}+\frac{d}{d\zeta_0}\Big(
\int_{\zeta_0}^{\zeta_0+1}e^{\bar{\psi}( \zeta,\zeta_0)
}\frac{\partial H(\zeta,\zeta_0)}{\partial\zeta_0} d\zeta\Big)=0\,.
\label{S7E36aN}
\end{equation}
On the other hand, using (\ref{S7E34N}), as well as (\ref{S7E35N})
we arrive at
\begin{align*}
&  \frac{d\bar{\psi}(\zeta_0+1,\zeta_0)}{d\zeta_0}\\
&  =-2\int_{\zeta_0}^{\zeta_0+1}\cos(2\pi(\eta-\zeta
_0-Z(\eta,\zeta_0)))e^{-\bar{\psi}(
\eta,\zeta_0)}\lambda(\eta)d\eta-\\
&  -2\int_{\zeta_0}^{\zeta_0+1}\cos(2\pi(\eta-\zeta
_0-Z(\eta,\zeta_0)))\lambda(
\eta)e^{-\bar{\psi}(\eta,\zeta_0)}\int_{\bar{\zeta}
}^{\eta}e^{\bar{\psi}(\xi,\zeta_0)}\frac{\partial
H(\xi,\zeta_0)}{\partial\zeta_0}d\xi d\eta\,.
\end{align*}
Substituting this formula into \eqref{S7E36}, we obtain the compatibility
condition
\begin{equation} \label{S7E38N}
\begin{aligned}
& e^{\bar{\psi}(\zeta_0+1,\zeta_0)}\int_{\zeta_0}
^{\zeta_0+1}\cos(2\pi(\eta-\zeta_0-Z(\eta,\zeta _0) )
)e^{-\bar{\psi}(\eta,\zeta_0)
}\lambda(\eta)d\eta\\
&+ e^{\bar{\psi}(\zeta_0+1,\zeta_0)}\int_{\zeta_0}
^{\zeta_0+1}\cos(2\pi(\eta-\zeta_0-Z(\eta,\zeta _0) )
)\lambda(\eta)e^{-\bar{\psi }( \eta,\zeta_0)}\\
&\times \Big[
\int_{\zeta_0}^{\eta}e^{\bar{\psi }(\xi,\zeta_0) }\frac{\partial
H( \xi,\zeta_0)
}{\partial\zeta_0}d\xi\Big]d\eta\\
& =\frac{1}{2}\frac{d}{d\zeta_0}\Big(\int_{\zeta_0}^{\zeta_0
+1}e^{\bar{\psi}(\zeta,\zeta_0)}\frac{\partial H(
\zeta,\zeta_0)}{\partial\zeta_0}d\zeta\Big)
\end{aligned}
\end{equation}
Equations (\ref{S7E29N}), (\ref{S7E36N}) and (\ref{S7E38N}) are
just the reformulation of the compatibility conditions for
$\lambda( \cdot)$ that we wanted to obtain.

As indicated above, the main goal of this Subsection is to show
that the compatibility conditions (\ref{S7E29N}), (\ref{S7E36N})
and (\ref{S7E38N}) are satisfied by a large class of functions
$\lambda(\cdot)$ for any $\zeta_0$ large enough. A first technical
problem is the following. Note that for $\zeta_0$ large enough,
the function $R(Z,\zeta ,\zeta_0)$ is not necessarily small.
Indeed, $R(Z,\zeta ,\zeta_0)$ is defined in (\ref{S7E15}),
(\ref{S7E22a}) and since $h_{1},h_{2}$ are just bounded functions
but $(f'(\zeta)) ^{2}\to\infty$ as $\zeta\to\infty$ it follows
that $R(Z,\zeta,\zeta_0)$ becomes large for some large values of
$Z$. In particular, due to this growth it is not obvious at all if
$Z(\zeta,\zeta_0) $ is a bounded function for $\zeta_0\to\infty$
even if $| \lambda|$ is assumed to be small. Moreover, for the
same reason it is not ''a priori'' obvious if the function
$H(\zeta,\zeta_0)$ defined in (\ref{S7E29aN}) is small for
$\zeta_0\to\infty$. The following Lemma shows that this is
actually the case:

\begin{lemma} \label{L7}
Suppose that $Z(\zeta;\bar{\zeta})$ solves
\eqref{S7E20}-\eqref{S7E22}. Let us assume also that
$|\lambda(\zeta)|\leq\varepsilon_0$ for
$\zeta\in[ \bar{\zeta },\bar{\zeta}+1]$. There exist $C>0$
and $\hat{\zeta}$ large such
that for any $\bar{\zeta}\geq\hat{\zeta}$ such that
\begin{equation}
|Z(\zeta;\bar{\zeta})|\leq C\varepsilon_0 \min\{
|\zeta-\bar{\zeta}|,|\zeta-\bar{\zeta }-1|\}
\,,\quad \text{for }\zeta\in[\bar{\zeta} ,\bar{\zeta}+1]\,.
\label{S7E56a}
\end{equation}
Moreover, for $\bar{\zeta}\geq\hat{\zeta}$, with $\hat{\zeta}$
large, the following inequality holds:
\begin{equation}
|H(\zeta,\bar{\zeta})|\leq\frac{C}{( L(\tau)
)^{3}}\,,\quad \text{for }\zeta\in[
\bar{\zeta},\bar{\zeta}+1]\,. \label{S7E57}
\end{equation}
\end{lemma}

\begin{proof}
We use a classical continuity argument. By assumption
(\ref{S7E56a}) holds for $\zeta=\bar{\zeta}$. As long as
\begin{equation}
|Z(\zeta;\bar{\zeta})|\leq\frac{1}{4} \min\left\{ |
\zeta-\bar{\zeta}|,|\zeta-\bar{\zeta }-1|\right\} \; \label{S7E58}
\end{equation}
we have that (\ref{S7E57}) holds true, due to (\ref{S7E22a}) as
well as the fact that $f'(\zeta)$ is approximately constant in
intervals of the form $[
\bar{\zeta},\bar{\zeta}+\frac{C}{f'( \bar{\zeta}) }]$.
Integrating the equation \eqref{S7E20} we recover (\ref{S7E58})
whence Lemma \ref{L7} follows.
\end{proof}

We need to rewrite the compatibility conditions (\ref{S7E36N}), (\ref{S7E38N})
in a more convenient form. We have
\begin{equation}
(e^{\bar{\psi}(\zeta_0+1,\zeta_0)}-1) +J(\zeta_0+1,\zeta_0)=0\,,
\label{S7E59}
\end{equation}
\begin{equation} \label{S7E60}
\begin{aligned}
&  \int_{\zeta_0}^{\zeta_0+1}\cos(2\pi(\eta-\zeta _0-Z(
\eta,\zeta_0)))e^{-\bar{\psi}(
\eta,\zeta_0)}\lambda(\eta)d\eta+\\
&  \int_{\zeta_0}^{\zeta_0+1}\cos(2\pi(\eta-\zeta _0-Z(
\eta,\zeta_0)))\lambda(\eta) e^{-\bar{\psi}(\eta,\zeta_0)}J(\eta
,\zeta_0)d\eta\\
&  =\frac{e^{-\bar{\psi}(\zeta_0+1,\zeta_0)}}{2}\frac
{d}{d\zeta_0}(J(\zeta_0+1,\zeta_0)),
\end{aligned}
\end{equation}
where
\begin{equation} \label{S7E61}
\begin{aligned}
J(\zeta,\zeta_0)&  \equiv\frac{d}{d\zeta_0}(
\int_{\zeta_0}^{\zeta}e^{\bar{\psi}(\eta,\zeta_0)
}H(\eta,\zeta_0)d\eta) -\int_{\zeta_0}^{\zeta}e^{\bar{\psi }(
\eta,\zeta_0)}\frac{\partial\bar{\psi}(\eta ,\zeta_0)
}{\partial\zeta_0}H(\eta,\zeta_0)
d\eta\\
&\equiv  \frac{d}{d\zeta_0}(F_{1}(\zeta,\zeta_0))
-F_{2}(\zeta,\zeta_0)
\end{aligned}
\end{equation}
A crucial step in all the forthcoming arguments is to derive
better estimates for $Z(\zeta,\zeta_0)$ and its derivatives as
those in Lemma \ref{L7}.

\begin{lemma} \label{L8}
Under the assumptions of Lemma \ref{L7} the following estimates
hold
\begin{gather}
|\frac{\partial^{k}Z(\zeta,\bar{\zeta})}{\partial
\bar{\zeta}^{k}}|\leq C\big[\varepsilon_0|\zeta-\bar {\zeta}|
^{3-k}+\frac{1}{( f'( \bar{\zeta}))^{\beta}}\big]\,,
\text{ for }\bar{\zeta}\leq\zeta\leq
\bar{\zeta}+\frac{1}{2}\label{S7E62a}\\
|\frac{\partial^{k}Z(\zeta,\bar{\zeta})}{\partial
\bar{\zeta}^{k}}|\leq C\big[\varepsilon_0|\zeta-\bar {\zeta}-1|
^{3-k}+\frac{1}{( f'( \bar{\zeta }+1))^{\beta}}\big]\,,
\text{ for }\bar{\zeta}+\frac {1}{2}\leq\zeta\leq\bar{\zeta}
+1 \label{S7E63}
\end{gather}
where $k=0,1,2,3$ and $\bar{\zeta}$ is large enough, and where
$\beta \in(0,1)$ might be chosen arbitrarily close to one.
\end{lemma}

\begin{proof}
Let us sketch the main argument in the proof of this result. The equation
\eqref{S7E20} might be rewritten in the form:
\begin{equation}
\frac{\partial Z}{\partial\zeta}=H(\zeta-\bar{\zeta}-Z)
\lambda(\zeta)+W(\zeta,\bar{\zeta},Z)
\label{S7E64}
\end{equation}
where $H(x)=\frac{\sin^{2}(\pi x)}{\pi^{2}}$. Due to the
\eqref{S7E20}, (\ref{S7E22a}) we have that
$W(\zeta,\bar{\zeta},Z)$ is a smooth function that has approximately
the form
\[
W(\zeta,\bar{\zeta},Z)=\Phi( f'(\zeta)(\zeta-\bar{\zeta}-Z)),
\]
 with $\Phi(x)=0$ for $0\leq x\leq L$ and
$|\Phi( x)|\leq\frac{C}{x}$ globally on $x$. Due to Lemma \ref{L7}
it follows that $Z$ is small and to the leading order can be
neglected. With this assumption it would be possible to
approximate $Z$ solution of \eqref{S7E21}, (\ref{S7E64}) as
\begin{equation}
Z(\zeta,\bar{\zeta})\approx Z_{1}(\zeta,\bar{\zeta
})+Z_{2}(\zeta,\bar{\zeta})\equiv\int_{\bar{\zeta}
}^{\zeta}H(\eta-\bar{\zeta})\lambda(\eta)
d\eta+\int_{\bar{\zeta}}^{\zeta}\Phi(f'(\eta)
(\eta-\bar{\zeta}))d\eta \,.\label{S7E65}
\end{equation}
Note that the function $Z_{2}(\zeta,\bar{\zeta})$ is small away
from a boundary layer close to $\zeta\approx\bar{\zeta}$.
Therefore, the results in Appendix A imply that in that region
$Z_{2}(\zeta,\bar {\zeta})$ might be approximated as
\[
Z_{2}(\zeta,\bar{\zeta})\approx\int_{\bar{\zeta}}^{\zeta}
\Phi(f'(\bar{\zeta})(\eta-\bar{\zeta
}))d\eta=\frac{1}{f'(\bar{\zeta})}
\int_0^{f'(\bar{\zeta})(\zeta-\bar{\zeta }) }\Phi(x)dx\,;
\]
i.e. $Z_{2}(\zeta,\bar{\zeta})$ is roughly of order
$\frac{1}{f'(\bar{\zeta})}$ in that boundary layer. On the other hand
$Z_{1}(\zeta,\bar{\zeta})$ is roughly of order
$\frac{\lambda(\bar{\zeta})}{3}( \zeta-\bar{\zeta })^{3}$ for
$\zeta\approx\bar{\zeta}$. Then, $Z_{2}(\zeta ,\bar{\zeta}) $ is
the leading term for $\zeta\approx\bar{\zeta}$ and
$Z_{1}(\zeta,\bar{\zeta})$ becomes the leading one for
$|\zeta-\bar{\zeta}|$ of order one. Note that
$Z_{2}(\zeta ,\bar{\zeta})$ is smooth due to the smoothness of $f$. In
particular its derivatives with respect to $\bar{\zeta}$ might be
estimated as $1/(f'(\bar{\zeta}))^{\beta}$ for
$\zeta\approx\bar{\zeta}$. On the contrary only three derivatives
of $Z_{1}(\zeta,\bar{\zeta})$ are bounded if the only assumption
made on $\lambda$ is boundedness. Therefore the decomposition
(\ref{S7E65}) would imply the estimate (\ref{S7E62a}). A similar
argument in the region $\zeta\approx\bar{\zeta}+1$ would imply
(\ref{S7E63}).

To illustrate how to make the argument above rigorous we
derive (\ref{S7E62a}) for $k=1$. Note that (\ref{S7E62a}),
(\ref{S7E63}) for $k=0$ are just a consequence of (\ref{S7E56a})
and the formula
\[
Z(\zeta,\bar{\zeta})=\int_{\bar{\zeta}}^{\zeta}H(
\eta-\bar{\zeta}-Z(\eta,\bar{\zeta}))\lambda( \eta)
d\eta+\int_{\bar{\zeta}}^{\zeta}W( \eta,\bar{\zeta
},Z(\eta,\bar{\zeta}))d\eta\,.
\]
The integral terms can then be estimated basically as the terms
$Z_{1}( \zeta,\bar{\zeta})$, $Z_{2}( \zeta,\bar{\zeta})$ above.
To prove (\ref{S7E62a}) we differentiate (\ref{S7E64}) with respect
to $\bar{\zeta}$, whence
\begin{align*}
\frac{\partial}{\partial\zeta}(\frac{\partial
Z}{\partial\bar{\zeta} })
&=\big[-H_{Z}(\zeta-\bar{\zeta}-Z)\lambda(\zeta) +\frac{\partial W(
\zeta,\bar{\zeta},Z)}{\partial Z}\big]
\frac{\partial Z}{\partial\bar{\zeta}}\\
&\quad +\big[-H_{\bar{\zeta}}(\zeta-\bar{\zeta}-Z)
\lambda(\zeta)+W_{\bar{\zeta}}( \zeta,\bar{\zeta},Z)\big]\,.
\end{align*}
Differentiating \eqref{S7E21} we obtain
$\frac{\partial Z}{\partial\bar{\zeta }}(\bar{\zeta},\bar{\zeta})=0$.
Then
\begin{equation}
\begin{aligned}
\frac{\partial Z}{\partial\bar{\zeta}}(\zeta,\bar{\zeta})
&=\int_{\bar{\zeta}}^{\zeta}e^{\int_{\bar{\zeta}}^{\eta}
\big[-H_{Z}(\xi-\bar{\zeta}-Z)\lambda(\xi)+\frac{\partial W(
\xi,\bar{\zeta},Z)}{\partial Z}\big]d\xi} \\
&\quad\times\Big[ -H_{\bar{\zeta}
}(\eta-\bar{\zeta}-Z)\lambda(\eta)+W_{\bar
{\zeta}}(\eta,\bar{\zeta},Z)\Big]d\eta\,.
\end{aligned}\label{S7E66}
\end{equation}
The exponential factor containing $H_{Z}$ might be estimated by
means of a constant using (\ref{S7E56a}). We then need to estimate
the term $\exp\big(\int_{\bar{\zeta}}^{\eta}\frac{\partial
W(\xi,\bar{\zeta},Z)}{\partial Z}d\xi\big)$. To this end we remark
that this term can be estimated as
$e^{\int_{\bar{\zeta}}^{\eta}f'(\zeta)|
\Phi'(f'(\zeta)(\zeta-\bar{\zeta }-Z) ) |d\xi}$ and this can be
estimated also by means of a constant (actually close to one if
$L$, $\bar{\zeta}$ are large). After estimating the exponential
factors in this manner the terms left in (\ref{S7E66}) can be
estimated as the derivatives of the functions $Z_{1}(
\zeta,\bar{\zeta}) ,\;Z_{2}(\zeta,\bar{\zeta })$ above. This
yields (\ref{S7E62a}) with $k=1$.

Higher order derivatives can be estimated in an analogous manner.
The main difference arises for $k=3$, because in that case
$\frac{\partial^{3}
Z}{\partial\bar{\zeta}^{3}}(\bar{\zeta},\bar{\zeta})
=-2\lambda(\bar{\zeta})$. In particular this yields a global term
$C\varepsilon_0$. Moreover, due to this higher derivatives cannot
be estimated in this manner unless additional regularity for
$\lambda(\cdot)$ is assumed.
\end{proof}

To show that there exist functions $\lambda(\cdot) $
satisfying (\ref{S7E29N}), (\ref{S7E59}), (\ref{S7E61}) for
$\zeta_0$ large we need to obtain estimates for $J(
\zeta,\zeta_0)$and some of its derivatives, or equivalently
$F_{1}(\zeta,\zeta_0)$, $F_{2}( \zeta,\zeta_0)$ and their
derivatives. To this end, we write
\begin{gather*}
F_{1}(\zeta,\zeta_0) =F_{1,1}(\zeta,\zeta _0)
+F_{1,2}(\zeta,\zeta_0)+F_{1,3}(
\zeta,\zeta_0)\,,\\
F_{2}(\zeta,\zeta_0) =F_{2,1}(\zeta,\zeta _0)
+F_{2,2}(\zeta,\zeta_0)+F_{2,3}( \zeta,\zeta_0),
\end{gather*}
where
\begin{gather*}
F_{1,1}(\zeta,\zeta_0)  \equiv\int_{\zeta_0} ^{\min\{
\zeta,\zeta_0+\frac{1}{(f(\zeta_0) ) ^{\alpha}}\}
}e^{\bar{\psi}( \eta,\zeta_0)
}H(\eta,\zeta_0)d\eta\,,\\
F_{1,2}(\zeta,\zeta_0) \equiv\int_{\min\{
\zeta,\zeta_0+\frac{1}{(f(\zeta_0))^{\alpha }}\}
}^{\min\{ \zeta,\zeta_0+1-\frac{1}{( f( \zeta_0+1)
)^{\alpha}}\} }e^{\bar{\psi}(
\eta,\zeta_0)}H(\eta,\zeta_0)d\eta\,,\\
F_{1,3}(\zeta,\zeta_0)  \equiv\int_{\min\{
\zeta,\zeta_0+1-\frac{1}{(f(\zeta_0+1)) ^{\alpha}}\}
}^{\min\{ \zeta,\zeta_0+1\} }e^{\bar{\psi
}(\eta,\zeta_0)}H( \eta,\zeta_0)d\eta\,,
\end{gather*}

\begin{gather*}
F_{2,1}(\zeta,\zeta_0) \equiv\int_{\zeta_0} ^{\min\{
\zeta,\zeta_0+\frac{1}{(f(\zeta_0) ) ^{\alpha}}\}
}e^{\bar{\psi}( \eta,\zeta_0)
}\frac{\partial\bar{\psi}(\eta,\zeta_0)}{\partial\zeta_0
}H(\eta,\zeta_0)d\eta\,,\\
F_{2,2}(\zeta,\zeta_0) \equiv\int_{\min\{
\zeta,\zeta_0+\frac{1}{(f(\zeta_0))^{\alpha }}\}
}^{\min\{ \zeta,\zeta_0+1-\frac{1}{( f( \zeta_0+1)
)^{\alpha}}\} }e^{\bar{\psi}(\eta,\zeta_0)
}\frac{\partial\bar{\psi}(\eta,\zeta_0)
}{\partial\zeta_0}H(\eta,\zeta_0)d\eta\,,\\
F_{2,3}(\zeta,\zeta_0) \equiv\int_{\min\{
\zeta,\zeta_0+\frac{1}{(f(\zeta_0))^{\alpha }}\}
}^{\min\{ \zeta,\zeta_0+1-\frac{1}{( f( \zeta_0+1)
)^{\alpha}}\} }e^{\bar{\psi}(\eta,\zeta_0)
}\frac{\partial\bar{\psi}(\eta,\zeta_0)
}{\partial\zeta_0}H(\eta,\zeta_0)d\eta\,.
\end{gather*}
and where $\alpha\in(0,1)\;$might be chosen arbitrarily close to
one.

The terms $F_{1,2}(\zeta,\zeta_0)$ and $F_{2,2}( \zeta,\zeta_0)$
might be easily estimated if $\zeta_0$ is large enough.

\begin{lemma}
\label{L9}Suppose that the assumptions of Lemma \ref{L7} are satisfied.
 Then
\begin{equation}
\big|\frac{d^{k}}{d\zeta_0^{k}}(F_{1,2}(\zeta,\zeta _0))\big|
+\big|\frac{d^{j}}{d\zeta_0^{j}}(F_{2,2}(\zeta,\zeta_0) ) \big|
\leq\frac{C}{( f(\zeta_0))^{\beta}} \label{S7E62}
\end{equation}
for $\zeta_0$ large enough, $\zeta_0\leq\zeta\leq\zeta_0+1$, where
$k=0,1,2,3$, $j=0,1,2$, and $\beta>0$ might be chosen arbitrarily
close to one.
\end{lemma}

\begin{proof}
This result is just a consequence of Lemma \ref{L8}, the
definitions of $F_{1,2}(\zeta,\zeta_0)$, $F_{2,2}( \zeta,\zeta _0)$
and the definitions of
$H( \zeta,\zeta_0)$, $\bar{\psi}(\zeta,\zeta_0)$ (cf. (\ref{S7E29aN}),
(\ref{S7E34N})).
\end{proof}

In other words, the integration in regions away from the
boundaries $\zeta=\zeta_0$, $\zeta=\zeta_0+1$ yields a negligible
contribution into $F_{1}(\zeta,\zeta_0)$, $F_{2}(\zeta,\zeta_0)$.
On the other hand in order to estimate the contributions to these
functions due to the regions close to the boundaries
$\zeta=\zeta_0$, $\zeta=\zeta _0+1$ we need some additional bounds
for $Z(\zeta,\bar{\zeta})$ and its derivatives.

\begin{lemma}
\label{L10} Under the assumptions of Lemma \ref{L7} the following estimates
hold
\begin{gather}
|\frac{d^{k}}{d\zeta_0^{k}}(F_{1,1}(\zeta,\zeta _0))|
+|\frac{d^{j}}{d\zeta_0^{j}}(F_{2,1}(\zeta,\zeta_0) ) |
\leq\frac{C}{(
f(\zeta_0))^{\beta}}\label{S7E62aa}\\
|\frac{d^{k}}{d\zeta_0^{k}}(F_{1,3}(\zeta,\zeta _0))|
+|\frac{d^{j}}{d\zeta_0^{j}}(F_{2,3}(\zeta,\zeta_0) ) |
\leq\frac{C}{( f(\zeta_0+1))^{\beta}} \label{S7E62b}
\end{gather}
for $\zeta_0$ large enough, $\zeta_0\leq\zeta\leq\zeta_0+1$, where
$k=0,1,2,\;j=0,1$ and $\beta>0$, might be chosen arbitrarily close
to one.
\end{lemma}

\begin{proof}
We rewrite
\begin{align*}
F_{1,1}(\zeta,\zeta_0)&\equiv\int_0^{\min\{
\zeta,\zeta_0+\frac{1}{(f(\zeta_0))^{\alpha
}}\}  -\zeta_0}H(\xi+\zeta_0,\zeta_0)d\xi\\
&\quad +\int_{\zeta_0}^{\min\{  \zeta,\zeta_0+\frac{1}{(f(
\zeta_0))^{\alpha}}\}  }[e^{\bar{\psi}(
\eta,\zeta_0)}-1]H(\eta,\zeta_0)d\eta\\
&\equiv F_{1,1,1}(\zeta,\zeta_0)+F_{1,1,2}(\zeta ,\zeta_0)\,.
\end{align*}
Since
\[
H(\xi+\zeta_0,\zeta_0)\equiv\frac{\sin^{2}( \pi(
\xi-Z(\zeta_0+\xi,\zeta_0)))R( Z(
\zeta_0+\xi,\zeta_0),\zeta_0+\xi,\zeta _0)}{\pi^{2}},
\]
 using Lemma \ref{L8} and  \eqref{S7E20}, we obtain
\[
\big|\frac{\partial^{k}(H(\xi+\zeta_0,\zeta_0) )
}{\partial\zeta_0^{k}}\big|\leq C\Big( \frac{1}{|\xi|
(f(\zeta_0))^{\beta}}\Big)^{k}
\]
for $k=0,1,2$, $0\leq\xi\leq\frac{1}{2}$. Then, using
(\ref{S7E22a}) we arrive at
\[
|\frac{\partial^{k}(F_{1,1,1}(\zeta,\zeta_0) )
}{\partial\zeta_0^{k}}|\leq C\frac{( f( \zeta_0)
)^{(1-\beta)k}}{(f( \zeta_0))^{\alpha}}
\]
for $k=0,1,2$, $0\leq\xi\leq\frac{1}{2}$.
On the other hand since $| e^{\bar{\psi}(\eta,\zeta_0)}-1| \leq
C\varepsilon_0|\zeta-\zeta_0|^{2}$ and using the fact that each
derivative of $H$ yields a contribution of order $f'(\zeta_0)$ we
obtain
\[
|\frac{\partial^{k}(F_{1,1,2}(\zeta,\zeta_0) )
}{\partial\zeta_0^{k}}|\leq\frac{C}{( f( \zeta_0))^{\alpha}}
\]
for $k=0,1,2$, $0\leq\xi\leq\frac{1}{2}$. This yields
\begin{equation}
|\frac{d^{k}}{d\zeta_0^{k}}(F_{1,1}(\zeta,\zeta _0))|
\leq\frac{C}{(f( \zeta_0) )^{\beta}} \label{S7E63a}
\end{equation}
for a new value of $\beta$ close to one.

On the other hand, in order to estimate $F_{2,1}(\zeta,\zeta _0)$
we use the fact that $|\bar{\psi}( \zeta ,\bar{\zeta})|\leq
C\varepsilon_0|\zeta-\zeta _0|$, whence, since each derivative of
$H$ yields a new multiplicative factor $f'( \zeta_0)$ we obtain
(\ref{S7E62aa}), using also (\ref{S7E63a}).
The proof of (\ref{S7E62b}) is similar.
\end{proof}

We can now prove the main result of this Subsection that shows
that it is possible to choose functions $\lambda(\cdot)$
satisfying the compatibility conditions (\ref{S7E29N}),
(\ref{S7E59}), (\ref{S7E60}) for $\zeta_0$ large enough in
infinite different manners.

\begin{proposition} \label{P2}
Suppose that $\lambda(\cdot)$ has the form
\begin{equation}
\lambda(\zeta)=\alpha_0+\alpha_{1}\cos(2\pi(
\zeta-\zeta_0))+\beta_{1}\sin(2\pi(\zeta -\zeta_0) )
+\tilde{\lambda}( \zeta-\zeta_0) \label{S7E64a}
\end{equation}
where:
\begin{gather*}
\int_{\zeta_0}^{\zeta_0+1}e^{2\pi\ell(\zeta-\zeta_0)
}\tilde{\lambda}(\zeta-\zeta_0) d\zeta=0\,,\quad \ell=0,\pm1,
\\
|\tilde{\lambda}(\zeta-\zeta_0)|\leq
\varepsilon_0\,,\quad \zeta\in[ \zeta_0,\zeta_0+1]\,.
\end{gather*}
Suppose that $L>0$ is large enough. Then, for any $\zeta_0$ large
enough there exist constants $\alpha_0$, $\alpha_{1}$, $\beta_{1}$
such that the function $\lambda(\cdot)$ in (\ref{S7E64a})
satisfies the compatibility conditions (\ref{S7E29N}),
(\ref{S7E59}), (\ref{S7E60}), as well as an estimate of the form
\[
|\lambda(\zeta)|\leq C\varepsilon_0
\,,\quad \zeta\in[\zeta_0,\zeta_0+1]\,.
\]
\end{proposition}

\begin{proof}
Formally linearizing the compatibility conditions (\ref{S7E29N}),
(\ref{S7E59}), (\ref{S7E60}), we obtain
\begin{gather*}
\int_{\zeta_0}^{\zeta_0+1}\sin^{2}(\pi(\zeta-\zeta
_0))\lambda(\zeta)d\zeta=-\int_{\zeta_0
}^{\zeta_0+1}H(\zeta,\zeta_0)d\zeta\equiv f_{1}(
\zeta_0)\,,\\
\int_{\zeta_0}^{\zeta_0+1}\sin(2\pi( \eta-\zeta_0) )
\lambda(\eta)d\eta=-J(\zeta_0+1,\zeta
_0)\equiv f_{2}(\zeta_0)\,,\\
\begin{aligned}
&\int_{\zeta_0}^{\zeta_0+1}\cos(2\pi( \eta-\zeta_0)
)\lambda(\eta)d\eta\\
&=-\int_{\zeta_0}^{\zeta_0+1}
\cos(2\pi(\eta-\zeta_0-Z(\eta,\zeta_0))) \lambda(\eta)
e^{-\bar{\psi}(\eta
,\zeta_0)}J(\eta,\zeta_0)d\eta\\
&\quad -\frac{e^{-\bar{\psi}(\zeta_0+1,\zeta_0)}}{2}\frac
{d}{d\zeta_0}(J(\zeta_0+1,\zeta_0))
\equiv f_{3}(\zeta_0)\,.
\end{aligned}
\end{gather*}
Using (\ref{S7E64a}) it follows that these equations can be rewritten as
\begin{equation}
\frac{\alpha_0}{2}=f_{1}(\zeta_0)\,,\quad
\frac{\alpha_{1}}{2}=f_{2}(\zeta_0)\,,\quad
\frac{\beta_{1}}{2}=f_{3}(\zeta_0)\,.
\label{S7E65a}
\end{equation}
Note that, due to the definition of $J(\zeta,\zeta_0)$ as well
as Lemmas \ref{L9}, \ref{L10} the functions $f_{\ell}(\zeta _0)$,
$\ell=1,2,3$ can be made arbitrarily small if $L$is large and
$\zeta_0$ is large enough, assuming that $| \lambda(\zeta)|
\leq2\varepsilon_0$ for $\zeta\in[ \zeta _0,\zeta_0+1]$. In
particular the left-hand sides of (\ref{S7E65a}) is larger than
the right-hand sides if $\sqrt{(\alpha_0)^{2}+(\alpha_{1})^{2}+(
\beta _{1}) ^{2}}=4\varepsilon_0$. Therefore, Proposition \ref{P2}
follows just using standard Degree Theory (cf. \cite{DT})
\end{proof}

\section{Analysis of the transition problem: Global well
posedness}\label{noncomp}

\subsection{Reducing the Transition Problem to a delay equation}

In the previous Sections we have proved that the Transition
Problem \eqref{S7E20}-\eqref{S7E22} might be solved, in infinite
different ways, in intervals $[ \zeta_0,\zeta_0+1+\delta]$ with
$\delta>0$ small. To conclude the proof of Theorem \ref{thm1} it
only remains to show that the solution of
\eqref{S7E20}-\eqref{S7E22} can be extended for arbitrarily large
values of $\zeta$.

Note that as long as $\lambda(\cdot)$ solves (\ref{S7E20}
)-\eqref{S7E22} the compatibility conditions (\ref{S7E29N}),
(\ref{S7E59}), (\ref{S7E60}) are satisfied with $\bar{\zeta}$
replacing $\zeta_0$. We need to rewrite (\ref{S7E52}) in a more
convenient manner. To this end we first rewrite (\ref{S7E60}) as
\begin{equation}
\int_{\bar{\zeta}}^{\bar{\zeta}+1}\cos(2\pi(\eta-\bar{\zeta
}-Z(\eta,\bar{\zeta})))e^{-\bar{\psi}( \eta,\bar{\zeta})
}\lambda(\eta)d\eta=Q(
\bar{\zeta}),\label{S8E1}
\end{equation}
where
\begin{align*}
Q(\zeta_0)&  \equiv\frac{e^{-\bar{\psi}(\zeta
_0+1,\zeta_0)}}{2}\frac{d}{d\zeta_0}(J(\zeta
_0+1,\zeta_0))\\
&\quad  -\int_{\zeta_0}^{\zeta_0+1}\cos(2\pi(\eta-\zeta
_0-Z(\eta,\zeta_0)))\lambda( \eta)
e^{-\bar{\psi}(\eta,\zeta_0)}J(\eta ,\zeta_0)d\eta\,.
\end{align*}
Differentiating (\ref{S8E1}), we obtain
\begin{align*}
&  e^{-\bar{\psi}(\bar{\zeta}+1,\bar{\zeta})}\lambda(
\bar{\zeta}+1)-\lambda(\bar{\zeta})\\
&+  2\pi\int_{\bar{\zeta}}^{\bar{\zeta}+1}\sin(2\pi(\eta
-\bar{\zeta}-Z(\eta,\bar{\zeta})))( 1+\frac{\partial
Z(\eta,\bar{\zeta})}{\partial\bar{\zeta}
})e^{-\bar{\psi}(\eta,\bar{\zeta})}\lambda(
\eta)d\eta\\
&-  \int_{\bar{\zeta}}^{\bar{\zeta}+1}\cos(2\pi( \eta-\bar{\zeta
}-Z(\eta,\bar{\zeta})))e^{-\bar{\psi}( \eta,\bar{\zeta})
}\frac{\partial\bar{\psi}( \eta,\bar{\zeta
})}{\partial\bar{\zeta}}\lambda(\eta)d\eta\\
&  =\frac{dQ(\bar{\zeta})}{d\bar{\zeta}}\,.
\end{align*}
Using (\ref{S7E35N}), this formula becomes
\[
e^{-\bar{\psi}(\bar{\zeta}+1,\bar{\zeta})}\lambda(
\bar{\zeta}+1)-\lambda(\bar{\zeta})+\int_{\bar{\zeta}
}^{\bar{\zeta}+1}K(\eta,\bar{\zeta})\lambda( \eta)
d\eta=W_{1}(\bar{\zeta})\,,
\]
where
\begin{align*}
  W_{1}(\bar{\zeta})
&  \equiv-2\pi\int_{\bar{\zeta}}^{\bar{\zeta}+1}\sin(2\pi(
\eta-\bar{\zeta}-Z(\eta,\bar{\zeta})))(
\int_{\bar{\zeta}}^{\eta}e^{\bar{\psi}(\xi,\bar{\zeta})}
\frac{\partial H(\xi,\bar{\zeta})}{\partial\bar{\zeta}}
d\xi)e^{-2\bar{\psi}(\eta,\bar{\zeta})}\lambda(
\eta)d\eta\\
&\quad  +\frac{dQ(\bar{\zeta})}{d\bar{\zeta}},
\end{align*}
\begin{align*}
K(\eta,\bar{\zeta})&  \equiv\Big[2\pi\sin( 2\pi(
\eta-\bar{\zeta}-Z(\eta,\bar{\zeta}))
)e^{-2\bar{\psi}(\eta,\bar{\zeta})} \\
& \quad +  \cos(2\pi(\eta-\bar{\zeta}-Z(\eta,\bar{\zeta })
))e^{-\bar{\psi}(\eta,\bar{\zeta})
}\frac{\partial\bar{\psi}(\eta,\bar{\zeta})}{\partial\bar
{\zeta}}\Big]
\end{align*}
On the other hand, using (\ref{S7E59}), we  obtain
\begin{equation}
\lambda(\bar{\zeta}+1)-\lambda(\bar{\zeta})
+\int_{\bar{\zeta}}^{\bar{\zeta}+1}K(\eta,\bar{\zeta})
\lambda(\eta)d\eta=W_{1}(\bar{\zeta})
+W_{2}(\bar{\zeta}),\label{S8E2}
\end{equation}
with
\[
W_{2}(\bar{\zeta})\equiv-e^{-\bar{\psi}(\bar{\zeta
}+1,\bar{\zeta})}J(\bar{\zeta}+1,\bar{\zeta})
\]
Equation (\ref{S8E2}) is well suited to prove global existence for
the solutions of the problem \eqref{S7E20}-\eqref{S7E22}. It turns
out that the right hand side of (\ref{S8E2}) converges to zero as
$\bar{\zeta} \to\infty$. On the other hand, the solution
obtained for the linearized transition problem in Subsection
\ref{subsectsim} shows that for $R=0$ the function $\lambda$ would
be periodic with period one. Since the right-hand side of
(\ref{S8E2}) vanishes for $R=0$, the integral term on the
left-hand side should vanish too. To see this more
clearly we rewrite the integral term as
\begin{align*}
&\int_{\bar{\zeta}}^{\bar{\zeta}+1}K(\eta,\bar{\zeta})
\lambda(\eta)d\eta \\
&  =\int_{\bar{\zeta}}^{\bar{\zeta} +1}\Big[
2\pi\sin(2\pi(\eta-\bar{\zeta}-Z(\eta
,\bar{\zeta})))e^{-2\bar{\psi}( \eta,\bar{\zeta
})} \\
&\quad  +  \cos(2\pi(\eta-\bar{\zeta}-Z(\eta,\bar{\zeta })
))e^{-\bar{\psi}(\eta,\bar{\zeta})
}\frac{\partial\bar{\psi}(\eta,\bar{\zeta})}{\partial\bar
{\zeta}}\Big]\lambda(\eta)d\eta\\
& =2\pi^{2}\int_{\bar{\zeta}}^{\bar{\zeta}+1}\frac{\partial\bar{\psi}(
\eta,\bar{\zeta})}{\partial\zeta}e^{-2\bar{\psi}( \eta
,\bar{\zeta})}d\eta
  -\frac{1}{\pi}\int_{\bar{\zeta}}^{\bar{\zeta}+1}d\eta\lambda(
\eta)\cos(2\pi(\eta-\bar{\zeta}-Z(\eta,\bar {\zeta})) ) \\
&\quad \times
e^{-\bar{\psi}(\eta,\bar{\zeta})}
\int_{\bar{\zeta}}^{\eta}\cos(2\pi( \xi-\bar{\zeta}-Z(
\xi,\bar{\zeta})))\big(1+\frac{\partial Z}
{\partial\bar{\zeta}}(\xi,\bar{\zeta})\big)\lambda(\xi)d\xi\,.
\end{align*}
Using (\ref{S7E35N}), we can write
\begin{align*}
\int_{\bar{\zeta}}^{\bar{\zeta}+1}K(\eta,\bar{\zeta})
\lambda(\eta)d\eta
&  =\pi^{2}(1-e^{-2\bar{\psi}( \bar{\zeta}+1,\bar{\zeta}) })\\
&\quad -\frac{1}{2\pi}\Big( \int_{\bar{\zeta}}^{\bar{\zeta}+1}\cos(2\pi(
\eta-\bar{\zeta}-Z(\eta,\bar{\zeta})))
e^{-\bar{\psi}(\eta,\bar{\zeta})}\lambda(\eta)d\eta\Big)
^{2}\\
&\quad -W_{3}(\bar{\zeta}),
\end{align*}
where
\begin{align*}
&  W_{3}(\bar{\zeta})\\
&  =-\frac{1}{\pi}\int_{\bar{\zeta}}^{\bar{\zeta}+1}d\eta\lambda(
\eta)\cos(2\pi(\eta-\bar{\zeta}-Z(\eta,\bar {\zeta})) )
e^{-\bar{\psi}(\eta,\bar{\zeta
})}\\
&\quad \times  \int_{\bar{\zeta}}^{\eta}\cos(2\pi( \xi-\bar{\zeta}-Z(
\xi,\bar{\zeta})))e^{-\bar{\psi}( \xi,\bar {\zeta})}\Big(
\int_{\bar{\zeta}}^{\xi}e^{\bar{\psi}( \upsilon,\bar{\zeta})
}\frac{\partial H( \upsilon,\bar{\zeta })
}{\partial\bar{\zeta}}d\upsilon\Big)\lambda(\xi)d\xi\,.
\end{align*}
Using (\ref{S7E59}) and (\ref{S7E60}), we obtain
\[
\int_{\bar{\zeta}}^{\bar{\zeta}+1}K(\eta,\bar{\zeta})
\lambda(\eta)d\eta=-W_{3}(\bar{\zeta})-W_{4}( \bar{\zeta})
-W_{5}(\bar{\zeta}),
\]
where
\begin{gather*}
W_{4}(\bar{\zeta}) =\pi^{2}(e^{-\bar{\psi}(
\bar{\zeta}+1,\bar{\zeta})}+1)e^{-\bar{\psi}( \bar
{\zeta}+1,\bar{\zeta})}J(\bar{\zeta}+1,\bar{\zeta}),\\
W_{5}(\bar{\zeta})  =\frac{1}{2\pi}(Q(
\bar{\zeta}))^{2}\,.
\end{gather*}
Therefore, we can finally write (\ref{S8E2}) as
\begin{equation}
\lambda(\bar{\zeta}+1)-\lambda(\bar{\zeta})=W( \bar{\zeta})
\equiv\sum_{i=1}^{5}W_{i}(\bar{\zeta
})\,.\label{S8E3}
\end{equation}
It turns out that the function $W(\bar{\zeta})$ approaches zero as
$\bar{\zeta}\to\infty$ fast enough. Using this fact it is
possible to show that $\lambda( \bar{\zeta})$ is globally bounded
as $\bar{\zeta}\to\infty$. In the rest of this Section we
will make precise this argument, and we will determine in which
sense the different terms in $W( \bar{\zeta})$ are small.

\subsection{Reformulating the delay equation as an integral equation}

The estimates so far derived yield bounds for the terms $W_{i}(
\bar{\zeta})$ with $i=2,\dots ,5$.

\begin{proposition}
\label{P3}Suppose that $|\lambda(\bar{\zeta}) | \leq\varepsilon_0$
\ for any $\zeta_0\leq\bar{\zeta}\leq\zeta_0+M$, for some $M$
large. Then:
\[
\big|\sum_{i=2}^{5}W_{i}(\bar{\zeta})\big|
\leq\frac {C}{(f(\bar{\zeta}))^{\beta}} \,,\quad \text{for }
\zeta_0\leq\bar{\zeta}\leq\zeta_0+M
\]
for some $\beta\in(0,1)$.
\end{proposition}

\begin{proof}
Lemmas \ref{L9}, \ref{L10} imply that $|J(\bar{\zeta}
+1,\bar{\zeta})|\leq\frac{C}{(f(\bar{\zeta }) ) ^{\beta}}$ for
$\zeta_0\leq\bar{\zeta}\leq\zeta_0+M$. Therefore
$|W_{2}(\bar{\zeta})|+|W_{4}( \bar{\zeta}) |\leq\frac{C}{(f(
\bar{\zeta}))^{\beta}}$. On the other hand, Lemmas \ref{L9},
\ref{L10} imply also that
$\sup_{\bar{\zeta}\leq\zeta\leq\bar{\zeta}+1}|J(
\zeta,\bar{\zeta})|\leq\frac{C}{(f( \bar{\zeta}) ) ^{\beta}}$, and
also $\sup_{\bar{\zeta}\leq \zeta\leq\bar{\zeta}+1}|\frac{\partial
J(\zeta,\bar{\zeta })}{\partial\bar{\zeta}}| \leq\frac{C}{(f(
\bar{\zeta}))^{\beta}}$. Moreover, (\ref{S7E61}) implies
\[
\frac{\partial J(\zeta,\bar{\zeta})}{\partial\zeta}
=\frac{\partial}{\partial\bar{\zeta}}(e^{\bar{\psi}(\zeta
,\bar{\zeta})}H(\zeta,\bar{\zeta}))
-e^{\bar{\psi}(\zeta,\bar{\zeta})}\frac{\partial\bar{\psi
}(\zeta,\bar{\zeta})}{\partial\bar{\zeta}}H(\zeta ,\bar{\zeta});
\]
whence due to (\ref{S7E29aN}) and the definition of
$h_{1}(W,\tau)$, $h_{2}(W,\tau)$ we have
$\frac{\partial J(\bar{\zeta}+1,\bar{\zeta})}{\partial\zeta}=0$.
Then $|\frac{dJ(\bar{\zeta}+1,\bar{\zeta})}{d\bar{\zeta}}|
\leq\frac{C}{(f(\bar{\zeta}))^{\beta}}$, whence
$|Q(\bar{\zeta})|\leq\frac{C}{(f(\bar{\zeta})) ^{\beta}}$. Therefore,
$|W_{5}(\bar{\zeta})|\leq\frac{C}{( f( \bar{\zeta}))^{\beta}}$.
To estimate $W_{3}(\bar{\zeta})$ we need to estimate the
term
\[
 \int_{\bar{\zeta}}^{\xi}e^{\bar{\psi}(\upsilon,\bar{\zeta})}
 \frac{\partial H( \upsilon,\bar{\zeta
 })}{\partial\bar{\zeta}}d\upsilon
=\frac{\partial}{\partial\bar{\zeta}
}(\int_{\bar{\zeta}}^{\xi}e^{\bar{\psi}(\upsilon,\bar{\zeta
})}H(\upsilon,\bar{\zeta})d\upsilon)-\int
_{\bar{\zeta}}^{\xi}e^{\bar{\psi}(\upsilon,\bar{\zeta})}
\frac{\partial\bar{\psi}(\upsilon,\bar{\zeta})}{\partial
\bar{\zeta}}H(\upsilon,\bar{\zeta})d\upsilon
\]
we can argue as in the estimate of $F_{1,1}$ in Lemma \ref{L9}.
Then $| \int_{\bar{\zeta}}^{\xi}e^{\bar{\psi}(\upsilon,\bar{\zeta})}
 \frac{\partial H(\upsilon,\bar{\zeta})}{\partial\bar{\zeta} }d\upsilon|
\leq\frac{C}{(f(\bar{\zeta}))^{\beta}}$, whence
$|W_{3}(\bar{\zeta})|\leq\frac{C}{(f(\bar{\zeta}))^{\beta}}$ and
Proposition \ref{P3} follows.
\end{proof}

The term $W_{1}(\bar{\zeta})$ on the right-hand side of
(\ref{S8E3}) contains some terms that cannot be neglected as
$\bar{\zeta }\to\infty$ in the study of the long time
asymptotics of (\ref{S8E3}). Using the definitions of $W_{1}(
\bar{\zeta})$, $Q(\bar{\zeta})$ we obtain
\begin{align*}
&W_{1}(\bar{\zeta})\\
&  =-2\pi\int_{\bar{\zeta}}^{\bar{\zeta}+1}\sin(2\pi(\eta
-\bar{\zeta}-Z(\eta,\bar{\zeta})))
\Big(\int_{\bar{\zeta}}^{\eta}e^{\bar{\psi}(\xi,\bar{\zeta})}
\frac{\partial H(\xi,\bar{\zeta})}{\partial\bar{\zeta}}
d\xi\Big) e^{-2\bar{\psi}(\eta,\bar{\zeta})}\lambda(
\eta)d\eta \\
&\quad -\frac{d}{d\bar{\zeta}}\Big(\int_{\bar{\zeta}}^{\bar{\zeta}+1}
\cos(2\pi(\eta-\bar{\zeta}-Z(\eta,\bar{\zeta})))
\lambda(\eta)e^{-\bar{\psi}(\eta
,\bar{\zeta})}J(\eta,\bar{\zeta})d\eta\Big)\\
& \quad -\frac{1}{2}\frac{d}{d\bar{\zeta}}(\bar{\psi}(\bar{\zeta
}+1,\bar{\zeta}))e^{-\bar{\psi}(\bar{\zeta}
+1,\bar{\zeta})}\frac{d}{d\bar{\zeta}}(J(\bar{\zeta
}+1,\bar{\zeta}))
+  e^{-\bar{\psi}(\bar{\zeta}+1,\bar{\zeta})}\frac{d^{2}}
{d\bar{\zeta}^{2}}(J(\bar{\zeta}+1,\bar{\zeta}))
\end{align*}
The three first terms on the right-hand side can be
bounded as $C/(f(\bar{\zeta}))^{\beta}$ arguing as in the
proof of Proposition \ref{P3}.

\begin{lemma} \label{L11}
Under the assumptions of Proposition \ref{P3},
\begin{equation}
W_{1}(\bar{\zeta})=W_{6}(\bar{\zeta})
+e^{-\bar{\psi}(\bar{\zeta}+1,\bar{\zeta})}\frac{d^{2}}
{d\bar{\zeta}^{2}}(J(\bar{\zeta}+1,\bar{\zeta})),
\label{S8E4}
\end{equation}
where
\begin{equation}
|W_{6}(\bar{\zeta})|\leq\frac{C}{(
f(\bar{\zeta}))^{\beta}}\,,\quad \text{for }
\zeta_0\leq\bar{\zeta}\leq\zeta_0+M \label{S8E5}
\end{equation}
for some $\beta\in(0,1)$.
\end{lemma}

To estimate $W_{1}(\bar{\zeta})$ then reduces to deriving
approximations for
$\frac{d^{2}}{d\bar{\zeta}^{2}}(J( \bar{\zeta}+1,\bar{\zeta}))$.
This requires basically to obtain rather
precise approximations for $Z( \zeta,\bar{\zeta})$ in the regions
$\zeta\approx\bar{\zeta}$, $\zeta\approx\bar{\zeta}+1$.
We can further simplify the terms in
$W_{1}(\bar{\zeta })$ to be
\begin{align*}
J(\bar{\zeta}+1,\bar{\zeta})
& =\frac{\partial}{\partial \bar{\zeta}}
\Big( \int_{\bar{\zeta}}^{\bar{\zeta}+1}e^{\bar{\psi}(
\eta,\bar{\zeta})}H(\eta,\bar{\zeta})d\eta\Big)
-\int_{\bar{\zeta}}^{\bar{\zeta}+1}e^{\bar{\psi}( \eta,\bar{\zeta
})}\frac{\partial\bar{\psi}(\eta,\bar{\zeta})}
{\partial\bar{\zeta}}H(\eta,\bar{\zeta})d\eta\\
&=
\int_{\bar{\zeta}}^{\bar{\zeta}+1}e^{\bar{\psi}(\eta,\bar{\zeta
})}\frac{\partial H(\eta,\bar{\zeta})}{\partial \bar{\zeta}}d\eta\\
& =\frac{\partial}{\partial\bar{\zeta}}\Big( \int
_{\bar{\zeta}}^{\bar{\zeta}+1}H(\eta,\bar{\zeta}) d\eta\Big)
+\int_{\bar{\zeta}}^{\bar{\zeta}+1}\bar{\psi}( \eta,\bar{\zeta})
\frac{\partial H(\eta,\bar{\zeta})}{\partial\bar{\zeta}}
d\eta\\
&\quad +  \int_{\bar{\zeta}}^{\bar{\zeta}+1}(e^{\bar{\psi}(\eta
,\bar{\zeta})}-1-\bar{\psi}(\eta,\bar{\zeta}))
\frac{\partial H(\eta,\bar{\zeta})}{\partial\bar{\zeta}}d\eta\\
&  \equiv J_{1}(\bar{\zeta}+1,\bar{\zeta})+J_{2}(
\bar{\zeta}+1,\bar{\zeta})+J_{3}( \bar{\zeta}+1,\bar{\zeta })\,.
\end{align*}
Our main goal now is to obtain suitable approximations for the
functions
$\frac{d^{2}}{d\bar{\zeta}^{2}}(J_{1}(\bar{\zeta}+1,\bar{\zeta }))$,
$\frac{d^{2}}{d\bar{\zeta}^{2}}(J_{2}(\bar{\zeta}+1,\bar{\zeta}))$.
We also wish to show that
$\frac{d^{2}}{d\bar{\zeta}^{2}}(J_{3}( \bar{\zeta}+1,\bar{\zeta
}))$
is small as $\bar{\zeta}\to\infty$. As indicated
above, this requires to derive good approximations for $Z(\zeta
,\bar{\zeta})$ in the regions $\zeta\approx\bar{\zeta}$, $\zeta
\approx\bar{\zeta}+1$.

The key result of this section is the following.

\begin{proposition} \label{P5}
Suppose that the assumptions of Proposition \ref{P3}
hold. There exist functions $K_{1}(\cdot)$, $K_{2}( \cdot)$
satisfying
\begin{equation} \label{S8E5a}
\begin{gathered}
|K_{1}(x)|+|K_{2}(x)|\leq\min\big\{
\frac{C}{1+x^{2}},\frac{1}{(L(\tau))^{3}}\big\}, \\
\int_0^{\infty}K_{1}(x)dx=\int_{-\infty}^{0}K_{2}( x)
dx=0
\end{gathered}
\end{equation}
such that
\begin{equation} \label{S8E6}
\begin{aligned}
&\frac{d^{2}}{d\bar{\zeta}^{2}}(J_{1}( \bar{\zeta}+1,\bar{\zeta
})+J_{2}(\bar{\zeta}+1,\bar{\zeta}))
\\
&=f'(\bar{\zeta}) \int_{\bar{\zeta}}^{\bar{\zeta}+\delta }K_{1}(
f'( \bar{\zeta})(\eta-\bar{\zeta
}))\lambda(\eta)d\eta\\
&\quad +f'(\bar{\zeta}+1)\int_{\bar{\zeta}+1-\delta}
^{\bar{\zeta}+1}K_{2}(f'(\bar{\zeta}+1)(
\eta-(\bar{\zeta}+1)))\lambda(\eta)d\eta
+W_{7}(\bar{\zeta}),
\end{aligned}
\end{equation}
where $\delta>0$ is a small fixed number and
\begin{equation}
|W_{7}(\bar{\zeta})|+|\frac{d^{2}}
{d\bar{\zeta}^{2}}(J_{3}(\bar{\zeta}+1,\bar{\zeta}))|
\leq\frac{C}{(f'(\bar{\zeta})
)^{\beta}}\,,\quad \zeta_0\leq\bar{\zeta}\leq\zeta_0+M \label{S8E7}
\end{equation}
with $\beta\in(0,1)$.
\end{proposition}

\begin{remark} \label{rm23} \rm
Note that Proposition \ref{P5} states that
$\frac{d^{2}}{d\bar{\zeta}^{2} }(J(\bar{\zeta}+1,\bar{\zeta}))$
might be approximated as two ``Dirac-mass approximating'' kernels
in the regions $\zeta\approx\bar{\zeta}$,
$\zeta\approx\bar{\zeta}+1$.
\end{remark}

\begin{proof}
Let us sketch the main ideas in the proof of Proposition \ref{P5}.
We write
\begin{gather*}
\frac{d^{2}}{d\bar{\zeta}^{2}}(J_{1}( \bar{\zeta}+1,\bar{\zeta
})+J_{2}(\bar{\zeta}+1,\bar{\zeta})) =Y_{1}(\bar{\zeta})
+Y_{2}(\bar{\zeta})+Y_{3}( \bar{\zeta}),
\\
Y_{1}(\bar{\zeta})\equiv\frac{d^{3}}{d\bar{\zeta}^{3}}(
\int_{\bar{\zeta}}^{\bar{\zeta}+\delta}H( \eta,\bar{\zeta})
d\eta)+\frac{d^{2}}{d\bar{\zeta}^{2}}(\int_{\bar{\zeta}}
^{\bar{\zeta}+\delta}\bar{\psi}(\eta,\bar{\zeta}) \frac{\partial
H(\eta,\bar{\zeta})}{\partial\bar{\zeta}}d\eta),
\\
Y_{2}(\bar{\zeta})\equiv\frac{d^{3}}{d\bar{\zeta}^{3}}(
\int_{\bar{\zeta}+\delta}^{\bar{\zeta}+1-\delta}H( \eta,\bar{\zeta
})d\eta)+\frac{d^{2}}{d\bar{\zeta}^{2}}(\int
_{\bar{\zeta}+\delta}^{\bar{\zeta}+1-\delta}\bar{\psi}(
\eta,\bar{\zeta })\frac{\partial H( \eta,\bar{\zeta}) }{\partial
\bar{\zeta}}d\eta),
\\
Y_{3}(\bar{\zeta})\equiv\frac{d^{3}}{d\bar{\zeta}^{3}}(
\int_{\bar{\zeta}+1-\delta}^{\bar{\zeta}+1}H(
\eta,\bar{\zeta})d\eta)
+\frac{d^{2}}{d\bar{\zeta}^{2}}(\int_{\bar{\zeta
}+1-\delta}^{\bar{\zeta}+1}\bar{\psi}( \eta,\bar{\zeta})
\frac{\partial H(\eta,\bar{\zeta})}{\partial\bar{\zeta}} d\eta)
\,.
\end{gather*}
Using Lemma \ref{L9} we obtain the estimate
$| Y_{2}(\bar{\zeta})|\leq C/( f'(\bar{\zeta}))^{\beta}$.

We now describe how to approximate $Y_{1}(\bar{\zeta})$. The
computation of $Y_{3}(\bar{\zeta})$ is completely similar.
Using the form of the function $R$ in (\ref{S7E22a}) it would be
natural to approximate $Z(\zeta,\bar{\zeta})$ in the region $\zeta
\approx\bar{\zeta}$ using the function
$\bar{Z}(\zeta,\bar{\zeta})$ solution of
\begin{equation} \label{S8E8}
\begin{gathered}
\begin{aligned}
\bar{Z}_{\zeta}
&=(\zeta-\bar{\zeta}-\bar{Z}(\zeta,\bar{\zeta
}))^{2}\lambda(\zeta)+\Phi_{1}(f'(\bar{\zeta})(\zeta
-\bar{\zeta}-\bar{Z}(\zeta,\bar{\zeta}))) \\
&\quad +\beta(f(\bar{\zeta}))\Phi_{2}(f'( \bar{\zeta})
(\zeta-\bar{\zeta}-\bar{Z}(
\zeta,\bar{\zeta}))),
\end{aligned}\\
\bar{Z}(\bar{\zeta}^{+},\bar{\zeta})=0,
\end{gathered}
\end{equation}
where $|x\Phi_{1}(x)|+|\frac{\Phi _{2}(x)}{x}|\leq C$, and
$\Phi_{1}(x)$, $\Phi_{2}(x) $ vanish if $|x| \leq 1/L$. We
have used the approximation
\begin{equation}
\begin{aligned}
&H(\zeta,\bar{\zeta})\\
&\approx\Phi_{1}( f'(\bar{\zeta})(\zeta-\bar{\zeta}
-\bar{Z}( \zeta,\bar{\zeta })))
+\beta(f(\bar{\zeta})) \Phi_{2}(f'(\bar{\zeta})(
\zeta-\bar{\zeta}-\bar{Z}(\zeta,\bar{\zeta})))\,.
\end{aligned}\label{S8E9}
\end{equation}
Let us assume for the moment that $Z(\zeta,\bar{\zeta})$ might be
approximated by means of $\bar{Z}( \zeta,\bar{\zeta})$. We can
then approximate $\bar{Z}(\zeta,\bar{\zeta})$ as
\begin{equation}
\bar{Z}(\zeta,\bar{\zeta})\approx\bar{Z}_{1}(\zeta
,\bar{\zeta})+\bar{Z}_{2}(\zeta,\bar{\zeta})+U(
\zeta,\bar{\zeta}),\label{S8E9a}
\end{equation}
where
\begin{gather*}
\bar{Z}_{1,\zeta}=\Phi_{1}(f'(\bar{\zeta})(
\zeta-\bar{\zeta}-\bar{Z}_{1}(\zeta,\bar{\zeta}))),\\
\begin{aligned}
\bar{Z}_{2,\zeta}
&=-f'(\bar{\zeta})\Phi_{1}'(f'(\bar{\zeta})(\zeta
  -\bar{\zeta} -\bar{Z}_{1}(\zeta,\bar{\zeta})))
  \bar{Z}_{2}(\zeta,\bar{\zeta})\\
&\quad +\beta(f(\bar{\zeta }))\Phi_{2}(f'(\bar{\zeta})(\zeta-\bar{\zeta}
 -\bar{Z}_{1}(\zeta,\bar{\zeta}))),
\end{aligned}\\
U_{\zeta}=(\zeta-\bar{\zeta}-\bar{Z}_{1}(\zeta,\bar{\zeta})
 -\bar{Z}_{2}(\zeta,\bar{\zeta}))^{2}\lambda(\zeta)-P(\zeta,\bar{\zeta})U,
\end{gather*}
where
\begin{equation} \label{S8E9b}
\begin{gathered}
\bar{Z}_{1}(\bar{\zeta}^{+},\bar{\zeta})=\bar{Z}_{2}(
\bar{\zeta}^{+},\bar{\zeta})=U( \bar{\zeta}^{+},\bar{\zeta
})=0,
\\
\begin{aligned}
P(\zeta,\bar{\zeta})&\equiv f'(\bar{\zeta })
\Phi_{1}'(f'(\bar{\zeta})(
\zeta-\bar{\zeta}-\bar{Z}_{1}(\zeta,\bar{\zeta})))\\
&\quad +\beta(f(\bar{\zeta}))f'( \bar{\zeta}) \Phi_{2}'(f'(
\bar{\zeta}) (\zeta-\bar{\zeta}-\bar{Z}_{1}(\zeta,\bar{\zeta})))\,.
\end{aligned}
\end{gathered}
\end{equation}
Note that
\begin{gather}
\bar{Z}_{1}(\zeta,\bar{\zeta})=\frac{W_{1}( f'(\bar{\zeta})(\zeta-\bar{\zeta}))
}{f'(\bar{\zeta})}, \label{S8E10}\\
\bar{Z}_{2}(\zeta,\bar{\zeta})=\frac{\beta(f(
\bar{\zeta}))}{f'(\bar{\zeta})}
W_{2}(f'(\bar{\zeta})(\zeta-\bar{\zeta })), \label{S8E11}
\end{gather}
where
\begin{gather*}
W_{1}'(x)=\Phi_{1}(x-W_{1}(x)),\\
W_{2}'(x)=-\Phi_{1}'(x-W_{1}(x))W_{2}+\Phi_{2}(x-W_{1}(x)),\\
W_{1}(0)=W_{2}(0)=0\,.
\end{gather*}
On the other hand
\begin{equation}
U(\zeta,\bar{\zeta})=\int_{\bar{\zeta}}^{\zeta}e^{-\int
_{\bar{\zeta}}^{s}P(\xi,\bar{\zeta})d\xi}( s-\bar{\zeta
}-\bar{Z}_{1}(s,\bar{\zeta})-\bar{Z}_{2}(s,\bar{\zeta
}))^{2}\lambda(s)ds \label{S8E12}
\end{equation}
Using (\ref{S8E9}) we can approximate
$\int_{\bar{\zeta}}^{\bar{\zeta}+\delta }H(\eta,\bar{\zeta})
d\eta$ as
\begin{align*}
\int_{\bar{\zeta}}^{\bar{\zeta}+\delta}H(\zeta,\bar{\zeta})
d\zeta
&\approx\int_{\bar{\zeta}}^{\bar{\zeta}+\delta}[\Phi_{1}(
f'(\bar{\zeta})( \zeta-\bar{\zeta}-\bar{Z}(\zeta,\bar{\zeta})))
\\
&\quad +\beta(f( \bar{\zeta }))\Phi_{2}(f'(\bar{\zeta})
(\zeta-\bar{\zeta}-\bar{Z}(\zeta,\bar{\zeta})))]d\zeta\,.
\end{align*}
Using (\ref{S8E9a}), we obtain the  approximation
\begin{equation} \label{S8E13}
\begin{aligned}
&\int_{\bar{\zeta}}^{\bar{\zeta}+\delta}H(\zeta,\bar{\zeta})
d\zeta \\
&\approx\int_{\bar{\zeta}}^{\bar{\zeta}+\delta}\Phi_{1}(f'(\bar{\zeta})(\zeta-\bar{\zeta}-\bar{Z}_{1}(
\zeta,\bar{\zeta})-\bar{Z}_{2}( \zeta,\bar{\zeta})
))d\zeta\\
&\quad +\int_{\bar{\zeta}}^{\bar{\zeta}+\delta}\beta(f(\bar{\zeta
}))\Phi_{2}(f'(\bar{\zeta})(
\zeta-\bar{\zeta}-\bar{Z}_{1}(\zeta,\bar{\zeta})
-\bar{Z}_{2}(\zeta,\bar{\zeta})))d\zeta
\\
&\quad -\int_{\bar{\zeta}}^{\bar{\zeta}+\delta}f'(\bar{\zeta})
\Phi_{1}'(f'(\bar{\zeta})(
\zeta-\bar{\zeta}-\bar{Z}_{1}(\zeta,\bar{\zeta})-\bar{Z}
_{2}(\zeta,\bar{\zeta})))U(\zeta
,\bar{\zeta})d\zeta\\
&\quad -\int_{\bar{\zeta}}^{\bar{\zeta}+\delta}\beta(f(\bar{\zeta
}))\Phi_{2}'(f'(\bar{\zeta })
(\zeta-\bar{\zeta}-\bar{Z}_{1}(\zeta,\bar{\zeta })
-\bar{Z}_{2}(\zeta,\bar{\zeta})))U( \zeta,\bar{\zeta})
d\zeta\,.
\end{aligned}
\end{equation}
The first two terms on the right of (\ref{S8E13}), as well as
their derivatives are bounded by
$O(\frac{1}{( f'(\bar{\zeta}))^{\beta}})$, $\beta>0$, as can
be seen using (\ref{S8E10}), (\ref{S8E11}) and the change
of variables $\zeta-\bar{\zeta}=s$.
On the other hand, using (\ref{S8E12}) the
definition of $P(\zeta,\bar{\zeta})$ we can rewrite the last two
terms as
\begin{align*}
&-\int_{\bar{\zeta}}^{\bar{\zeta}+\delta}d\zeta P(\zeta,\bar{\zeta
})\int_{\bar{\zeta}}^{\zeta}e^{-\int_{\bar{\zeta}}^{\eta}P(
\xi,\bar{\zeta})d\xi}( \eta-\bar{\zeta}-\bar{Z}_{1}(
\eta,\bar{\zeta})-\bar{Z}_{2}(\eta,\bar{\zeta}))
^{2}\lambda(\eta)d\eta\\
&\equiv h_{1}( \bar{\zeta})
\end{align*}
We need to compute three derivatives of $h_{1}(\bar{\zeta}) $. The
contributions due to the extremes of integration vanish, since
$P(\bar{\zeta},\bar{\zeta})=P( \bar{\zeta}+1,\bar{\zeta}) =0$.
Then
\begin{align*}
\frac{d^{3}h_{1}(\bar{\zeta})}{d\bar{\zeta}^{3}}
&=-\int_{\bar{\zeta}}^{\bar{\zeta}+\delta}d\eta\lambda(\eta)\\
&\quad\times \frac{d^{3}}{d\bar{\zeta}^{3}}
\Big(\int_{\eta}^{\bar{\zeta}+\delta}P(\zeta,\bar{\zeta})
e^{-\int_{\bar{\zeta}}^{\eta}P(\xi,\bar {\zeta})d\xi}(\eta
 -\bar{\zeta}-\bar{Z}_{1}(\eta ,\bar{\zeta})
-\bar{Z}_{2}(\eta,\bar{\zeta}))^{2}d\zeta\Big)\,.
\end{align*}
The contributions of $P(\zeta,\bar{\zeta})$ and its derivatives at
$\zeta=\bar{\zeta}+\delta$ are bounded as
$O(1/(f'(\bar{\zeta}))^{\beta})$. Then
\begin{align*}
\frac{d^{3}h_{1}(\bar{\zeta})}{d\bar{\zeta}^{3}}
&=-\int_{\bar{\zeta}}^{\bar{\zeta}+\delta}d\eta\lambda(\eta)
\int_{\eta}^{\bar{\zeta}+\delta}\frac{d^{3}}{d\bar{\zeta}^{3}}\Big(
\big[\int_{\eta}^{\bar{\zeta}+\delta}P( \zeta,\bar{\zeta})d\zeta\big]\\
&\quad \times
e^{-\int_{\bar{\zeta}}^{\eta}P(\xi,\bar{\zeta})d\xi}(
\eta -\bar{\zeta}-\bar{Z}_{1}(\eta,\bar{\zeta})
-\bar{Z}_{2}(\eta,\bar{\zeta}))^{2}\Big)
+O(1/(f'(\bar{\zeta}))^{\beta})
\end{align*}
Using (\ref{S8E9b}) and (\ref{S8E10}), it follows that
\begin{align*}
&\int_{\bar{\zeta}}^{\eta}P(\xi,\bar{\zeta})d\xi\\
&=\int_0^{f'(\bar{\zeta})( \eta-\bar{\zeta})
}\Phi_{1}'(x-W_{1}(x))dx+\beta( f( \bar{\zeta}) )
\int_0^{f'(\bar{\zeta })(\eta-\bar{\zeta}) }\Phi_{2}'(
x-W_{1}(x))dx\,.
\end{align*}
Using (\ref{S8E9b}), (\ref{S8E10}), (\ref{S8E11}), we obtain
\begin{align*}
\int_{\eta}^{\bar{\zeta}+\delta}P(\zeta,\bar{\zeta})d\zeta
&=f'(\bar{\zeta})\int_{\eta}^{\bar{\zeta}+\delta}
\Phi_{1}'(f'(\bar{\zeta})( \zeta-\bar{\zeta})-W_{1}(f'(
\bar{\zeta})
(\zeta-\bar{\zeta})))d\zeta\\
&\quad +\beta(f(\bar{\zeta}))\int_{\eta}^{\bar{\zeta
}+\delta}f'(\bar{\zeta})\Phi_{2}'\big(
f'(\bar{\zeta})(\zeta-\bar{\zeta})\\
&\quad -W_{1}(f'(\bar{\zeta})(\zeta-\bar{\zeta }))\big)
d\zeta+O(1/(f'( \bar{\zeta})) ^{\beta}),
\end{align*}
\begin{align*}
&(\eta-\bar{\zeta}-\bar{Z}_{1}(\eta,\bar{\zeta})-\bar
{Z}_{2}(\eta,\bar{\zeta}))^{2}\\
&=\frac{1}{(f'(\bar{\zeta}))^{2}}(
f'(\bar{\zeta})(\eta-\bar{\zeta})
-W_{1}(f'(\bar{\zeta})(\eta-\bar{\zeta }))
 -\beta(f(\bar{\zeta}))W_{2}( f'(\bar{\zeta}) (\eta-\bar{\zeta
})))^{2}
\end{align*}
Therefore, we arrive to an approximation of the form
\begin{align*}
&\int_{\eta}^{\bar{\zeta}+\delta}P(\zeta,\bar{\zeta})d\zeta
e^{-\int_{\bar{\zeta}}^{\eta}P(\xi,\bar{\zeta})d\xi}(
\eta-\bar{\zeta}-\bar{Z}_{1}(\eta,\bar{\zeta})-\bar{Z}
_{2}(\eta,\bar{\zeta}))^{2}\\
&=\frac{1}{(f'(\bar{\zeta}))}S( f'(\bar{\zeta})(\eta-\bar{\zeta})
,\beta(f(\bar{\zeta})))
\end{align*}
and
\begin{align*}
\frac{d^{3}h_{1}(\bar{\zeta})}{d\bar{\zeta}^{3}}
&=-\int_{\bar{\zeta}}^{\bar{\zeta}+\delta}d\eta\lambda(\eta)
\int_{\eta}^{\bar{\zeta}+\delta}\frac{d^{3}}{d\bar{\zeta}^{3}}\Big(
\frac {1}{(f'(\bar{\zeta}))}\\
&\quad \times \tilde{S}(f'(\bar{\zeta})(\eta-\bar{\zeta}) ,
\beta(f(\bar{\zeta})))\Big)d\zeta
+O\big(1/(f'(\bar{\zeta}))^{\beta}\big)\,.
\end{align*}
We now remark that if $\delta=1/(f'( \bar{\zeta}))^{\alpha}$
with $\alpha<1$ all the contributions due to terms
like $\Phi_{2}$, $W_{2}$ and analogous ones yield relative
corrections of order $1/( f'(\bar{\zeta}) ) ^{\beta}$,
with $\beta>0$, perhaps small if $\alpha$ is close to zero, but in
any case strictly positive. In particular this implies that if the
size of the final terms obtained is of order one the correction
due to the presence of $\beta(f(\bar{\zeta}))$ would be
negligible. We then write
\begin{equation} \label{S8E14}
\begin{aligned}
\frac{d^{3}h_{1}(\bar{\zeta})}{d\bar{\zeta}^{3}}
&=-\int_{\bar{\zeta}}^{\bar{\zeta}+\delta}d\eta\lambda(\eta)
\Big(\int_{\eta}^{\bar{\zeta}+\delta}\frac{d^{3}}{d\bar{\zeta}^{3}}
\Big(\frac{1}{(f'(\bar{\zeta}))}S_{1}(
f'(\bar{\zeta})(\zeta-\bar{\zeta}))\Big)d\zeta\Big)\\
&\quad\times \Big(1+O(1/(f'(\bar{\zeta}))^{\beta})\big)
+O(1/f'(\bar{\zeta}))^{\beta}),
\end{aligned}
\end{equation}
where
\[
S_{1}(x)=\Big[\int_{x}^{\infty}\Phi_{1}'(
\xi-W_{1}(\xi))d\xi\Big]e^{-\int_0^{x}\Phi _{1}'(\xi-W_{1}(\xi))d\xi}(
x-W_{1}(x))^{2}
\]
Since $\log(f^{k)}(\zeta))\sim\log(f( \zeta))$ as
$\zeta\to\infty$ (cf. Appendix A), it turns out that the
derivatives of $f(\bar{\zeta})$ in (\ref{S8E14}) would yield
smaller contributions that the derivatives of the term
$\bar{\zeta}$ that, roughly, multiplies the different terms by
$f'(\bar{\zeta})$. Then
\begin{align*}
\frac{d^{3}h_{1}(\bar{\zeta})}{d\bar{\zeta}^{3}}
&=(f'(\bar{\zeta}))^{2}\int_{\bar{\zeta}
}^{\bar{\zeta}+\delta}d\eta\lambda(\eta)
\Big(\int_{\eta
}^{\bar{\zeta}+\delta}S_{1}'''( f'(
\bar{\zeta})(\zeta-\bar{\zeta}))d\zeta\Big)\\
&\quad\times \Big(1+O(1/(f'(\bar{\zeta}))^{\beta})\Big)
+O(1/(f'(\bar{\zeta}))^{\beta}),
\end{align*}
whence
\begin{align*}
\frac{d^{3}h_{1}(\bar{\zeta})}{d\bar{\zeta}^{3}}
&=-f'(\bar{\zeta})\Big[\int_{\bar{\zeta}}^{\bar
{\zeta}+\delta}d\eta\lambda(\eta)S_{1}''(
f'(\bar{\zeta})(\eta-\bar{\zeta}))d\zeta\Big]\\
&\quad\times \Big(1+O(1/(f'(\bar{\zeta}))^{\beta})\Big)
+O(1/(f'(\bar{\zeta}))^{\beta})
\end{align*}
and this formula yields the sough-for structure (\ref{S8E6}).
It remains to estimate in a similar manner the term
\[
\frac{d^{2}}{d\bar{\zeta}^{2}}(\int_{\bar{\zeta}}^{\bar{\zeta}+\delta
}\bar{\psi}(\eta,\bar{\zeta})\frac{\partial H(\eta
,\bar{\zeta})}{\partial\bar{\zeta}}d\eta)
\]
Arguing as in the previous case it would follow that this term might be
approximated as
\[
\frac{d^{2}}{d\bar{\zeta}^{2}}(\int_{\bar{\zeta}}^{\bar{\zeta}+\delta
}\bar{\psi}(\eta,\bar{\zeta})\frac{\partial H(\eta
,\bar{\zeta})}{\partial\bar{\zeta}}d\eta)=\frac{d^{2}
h_{2}(\bar{\zeta})}{d\bar{\zeta}^{2}}
+O(1/(f'(\bar{\zeta}))^{\beta}),
\]
where
\begin{gather*}
h_{2}(\bar{\zeta})=2f'(\bar{\zeta})
\int_{\bar{\zeta}}^{\bar{\zeta}+\delta}\lambda(\eta)(
\eta-\bar{\zeta}-Z_{1}(\eta,\bar{\zeta}))[
\int_{\eta}^{\bar{\zeta}+\delta}\Psi(f'(\bar{\zeta
})(\zeta-\bar{\zeta}))d\zeta]d\eta, \\
\Psi(x)\equiv\frac{d(\Phi_{1}(x-W_{1}(x)))}{dx};
\end{gather*}
whence
\[
h_{2}(\bar{\zeta})=\frac{1}{f'(\bar{\zeta })
}\int_{\bar{\zeta}}^{\bar{\zeta}+\delta}\lambda(\eta)
S_{2}(f'(\bar{\zeta})(\eta-\bar{\zeta }))d\eta,
\]
where
\[
S_{2}(x)=2(x-W_{1}(x))\int _{x}^{\infty}\Psi(\xi)d\xi\,.
\]
Then
\[
\frac{d^{2}h_{2}(\bar{\zeta})}{d\bar{\zeta}^{2}}
=f'(\bar{\zeta})\Big[\int_{\bar{\zeta}}^{\bar{\zeta}+\delta
}\lambda(\eta) S_{2}''(f'(\bar{\zeta}) (
\eta-\bar{\zeta}))d\eta\Big]
\Big(1+O( 1/(f'(\bar{\zeta}))^{\beta} )\Big)
\]
and this yields also the structure in (\ref{S8E6}). The bounds for
$\frac{d^{2}}{d\bar{\zeta}^{2}}(J_{3}( \bar{\zeta}+1,\bar{\zeta
}))$ might be obtained in a similar manner, using the fact that
the presence of an additional term $( \eta-\bar{\zeta}) ^{2}$
yields smallness. The approximations near the value
$\zeta=\bar{\zeta }+1$ can be derived in a similar manner, whence
Proposition \ref{P5} follows.
\end{proof}

\subsection{Bounds for the solutions of the integral equation}

\begin{proof}[Proof of Theorem \ref{thm1}]
In summary, using Propositions
\ref{P3} and \ref{P5} and Lemma \ref{L11}, we can rewrite
(\ref{S8E3}) as
\begin{align*}
  \lambda(\bar{\zeta}+1)-\lambda(\bar{\zeta})
&  =e^{-\bar{\psi}(\bar{\zeta}+1,\bar{\zeta})}\Big[ f'(\bar{\zeta})
\int_{\bar{\zeta}}^{\bar{\zeta}+\delta }K_{1}( f'(
\bar{\zeta})(\eta-\bar{\zeta
}))\lambda(\eta)d\eta \\
&\quad  +  f'(\bar{\zeta}+1)\int_{\bar{\zeta}
+1-\delta}^{\bar{\zeta}+1}K_{2}(f'( \bar{\zeta}+1)
(\eta-(\bar{\zeta}+1)))\lambda(
\eta)d\eta\Big]\\
&\quad +O(1/(f'(\bar{\zeta}))^{\beta})
\end{align*}
On the other hand, using the compatibility condition
(\ref{S7E36N}) we can replace the exponential factor
$e^{-\bar{\psi}(\bar{\zeta}+1,\bar {\zeta})}$ by one, introducing
in turn a corrective term
\begin{equation} \label{S8E15}
\begin{aligned}
  \lambda(\bar{\zeta}+1)-\lambda( \bar{\zeta})
&  =\Big[f'(\bar{\zeta})\int_{\bar{\zeta}}
^{\bar{\zeta}+\delta}K_{1}(f'(\bar{\zeta})(
\eta-\bar{\zeta}))\lambda(\eta) d\eta \\
&\quad  + f'(\bar{\zeta}+1)\int_{\bar{\zeta}
+1-\delta}^{\bar{\zeta}+1}K_{2}(f'( \bar{\zeta}+1)
(\eta-(\bar{\zeta}+1)))\lambda(
\eta)d\eta\Big]\\
& \quad +O(1/(f'(\bar{\zeta}))^{\beta})
\end{aligned}
\end{equation}
Equation (\ref{S8E15}) is satisfied as long as $|\lambda(
\bar{\zeta}+1)|\leq\varepsilon_0$. The local existence Theorem
(cf. Theorem \ref{Th2}) implies that it is then possible extend
the solution in a larger interval. It remains to show that the
function $\lambda$ solution of (\ref{S8E15}) is globally defined
in time and that the estimate $| \lambda(\bar{\zeta}+1)|
\leq\varepsilon_0$ remains being valid for arbitrarily large
values of $\bar{\zeta}$.
To this end, we define
\[
\psi(\bar{\tau})  =\lambda(\bar{\zeta}+1),\quad
\bar{\tau}    =f(\bar{\zeta}+1)\,.
\]
Using the fact that $S(f(\bar{\zeta}+1))=f( \bar{\zeta})$ (cf.
(\ref{S5E5})), as well as the asymptotics
of $f$ (cf. Appendix A), (\ref{S8E15}) becomes
\begin{align*}
\psi(\bar{\tau})-\psi(S(\bar{\tau}))
&=\int_{f(f^{-1}(\bar{\tau})-\delta)}^{\bar{\tau}}
 K_{2}(\bar{\tau}-s)\psi(s)ds\\
&\quad +\int_{S(\bar{\tau})}^{f(f^{-1}
 (S(\bar{\tau})) +\delta)}K_{1}(s-S(\bar {\tau}))\psi(s)ds
 +O(\frac{1}{(S(\bar{\tau}))^{\beta} })
\end{align*}
where the value of $\beta$ might change from one formula to the other,
but it is always a positive number.

Using (\ref{S8E7}) we can then obtain the  inequality
\[
\sup_{\tau_{n}\leq\tau\leq\tau_{n+1}}|\psi(\tau) |
\leq\Big(1+\frac{C}{(L(\tau_{n-1}))^{3}
}\Big)\sup_{\tau_{n-1}\leq\tau\leq\tau_{n}}|\psi( \tau)|
+\frac{C}{(\tau_{n-1})^{\beta}}
\]
where $\tau_{n}=S^{-1}(\tau_{n-1})$, and $\tau_0$ is the initial
time for $\bar{\tau}$. Using (\ref{Lhyp}) it then follows that
\[
\sup_{\tau_{n}\leq\tau\leq\tau_{n+1}}|\psi(\tau) |
\leq\big(1+\frac{C}{n-1}\big)\sup_{\tau_{n-1}\leq\tau\leq\tau_{n} }|
\psi(\tau)|+\frac{C}{(\tau _{n-1})^{\beta}}
\]
whence, due to the very fast growth of $\tau_{n}$ we obtain,
upon iteration
\begin{equation}
\sup_{\tau_{n-1}\leq\tau\leq\tau_{n}}|\psi(\tau) |\leq C\epsilon_0,
\label{S8E15a}
\end{equation}
where $\epsilon_0$ might be arbitrarily small if $\tau_0$ is
large enough and $\sup_{\tau_0\leq\tau\leq\tau_{1}}| \psi(
\tau) |$ is small.

Formula (\ref{S8E15a}) implies that $\psi(\tau)$ remains small for
arbitrarily long times. In particular the assumptions required in
Lemma \ref{L7} and in the subsequent arguments are satisfied.
Moreover, (\ref{S8E15a}) implies that the solution of the problem
(\ref{S7E20} )-\eqref{S7E22}, or equivalently (\ref{S6E20}),
(\ref{S6E36}), (\ref{S6E37}) can be extended to arbitrarily long
times. Theorem \ref{thm1} is then a consequence of Theorem
\ref{T2}.
\end{proof}


\section{Appendix: Some properties of the function $f(\zeta)$}

In this Appendix we collect several properties of the function
$f(\zeta)$ that have been used repeatedly in Section \ref{noncomp}.
The function $f(\zeta)$ is defined by means of (\ref{S5E5}),
(\ref{S5E6}) as well as the compatibility conditions (\ref{S5E7}
)-(\ref{S5E11}). Although the properties described in this
Appendix could be generalized to more general functions $S$, we
will assume by definiteness that (\ref{S4E29}) holds, as well as
similar asymptotic formulae for the derivatives of $S$. Under
these assumptions the function $f$ has the following properties
\begin{equation}
f(\zeta)\gg \exp(\exp(\exp(\dots
\exp(\zeta))))\quad\text{as }\zeta \to\infty\label{F1}
\end{equation}
for any finite number of iterated exponentials.

\begin{equation}
f^{(k-1)}(\zeta)\ll f^{(k)}( \zeta)\ll (f(\zeta)) ^{1+\varepsilon
}\quad\text{as }\zeta\to\infty\label{F2}
\end{equation}
for any $\epsilon>0$, and any $k=1,2,3$.
\begin{equation}
f(\zeta+\delta)\gg f(\zeta) \quad\text{as }
\zeta\to\infty\label{F3}
\end{equation}
for any $\delta>0$.
\begin{equation}
f(\zeta+\frac{C}{f(\zeta)})\sim f(\zeta)\quad\text{as }
\zeta\to\infty\label{F4}
\end{equation}
for any $C>0$.
Also there holds
\begin{gather}
\log(f'(\zeta))\sim\log(f( \zeta))\quad
\text{as }\zeta\to\infty \label{F10}
\\
\log(\beta(\tau))\sim-2\log(\tau)
\quad \text{as }\tau\to\infty\label{F11}
\end{gather}
Property (\ref{F1}) follows from iterating (\ref{S5E5}) more
than $k$ times. Property (\ref{F3}) can be proved in a similar
manner. Indeed, iterating (\ref{S5E5}) by means of an exponential
function it follows that,since $f(\zeta)\to\infty$, that
$f(\zeta+\delta)-f(\zeta)\to\infty$, Then
\begin{align*}
f(\zeta+\delta)
& =S^{-1}(f(\zeta+\delta -1))\\
&=S^{-1}([f(\zeta+\delta-1)-f(\zeta-1)]+f(\zeta-1))\\
&  \gg S^{-1}(f(\zeta-1))\\
&=f(\zeta)
\end{align*}
as $\zeta\to\infty$.
Property (\ref{F2}) follows from differentiating (\ref{S5E5}), which
yields
\begin{equation}
f'(\zeta)=S'(f(\zeta+1)) f'(\zeta+1) \sim\frac{a}{f(
\zeta+1)}f'(\zeta+1)\,. \label{F5}
\end{equation}
Iterating (\ref{F5}) to estimate $f'(\zeta+1)$ it follows from
(\ref{F1}) that $f'(\zeta)\to \infty$. Combining this with
(\ref{F5}) we obtain the first inequality in (\ref{F2}) with
$k=1$. On the other hand, combining (\ref{S5E5}) and (\ref{F5}) we
obtain
\begin{equation}
\frac{f'(\zeta+1)}{f(\zeta+1)}=\Big[ \frac{1}{S'(f(\zeta+1))}\frac{f(
\zeta) }{S^{-1}(f(\zeta))}\Big]\frac{f'(\zeta)}{f(\zeta)}\,. \label{F6}
\end{equation}
The term between brackets is bounded by $Cf(\zeta)$. Iterating
(\ref{F6}) we obtain
\begin{equation}
\frac{f'(\zeta_0+n)}{f(\zeta_0+n)
}=\prod_{\ell=0}^{n-1}\Big[\frac{1}{S'(f(\zeta
_0+1+\ell))}\frac{f(\zeta_0+\ell)} {S^{-1}(f(\zeta_0+\ell))}\Big]
\frac{f'(\zeta_0)}{f(\zeta_0)} \,.\label{F7}
\end{equation}
The product in (\ref{F7}) can be bounded by
\begin{equation}
\prod_{\ell=0}^{n-1}[Cf(\zeta_0+\ell)]\,.\label{F8}
\end{equation}
Taking the logarithm and using that $f(\zeta+1)/f(\zeta)\geq2$ for
$\zeta$ large enough we obtain, after adding a geometric series, an upper
estimate for the product in (\ref{F8}), of the form
\[
\exp(B\log(f(\zeta_0+n-1))) =(f(\zeta_0+n-1))^{B}
\]
for some $B>0$; whence (\ref{F7}) yields
\begin{equation}
f'(\zeta)\leq Cf(\zeta)(f( \zeta-1))^{B} \label{F9}
\end{equation}
and since (\ref{S5E5}) implies that $f(\zeta-1)\leq
C\log(f(\zeta))$ we obtain (\ref{F2}) for $k=1$.

The proof of (\ref{F2}) for $k=2,3$ is similar.
To show (\ref{F4}) we iterate (\ref{S5E5}) to obtain
\[
f\big(\zeta+\frac{C}{f(\zeta)}\big)
=S^{-1}\Big( S^{-1}\Big( \dots
S^{-1}\Big(f\big(\zeta+\frac{C}{f(\zeta)}-n\big)\Big)\Big)\Big)
\]
where the number of iterations $n$ is such that
$\zeta+\frac{C}{f(\zeta)}-n\in[\zeta_0,\zeta_0+1]$.
Since $f(\zeta)$ is huge we can
approximate the terms $S^{-1}(f(\zeta+\frac{C}{f(\zeta)}-n))$ as
$S^{-1}(f(\zeta-) n)+\frac{C(S^{-1})'(f(\zeta-n))}{f(\zeta)}$.
Using this approximation in $n-1$ iterations, as well as
(\ref{F5}) we obtain the approximation
\[
f\big(\zeta+\frac{C}{f(\zeta)}\big)=S^{-1}\Big(f( \zeta-1)
+\frac{Cf'(\zeta-1)}{f( \zeta)}\Big)
\]
and using (\ref{F9}), (\ref{F4}) follows. A more rigorous proof would just
replace the approximation using derivatives by upper bounds. Similar
approximations might be derived for the derivatives.

Finally, (\ref{F10}) follows from (\ref{F3}), (\ref{F5}) and
(\ref{F11}) is a consequence of (\ref{S5E15}),
(\ref{F1})-(\ref{F4}). The detailed derivation of (\ref{F11}) as
well as more detailed asymptotics for $\beta( \tau)$ have been
given in (\cite{V2}).


\subsection*{Acknowledgements}
The author wants to thank B. Niethammer for several illuminating
discussions during the writing of this paper.


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\end{document}