\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 54, pp. 1--22.\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 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/54\hfil Local stability of spike steady states]
{Local stability of spike steady states in a simplified
Gierer-Meinhardt system}
\author[G. E. Karadzhov, D. E. Edmunds, P.P.N. de Groen\hfil EJDE-2005/54\hfilneg]
{Georgi E. Karadzhov, David Edmunds, Pieter de Groen}  

\address{Georgi E. Karadzhov \hfill\break
Institute of Mathematics and Informatics \\
Bulgarian Academy of Sciences\\
 1113 Sofia, Bulgaria}
\email{geremika@math.bas.bg}

\address{David E. Edmunds \hfill\break
Department of Mathematics,
University of Sussex\\
Brighton BN1 9RF, U. K.}
\email{d.e.edmunds@sussex.ac.uk}

\address{Pieter P.N. de Groen \hfill\break
Department of Mathematics, Vrije Universiteit Brussel \\
Pleinlaan 2, B-1050 Brussels, Belgium}
\email{pdegroen@vub.ac.be}

\date{}
\thanks{Submitted March 3, 2005. Published May 23, 2005.}
\subjclass[2000]{35B25, 35K60}
\keywords{Spike solution; singular perturbations;
reaction-diffusion equations; \hfill\break\indent
Gierer-Meinhardt equations}

\begin{abstract}
 In this paper we study the stability of the single internal spike
 solution of a simplified Gierer-Meinhardt' system of equations
 in one space dimension.
 The linearization around this spike consists of a selfadjoint
 differential operator plus a non-local term, which is a non-selfadjoint
 compact integral operator.
 We find the asymptotic behaviour  of the small eigenvalues and we prove
 stability of the steady state for the parameter $(p,q,r,\mu)$ in a
 four-dimensional region (the same as for the shadow equation,
 \cite{gk1}) and for any finite $D$ if $\varepsilon$ is sufficiently small.
 Moreover, there exists an exponentially large $D(\varepsilon)$ such that
 the stability is still valid for $D<D(\varepsilon)$. Thus we extend the
 previous results known only for the case $r=p+1$ or $r=2, 1<p<5$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks


\section{Introduction \label{intro}}

Based on pioneering ideas of Turing \cite{turing}
about pattern formation by interaction of
diffusing chemical substances, Gierer \& Meinhardt
proposed and studied in \cite{gierer} the following system of reaction
diffusion equations on a spatial domain $\Omega$
\begin{equation} \label{gm}
\begin{gathered}
 U_t=\varepsilon^2\Delta U-U+U^p H^{-q} \quad x\in\Omega,\; t>0,\\
\tau H_t= D \Delta H-\mu H+U^r H^{-s}\quad x\in\Omega,\; t>0\,,\\
\partial_nU=0=\partial_nH \quad x\in\partial\Omega,\; t\ge 0,
\end{gathered}
\end{equation}
where $U$ and $H$ represent activator and inhibitor
concentrations, $\varepsilon$ and $D$  their diffusivities, and where
$\tau$ and $\mu$ are the reaction time rate and the decay rate of
the inhibitor; $D$ is assumed to be positive and $\varepsilon$ and $\tau$
small (positive). $\Omega$ is a bounded domain; we shall restrict
our analysis to one space dimension and choose $\Omega:=[-1,1]$.
The exponents $\{p>1\,, ~q>0\,,~r>1\}$
satisfy the inequality
\begin{equation} \label{eq1s1}
\gamma_r:=\frac{qr}{p-1}>1\,.
\end{equation}
Iron, Ward \& Wei \cite{ward2} analyze by formal asymptotic expansions the
stability of approximate $N-$spike solutions for the simplified system,
namely, that obtained by taking
$\tau=0$. Rigorous results are obtained in \cite{gk1} for the case
of the so-called shadow equation
$$
U_t=
\varepsilon^2 U_{xx}-U+2^q U^{p}\Bigl(\int_{-1}^1 U^r dx\Bigr)^{-q} \,,\quad
U_x(-1,t)=U_x(1,t),
$$
derived from the system in the limit $D\to \infty$ and $\tau\to 0$.


In this paper we propose to study rigourously the simplified system when $s=0$.
 After rescaling $U \to \varepsilon^{-\nu_1} U$, $H \to \varepsilon^{-\nu_2} H$,
$\nu_1:=\frac{q}{1-p+qr}$, $\nu_2:=\frac{p-1}{1-p+qr}$, we get the system
\begin{equation} \label{simplgm}
\begin{gathered}
 U_t= \varepsilon^2\Delta U-U+U^p H^{-q} \quad x\in\Omega,\; t>0,\\
 0= D\Delta H-\mu H+\varepsilon^{-1}U^r  \quad x\in\Omega,\; t>0\\
\partial_nU=0=\partial_nH \quad x\in\partial\Omega,\; t\ge 0
\end{gathered}
\end{equation}
Our goal is a rigorous study of stability of the single
internal spike solution via the spectrum of the linearized operator.
In \cite{takagi} it is shown by the implicit function theorem, that such
a spike solution exists for
\begin{quote}
$p>1$, $r>1$, $q>0$, $qr\neq (p-1)(s+1)$\\
 and $D$ exponentially
large w.r.t. $\varepsilon>0$,
\end{quote}
which is close to the shadow spike corresponding to
$D=\infty$. A different approach, based on geometric singular perturbation
theory, is applied in \cite{doelman1} for the same problem on the whole line.
In \cite{ww} a rigorous treatment of the stability of multiple-peaked spike
solutions is given, based on the Liapunov-Schmidt reduction method. See also
\cite{ward3}, \cite{ward4}, where stability and Hopf bifurcation of the
one-spike solution is studied.
In this paper we construct a single spike solution
(on a bounded interval) by fix-point iteration
and we establish stability by a rigorous analysis of the spectrum of
the first variation around this spike.

In section \ref{spike} we construct a positive (stationary) solution with a single
internal spike
 for $p>1$, $r>1$, $q>0$, $qr\neq (p-1)(s+1)$ and for any fixed $D$,
using contraction
around another shadow spike that exists for all $D>0$. The existence of such
a solution is proved in \cite{wei} in a larger domain $\sqrt D \gg \varepsilon$.
In section \ref{reference} we study the spectrum of the differential
operator $L_\varepsilon$.
The eigenvalues are estimated using  Rayleigh's quotient.
In section \ref{perturb} we make a detailed study of the influence
of the nonlocal term on the eigenvalues as a function of the
parameters $p$, $q$, $r$ and $D$ using perturbational methods.
We construct an
asymptotic approximation of the small eigenvalue $\lambda_\varepsilon$ of the perturbed
non-selfadjoint operator and show that $Re\lambda_\varepsilon >0$ for any finite $D$ and
for sufficiently small $\varepsilon$. We cover not only the usual known cases
$r=p+1$, or $r=2$, $1<p<5$ (see \cite{ward2}), but also $r=(p+3)/2$, and using
perturbational techniques, some wide areas around all these cases (see details
in \cite{gk1}).
Moreover, we show that there is a critical value $D(\varepsilon)$,
which is exponentially large w.r.t. $\varepsilon\in (0,\varepsilon_0)$,
such that $Re \lambda_\varepsilon >0$ for $D<D(\varepsilon)$ an may be negative
above $D(\varepsilon)$, implying that stability may be lost for too
large $D$. The same value for $D(\varepsilon)$ was obtained in \cite{ward2} by formal
asymptotic methods.
Finally,
we study in section \ref{contrac} the stability
of the spike solution  along the lines of \cite{gk1}.

\newpage %P% avoids the beginning of a section on the last line of a page

\section{A spike solution and linearization around it} \label{spike}


\subsection{Existence of a stationary one-spike solution and its
asymptotics} \label{subspike}
Let $(S(x), H(X))$ be a steady state of equations \ref{simplgm}, i.e.
\begin{equation} \label{takagi1}
\begin{gathered}
\varepsilon^2 S''-S +S^p H^{-q}=0\,,\quad S'(\pm 1)=0\,,\\
 H''-\delta^2 H +\delta^2 \mu^{-1}\varepsilon^{-1}
 S^r=0\,,\quad H'(\pm 1)=0\,,
\end{gathered}
\end{equation}
where $\delta^2:=\mu/D$.
We shall prove the existence of such a spike solution for any fixed
$\delta>0$ and all $\varepsilon$, $0<\varepsilon<\varepsilon_0(\delta)$ if
$p>1 ,r>1, q>0, qr\neq p-1$. By definition, the (single) spike solution
(or spike) is such a steady state for which $S(x)=O(1)$ as $\varepsilon\to 0$ in
a neighbourhood of the origin and $S(x)$ is exponentially small outside.

Let $h$ be the solution of the linear equation
$$
h''-\delta^2 h =-f,\; h'(\pm1)=0\,,
$$
then
$$
h(x)=\int_{-1}^1 {\widetilde G}_\delta (x,y)f(y)dy,
$$
where Green's function ${\widetilde G}_\delta$ is given by
\begin{equation} \label{green3}
  {\widetilde G}_\delta (x,y)=\frac{1}{\delta \sinh 2\delta}\cosh \delta(1+x)
\cosh\delta(1-y)\quad \mbox{if }x<y .
\end{equation}
This function is even: ${\widetilde G}_\delta (-x,-y)={\widetilde G}_\delta(x,y)$.
We can solve the second equation of the system \eqref{takagi1} using Green's
function and eliminate  $H$ from the first equation by
\[
H(x)=\frac{1}{\varepsilon}\int_{-1}^{1} G_\delta (x,y) S^r(y) dy ,
\]
where
\[
G_\delta:={\delta^2{\widetilde G}_\delta \over\mu},\quad
g_\delta:=G_\delta(0,0)=\frac{\delta \cosh^2 \delta}
{\mu \sinh 2\delta}={1\over 2\mu}+O(\delta^2).
\]
Hence the spike solution $S$ satisfies the
equation
\begin{equation} \label{spike0}
\varepsilon^2 S''(x)-S(x)+S^p(x) \Bigl(
\frac{1}{\varepsilon}\int_{-1}^{1} G_\delta (x,y) S^r(y) dy\Bigr)^{-q}=0,\quad
S'(\pm1) =0.
\end{equation}
It should be positive; however, it is more convenient
first to construct a solution of the equation
$$
\varepsilon^2 S''(x)-S(x)+|S(x)|^p \Bigl(
\frac{1}{\varepsilon}\int_{-1}^{1} G_\delta (x,y) |S(y)|^r dy\Bigr)^{-q}=0,
\quad S'(\pm1)=0
$$
and prove a posteriori that this solutionis positive and hence coincides with
the solution of \eqref{spike0}.
In order to find the limit as $\varepsilon \to 0$ we use the stretched variable
$\xi=x/\varepsilon$. Setting $\varphi_{\varepsilon,\delta}(\xi):=S(\varepsilon\xi)$ we find
\begin{equation} \label{spike1}
\begin{gathered}
\varphi_{\varepsilon,\delta}''(\xi)-\varphi_{\varepsilon,\delta}(\xi)+|\varphi_{\varepsilon,
\delta}(\xi)|^p \Bigl(\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta (\varepsilon\xi,\varepsilon\eta) |
\varphi_{\varepsilon,\delta}(\eta)|^r d\eta \Bigr)^{-q}=0\,,
 \\
 \varphi_{\varepsilon,\delta}'(\pm 1/\varepsilon)=0\,.
\end{gathered}
\end{equation}
Taking the (formal) limit $\varepsilon \to 0$ we get the equation
\begin{equation} \label{limspike}
\varphi_{0,\delta}''(\xi)-\varphi_{0,\delta}^{}(\xi)+\varphi_{0,\delta}^p(\xi)
\Bigl(
\int_{-\infty}^{\infty} g_\delta^{}  \varphi_{0,\delta} ^r (\eta)
d\eta\Bigr)^{-q}=0\,, \quad \varphi_{0,\delta}'(\pm \infty)=0\,.
\end{equation}
Thus
$$\varphi_{0,\delta}^{} =w_p\,\Bigl(
\int_{-\infty}^{\infty} g_\delta^{}  w_p ^r (\eta)
d\eta\Bigr)^{-\alpha_r}\,, \quad \alpha_r=\frac{q}{1-p+rq}\,,
$$
where $w_p$ satisfies
\begin{equation} \label{eq4s1}
w_p''-w_p^{}+w_p^p=0\,, \quad
w_p'((\pm \infty)=0\,.
\end{equation}
For all $p>1$ this equation happens to have the closed form solution,
cf. \cite{gk1},
\begin{equation} \label{eq5s1}
w_p(\xi):={\textstyle\left(\frac{p+1}{2}\right)}^{1\over p-1}
\left(\cosh({\textstyle{\frac{p-1}{2}}}\,\xi)\right)^{-{2\over p-1}}\,,
\end{equation}
which for large $|\xi|$  has the asymptotic behaviour
\begin{equation} \label{eq6s1}
w_p(\xi)=\alpha\,e^{-|\xi|}\,(1+O(e^{-(p-1)|\xi|}))\,, \quad
\alpha:=(2p+2)^{1\over p-1}\,.
\end{equation}
Now we want to solve the equation \eqref{spike1} for all $\delta>0$. To this
end we introduce an extra parameter $\nu\leq \varepsilon$ in the non-linear part of
\eqref{spike1} defining
\begin{equation} \label{modspike0}
Q_\nu[\varphi](\xi)   :=|\varphi(\xi)|^p \Bigl(
\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta (\nu\xi,\nu\eta) |\varphi(\eta)|^rd\eta\Bigr)^{-q}
\end{equation}
and rewrite \eqref{spike1} in the form
\begin{equation} \label{modspike1}
\varphi''(\xi)-\varphi(\xi)+Q_\varepsilon[\varphi]=0,\;\;\;
 \varphi'(\pm 1/\varepsilon)=0.
\end{equation}
Setting the parameter $\nu$ to zero, we get a simplified equation, that we
shall call the corresponding
{\it shadow equation} (and which differs from Takagi's \cite{takagi} by a
multiplicative constant):
$$
\widetilde\varphi''(\xi)-\widetilde\varphi(\xi)+Q_0[\widetilde\varphi]=0,\;
\widetilde\varphi'(\pm 1/\varepsilon)=0.
$$
The solution is given  by
\begin{equation} \label{shad2}
\widetilde\varphi:=\Big(\int_{-1/\varepsilon}^{1/\varepsilon} g_\delta
\psi_\varepsilon^r(\eta)d\eta\Big)^{-\alpha_r}\psi_\varepsilon,
\end{equation}
where
\[
\psi_\varepsilon''-\psi_\varepsilon+\psi_\varepsilon^p=0,\quad
\psi_\varepsilon'(\pm1/\varepsilon)=0.
\]
This equation has a unique solution with a single spike located in the interior
of the domain; its properties are well known \cite{gk1} Section 2.1 and
\cite{takagi}. Thus we have constructed a shadow spike solution for any
fixed $\delta>0$,
which coincides with the shadow solution from \cite{gk1}, \cite{takagi} if
$\delta=0$.

The main idea is to find a solution of the problem \eqref{modspike1} in a small
neighbourhood of our shadow spike solution $\widetilde\varphi$.
If $\varphi=\widetilde\varphi+u$ we get an equation for $u$:
\begin{equation} \label{int1a}
\begin{gathered}
u''-u+\{Q_0[{\widetilde \varphi}+u]-
Q_0[\widetilde\varphi]\}+\{Q_\varepsilon[\widetilde\varphi+u]-
Q_0[\widetilde\varphi+u]\}=0\,,\\
u'(\pm 1/\varepsilon)=0\,.
\end{gathered}
\end{equation}
Using the Taylor formula we can write
\begin{equation} \label{int2}
\begin{gathered}
Q_0[\widetilde\varphi+u]-Q_0[\widetilde\varphi]=Q'_0[\widetilde\varphi]u+f(u)\,,\\
f(u):=\int_0^1\{\partial_\sigma Q_0[\widetilde\varphi+\sigma u]-
\partial_\sigma Q_0[\widetilde\varphi+\sigma u]_{\sigma=0}\} d\sigma,
\end{gathered}
\end{equation}
and
\begin{equation} \label{int2a}g_\varepsilon(u):=
Q_\varepsilon[\widetilde\varphi+u]-Q_0[\widetilde\varphi+u]=
\int_0^\varepsilon \partial_\nu Q_\nu[\widetilde\varphi+ u] d\nu,
\end{equation}
where $Q'_0[\widetilde\varphi]$ is the (non-local) linear operator
\begin{align*}
Q'_0[\widetilde\varphi] u
&:=p{\widetilde\varphi}^{p-1} u
\Big(\int_{-1/\varepsilon}^{1/\varepsilon} g_\delta {\widetilde\varphi}^r(\eta)d\eta\Big)^{-q}\\
&\quad -rq{\widetilde\varphi}^p
\Big(\int_{-1/\varepsilon}^{1/\varepsilon} g_\delta {\widetilde\varphi}^r(\eta)d\eta\Big)^{-q-1}
\int_{-1/\varepsilon}^{1/\varepsilon} g_\delta {\widetilde\varphi}^{r-1}(\eta)u(\eta)d\eta.
\end{align*}
The linear part of the operator in equation \eqref{int1a} is given by
$\widetilde A $,
\begin{align*}
\widetilde A u&:=-u''+u-Q_0'[\widetilde\varphi]u\\
&=-u''+u-p\widetilde\varphi^{p-1}
\Bigl(\int_{-1/\varepsilon}^{1/\varepsilon}
g_\delta\widetilde\varphi^r(\eta)d\eta\Bigr)^{-q}u\\
&\quad + rq\widetilde\varphi^p
\Bigl(\int_{-1/\varepsilon}^{1/\varepsilon} g_\delta\widetilde\varphi^r(\eta)d\eta\Bigr)^{-q-1}
\int_{-1/\varepsilon}^{1/\varepsilon} g_\delta\widetilde\varphi^{r-1}(\eta)u(\eta)d\eta,
\end{align*}
or
\begin{equation} \label{shad1a}
\widetilde A u=-u''+u-p\psi^{p-1}u+\frac{rq  \psi_\varepsilon^p
\langle  u,\psi_\varepsilon^{r-1}\rangle}{\langle  1,\psi_\varepsilon^{r}\rangle}.
\end{equation}
It is equal to the operator associated with the shadow equation as in
\cite[eqs. (1.3), (2.1), (2.20), (2.21)]{gk1}. In \cite{gk1} it is shown that
$\widetilde L u:=-u''+u-p\psi^{p-1}u $ restricted to even functions is
invertible in $L^2$. Now we remark that $\widetilde A$ is also invertible if
$qr\neq p{-}1$. Indeed, it is sufficient to show that zero is not an eigenvalue
of $\widetilde A$. Suppose, on the contrary, that $\widetilde A u=0$. Then, since
$\widetilde L \psi_\varepsilon =(1-p)\psi_\varepsilon^p$, we see that $u=c\psi_\varepsilon$ and
$\widetilde A \psi_\varepsilon =(1-p+qr)\psi_\varepsilon^p\neq 0$. Thus
 $\widetilde A^{-1}$, restricted to even functions,
 is a bounded operator in $L^2$, uniformly w.r.t. $\varepsilon$ (cf. \cite{gk1}).
Here $L^2$ is the space of quadratically integrable functions on the interval
$(-1/\varepsilon,1/\varepsilon)$. Let $H^2$ be  the associated Sobolev
space, equipped with  the usual norm $\|u\|_2:=\|u''\|+\|u\|$.
Since $\|u\|_2 \asymp \|(A+c)u\|$ for some large constant $c>0$, we conclude
that $\widetilde A^{-1}$, restricted to even functions, is a bounded operator
from $L^2$ to $H^2$, uniformly w.r.t. $\varepsilon$.
In this way we reduce the problem \eqref{int1a}  to the integral equation
\begin{equation} \label{int1}
u = M u,\quad \mbox{where}\quad
Mu:= {\widetilde A}^{-1}[f(u)+g_\varepsilon(u)]\,,
\end{equation}
with $f$ and $g$ as defined in \eqref{int2}, \eqref{int2a}.


We are going to apply the contraction method in the ball
\begin{align*}
X_\varepsilon:=\{&u\in H^2(-1/\varepsilon,1/\varepsilon) :
\mbox{$u$ is even, },
u'(\pm1/\varepsilon)=0\,,\\
&\|u\|_\omega:=\|u\|_2+\max |u(\xi)|/\omega(\xi)\leq
\varrho\}\,,
\end{align*}
where $0<\varepsilon<\varepsilon_0(\delta)$ and where $\omega(\xi)$ is the weight
function
\begin{equation} \label{shad2aa}
\omega(\xi):=\begin{cases}
e^{-(p-1)|\xi|}& \mbox{if } 1<p<2\,,\\
(1+|\xi|)e^{-|\xi|}& \mbox{if } p=2\,,\\
 e^{-|\xi|}& \mbox{if } p>2\,. 
 \end{cases}
\end{equation}
Since by \cite[eq. (2.11)]{gk1},
\begin{equation} \label{shad2a}
|\psi_\varepsilon(\xi)-w_p(\xi)|\leq c\,e^{-1/\varepsilon},\quad  |\xi|\leq 1/\varepsilon,
\end{equation}
we can find a constant $\xi_\varepsilon=\log(C/\varepsilon)$ (where $C$ is a generic positive constant
in the sequel)
so that $\psi_\varepsilon>\varepsilon^\kappa$ on $[-\xi_\varepsilon,\xi_\varepsilon]$, where $\kappa$
satisfies $\max(1/2,1/r)<\kappa<1$. Then
$$
\widetilde\varphi(\xi)>C\,g_\delta^{-\alpha_r} \varepsilon^\kappa,\quad\mbox{if }
|\xi|\le\xi_\varepsilon\,.
$$
Therefore, choosing
\begin{equation} \label{choice1}
\varrho:=C\,g_\delta^{-\alpha_r} \varepsilon^\kappa,
\end{equation}
we get
$$\widetilde\varphi+\sigma u>0
\quad \mbox{on }[-\xi_\varepsilon,\xi_\varepsilon] \mbox{ for any }
 u\in X_\varepsilon \mbox{ and } 0<\sigma<1.
$$
Hence
\begin{align*}
V&:= \int_{-1/\varepsilon}^{1/\varepsilon} G_\delta(\nu\xi,\nu\eta)|\widetilde
\varphi(\eta)+\sigma u(\eta)|^r d\eta>\\
&> \int_{-\xi_\varepsilon}^{\xi_\varepsilon} G_\delta(\nu\xi,\nu\eta)(\widetilde \varphi(\eta)
+\sigma u(\eta))^r d\eta-
\int_{|\xi|>\xi_\varepsilon} G_\delta(\nu\xi,\nu\eta)|\widetilde \varphi(\eta)
+\sigma u(\eta)|^r d\eta.
\end{align*}
Since $\widetilde \varphi +\sigma u>C g_\delta^{-\alpha_r}$ on $[-1,1]$, since
$$
\frac{g_\delta}{\cosh^2 \delta} =\frac{\delta}{\mu\sinh 2\delta}\leq G_\delta(x,y)
\leq 2g_\delta,\quad \mbox{if } -1\leq x,y\leq 1\,,
$$
and since $\xi_\varepsilon>1$ for $\varepsilon<\varepsilon_0(\delta)$, we get
$$
 V>C g_\delta^{1-r\alpha_r} [1-C_\delta \varepsilon^{\kappa r-1}]\cosh^{-2} \delta
 >C g_\delta^{1-r\alpha_r} \cosh^{-2} \delta
$$
if $\kappa r-1>0$ and if $0<\varepsilon<\varepsilon_0(\delta)$. Therefore,
\begin{equation} \label{positiv1}
V>C_\delta\quad \mbox{if }  0<\varepsilon<\varepsilon_0(\delta),
\end{equation}
uniformly w.r.t. $\varepsilon$,
where the positive quantities $C_\delta$ and $\varepsilon_0(\delta)$ are equivalent
to $1$ w.r.t. $\delta$ on any compact interval $[0,\delta_0]$.
All other estimates below
will be uniform in the same sense.
We shall first prove the estimates for $ u\in X_\varepsilon$,
\begin{equation} \label{contr1}
\|f(u)\|\leq C_\delta \varrho^\gamma \|u\|_2^2
\end{equation}
and
\begin{equation} \label{contr2}
\|g_\varepsilon(u)\|\leq C_\delta \varepsilon  ,
\end{equation}
where here and below $\gamma$ is a generic positive number that depends on
$p$ and $r$.

To prove \eqref{contr1} we use the definition of $f(u)$ in \eqref{int2}
and write it as a sum of five terms
$f(u)=\sum _{j=1}^5 f_j(u)\,$, where
$$
\begin{aligned}
f_1(u):=&\, p u\int_{0}^{1}\langle |\widetilde\varphi+\sigma u|^{r},g_\delta\rangle^{-q}
\left[|\widetilde\varphi +\sigma u|^{p-1} \mathop{\rm sign}
 (\widetilde\varphi+\sigma u)-\widetilde\varphi^{p-1}\right]
\, d\sigma\,,\\
f_2(u):=&\, p \widetilde\varphi^{p-1} u \int_{0}^{1}\left[\langle  |\widetilde\varphi+
\sigma u|^{r},g_\delta\rangle^{-q}-
\langle \widetilde\varphi^r,g_\delta\rangle^{-q}\right]\, d\sigma\,,\\
f_3(u):=& {}-qr\int_{0}^{1}
\langle  |\widetilde\varphi+\sigma u|^r,g_\delta\rangle^{-q-1}
|\widetilde\varphi +\sigma u|^{p}\\
&\quad \quad \quad \quad \quad \times \langle |\widetilde\varphi+\sigma u|^{r-1}
\mathop{\rm sign} (\widetilde\varphi+\sigma u)-\widetilde\varphi^{r-1},g_\delta u\rangle
\, d\sigma\,,
\\
 f_4(u):=& {} -qr\int_{0}^{1}\langle  |\widetilde\varphi+\sigma u|^r,g_\delta\rangle^{-q-1}
[|\widetilde\varphi +\sigma u|^{p}-\widetilde\varphi^p] \langle
\widetilde\varphi^{r-1},g_\delta
u\rangle \, d\sigma\,,\\
f_5(u):=& {}-qr\int_{0}^{1}\left[\langle  |\widetilde\varphi+\sigma u|^r,
g_\delta\rangle^{-q-1}
-\langle \widetilde\varphi^r,g_\delta\rangle^{-q-1}\right]  \langle
\widetilde\varphi^{r-1},g_\delta
u\rangle \,\widetilde \varphi^p
\, d\sigma\,.
\end{aligned}
$$
Denote the second factor in the integrand of $f_1$
by
$$
f_0(u):=|\widetilde\varphi+\sigma u|^{p-1} \mathop{\rm sign}(\widetilde\varphi+
\sigma u) -\widetilde\varphi^{p-1}.
$$
For all $\varepsilon>0$, $0\le\sigma\le 1$ and all functions $u$ it satisfies
\begin{equation} \label{est0}
|f_0(u)|\leq \begin{cases}
2\min\{\widetilde\varphi^{p{-}2}| u|,| u|^{p-1}\}& \text{ if }1<p\le 2\\
2^{p-1} \max\{\widetilde\varphi^{p{-}2}| u|,| u|^{p-1}\}
&\text{if } p> 2
\end{cases}
\end{equation}
This is a consequence of the following inequalities:\\
 If $a>0$ and $0<b<1$ then
\begin{enumerate}
\item  $0 \leq (a+x)^b - a^b \leq bxa^{b-1}$ and
$0 \leq (a+x)^b - a^b \leq x^b$ for all $x\geq 0$
\item  $0 \leq a^b-(a-y)^b \leq y a^{b-1}$ and
$0 \leq a^b-(a-y)^b \leq y^b$ for all $0\leq y \leq a$
\item $0 \leq a^b + t^b \leq 2(a+t)^b$ and
$0 \leq a^b + t^b \leq 2a^{b-1}(a+t)$ for all $t \geq 0$.
\end{enumerate}
Note that the inequalities (1) and (2) above are sharp.\\
If $a>0$ and $b>1$ then
\begin{enumerate}
\item  $0 \leq (a+x)^b - a^b \leq
\begin{cases}
   2^bxa^{b-1}&\text{if } 0\le x\le a\,,\\
   2^bx^b&\text{if } x\ge a\,,
\end{cases}$
\item  $0 \leq a^b-(a-y)^b \leq by a^{b-1}$ if $0\leq y \leq a$
\item  $0 \leq a^b + t^b \leq (a+t)^b$ if $t \geq 0$
\end{enumerate}
Substituting $y=-x$ and $t=-a{-}x$, $b=p{-}1$, $a=\widetilde\varphi$ and
 $x=\sigma u$ this proves \eqref{est0}.
Restricting this inequality to functions $u\in X_\varepsilon$,
which are uniformly bounded by $\varrho\omega$ $\bigl($with $\varrho<1$, cf.
\eqref{choice1}$\bigr)$,
we find the estimate
\begin{equation} \label{est0a}
|f_0(u)|\leq C|u|^{\sigma_p},\quad  \sigma_p:=\min(1,p-1),\; u\in X_\varepsilon\,,
\end{equation}
for some $C>0$ not depending on $\delta$ or $\varepsilon$, cf. \eqref{positiv1}.
Essentially, the restriction $| u|\le\varrho\omega$ in this
inequality is necessary only if $p>2$.

Using \eqref{est0a}, \eqref{choice1} and the definitions of $f_j(u)$ we find the
following uniform estimates if $|u|\leq \varrho\omega$:
\setlength{\arraycolsep}{.2em}
\def\vsep#1{{\vrule height #1 depth 0pt width 0pt}}
\begin{eqnarray}
\label{est1}|f_1(u)|&\leq&\displaystyle
 C\,|u|^{1+\sigma_p}\leq C_\delta
\rho^{1+\sigma_p}\omega^{1+\sigma_p}
\leq C\, \rho^{1+\sigma_p}\widetilde\omega,
\\
&&\displaystyle\vsep{1.6em}
\mbox{where }\widetilde\omega(\xi):=\omega(\xi)\ \mbox{if}\  1<p<2\ \mbox{and}
\ \widetilde\omega(\xi):=e^{-|\xi|}\ \mbox{if}\ p\geq 2\,,\nonumber\hspace*{2em}
\\  \label{est2}
|f_2(u)|&\leq&\displaystyle\vsep{1.9em}
C\,\widetilde\phi^{p-1}|u|\rho\leq C\, \rho^{2}\omega
\widetilde\phi^{p-1}
\leq C\, \rho^{2}\widetilde\omega,
\\  \label{est3}
|f_3(u)|&\leq&\displaystyle\vsep{1.9em}
C\,(\widetilde\phi^{p}+|u|^p)\rho^{1+\sigma_r}\leq C\,
\rho^{1+\sigma_r}
\omega (\widetilde\phi^{p}+\omega^p)
\leq C\, \rho^{1+\sigma_r}\widetilde\omega,
\\  \label{est4}
|f_4(u)|&\leq&\displaystyle\vsep{1.9em}
C\, \rho |u|(\widetilde\phi^{p-1}+|u|^{p-1})\leq C\, \rho^{2}
\omega (\widetilde\phi^{p-1}+\omega^{p-1})
\leq C\, \rho^{2}\widetilde\omega,
\\  \label{est4a}
|f_5(u)|&\leq&\displaystyle\vsep{1.9em}
C\, \rho^2 \widetilde\phi^{p}
\leq C\, \rho^{2}\widetilde\omega,
\end{eqnarray}
These estimates are uniform w.r.t. $\xi \in [-\frac{1}{\varepsilon}$,
$\frac{1}{\varepsilon}]$, $\delta\in[0,\delta_0^{}]$ and
$0<\varepsilon\le\varepsilon_0^{}(\delta)$.
This proves \eqref{contr1} for $u\in X_\varepsilon$.


To prove \eqref{contr2}, we use the
estimates
\begin{equation} \label{est2a}
|\partial_x G_\delta|\leq \delta^2 \mu^{-1}\cosh 2\delta, \quad
|\partial_y G_\delta|\leq \delta^2 \mu^{-1}\cosh 2\delta,
\end{equation}
whence we get the uniform estimate
\begin{equation} \label{est4b}
\begin{aligned}
|g_\varepsilon(u)|&\leq
 C\varepsilon  (1+|\xi|)(\widetilde\varphi^p+|u|^p)\Bigl(1+
 \int_{-1/\varepsilon}^{1/\varepsilon} |\eta|\,|u|^r d\eta\Bigr)\\
 &\leq
 C\varepsilon  (1+|\xi|)(\widetilde\varphi^p+\omega^p)\leq
 C\varrho^{1+\gamma}\widetilde  \omega.
\end{aligned}
\end{equation}
Thus \eqref{contr2} follows.
The estimates \eqref{contr1}, \eqref{contr2} imply the uniform estimate for
$u\in X_\varepsilon$
\begin{equation} \label{contr3}
\|Mu\|_2\leq C_\delta \varrho^{1+\gamma},\quad 0<\varepsilon<\varepsilon_0(\delta).
\end{equation}
Now we prove that for $u\in X_\varepsilon$ the following uniform (pointwise) estimate
holds
\begin{equation} \label{contr1a}
| Mu| \leq C_\delta\varrho^{1+\gamma}\omega,\quad 0<\varepsilon<\varepsilon_0(\delta).
\end{equation}
To this end we write the equation \eqref{int1} in the form
\begin{equation} \label{int1c}
(-\partial^2+1)Mu=F(u),\quad F(u):=p\psi_\varepsilon^{p-1}Mu +B_\varepsilon(Mu)+f(u)+g_\varepsilon(u).
\end{equation}
We claim that if $u\in X_\varepsilon$ then
\begin{equation} \label{est1a}
|F(u)|\leq C_\delta\widetilde\omega \varrho^{1+\gamma}.
\end{equation}
Indeed, by \eqref{contr3} we know that $|Mu|\leq C_\delta\varrho^{1+\gamma}$, hence
$$
\psi_\varepsilon^{p-1}|Mu|\leq C_\delta\varrho^{1+\gamma}e^{-(p-1)|\xi|}\leq
 C_\delta\varrho^{1+\gamma}\widetilde\omega.
$$
 From estimates \eqref{est1}--\eqref{est4}, \eqref{est4a}, \eqref{est4b} and
 $|u|\leq \varrho\omega$ we get
$$
|B_\varepsilon(Mu)|\leq C_\delta\varrho^{1+\gamma}\widetilde\varphi^p\leq C_\delta
\varrho^{1+\gamma}\widetilde\omega.
$$
Hence \eqref{est1a} follows.
Further, from \eqref{int1c} we have that
$$
Mu(\xi)=\varepsilon^{-1}\int_{-1/\varepsilon}^{1/\varepsilon} {\widetilde G}_{1/\varepsilon} (\varepsilon,\xi,\varepsilon\eta)
F(u)(\eta)d\eta.
$$
Using the explicit formula \eqref{green3} and the estimate \eqref{est1a} we can
calculate the above integral and derive \eqref{contr1a}.
Finally, from \eqref{contr3} and \eqref{contr1a} it follows that $M$ maps
$ X_\varepsilon$ into
$X_\varepsilon$ uniformly for $0<\varepsilon<\varepsilon_0(\delta)$.

To prove that $M$ is a contraction we estimate the difference
$$
f(u_1)-f(u_2)=\int_0^1\left\{Q_0^{\prime}[\widetilde\varphi+u_\sigma]-
Q_0^{\prime}[\widetilde\varphi]\right\}
d\sigma (u_1-u_2),
$$
where  $u_\sigma:= u_2+\sigma(u_1-u_2)$.
As before we write this difference as a sum of five terms,
$f(u_1)-f(u_2)=\sum_{j=1}^5 g_j(u_1,u_2)$,
\begin{eqnarray*}
g_1(u_1,u_2)&:=&\displaystyle
p (u_1{-}u_2)\int_{0}^{1}\langle\, |\widetilde\phi{+}
u_\sigma|^{r},g_\delta\,\rangle^{-q}
\left\{|\widetilde\phi {+}u_\sigma|^{p-1} sign (\widetilde\phi{+}u_\sigma)-\widetilde\phi^{p-1}
\right\}d\sigma ,
\\
g_2(u_1,u_2)&:=&\displaystyle\vsep{1.9em}
p\widetilde\phi^{p-1}(u_1{-}u_2)  \int_{0}^{1}\left[\langle\, |\widetilde\phi+
u_\sigma|^{r},g_\delta\,\rangle^{-q}-
\langle\,\widetilde\phi^r,g_\delta\,\rangle^{-q}\right] d\sigma ,
\\
g_3(u_1,u_2)&:=&\displaystyle\vsep{1.9em}
-\,qr\int_{0}^{1}\langle\, |\widetilde\phi+u_\sigma|^r,g_\delta\,\rangle^{-q-1}
|\widetilde\phi +u_\sigma|^{p}\times{}
\cr&&\displaystyle\hspace*{6em}
 \langle\,|\widetilde\phi+u_\sigma|^{r-1}
sign (\widetilde\phi+u_\sigma)-\widetilde\phi^{r-1},g_\delta (u_1-u_2)\,\rangle
 d\sigma,\vsep{1.5em}
\end{eqnarray*}   %%% This is an artificial split of the array over two pages. These lines should
\begin{eqnarray*} %%% be replaced by "\cr\vsep{1.9em}" if both parts end up on one page!!
 g_4(u_1,u_2)&:=&\displaystyle
-\,qr\int_{0}^{1}\langle\, |\widetilde\phi+\sigma u|^r,g_\delta\,\rangle^{-q-1}
\bigl[|\widetilde\phi +u_\sigma|^{p}-\widetilde\phi^p\bigl]
 \langle\, \widetilde\phi^{r-1},g_\delta
(u_1-u_2)\,\rangle
 d\sigma,
\\
g_5(u_1,u_2)&:=&\displaystyle
-\,qr\int_{0}^{1}\bigl[\langle\, |\widetilde\phi{+}
u_\sigma|^r,g_\delta\,\rangle^{-q{-}1}
{-}\langle\,\widetilde\phi^r,g_\delta\,\rangle^{-q{-}1}\bigr]
\langle\, \widetilde\phi^{r-1},g_\delta
(u_1{-}u_2)\,\rangle \widetilde \phi^p
 d\sigma.\vsep{1.9em}
\end{eqnarray*}
Using \eqref{est0a}, \eqref{positiv1} we find the following uniform
(pointwise) estimates if both $u_1$ and $u_2$ are in $X_\varepsilon$
and satisfy $|u_j|\leq \varrho\omega$ (pointwise):
\begin{eqnarray}\label{est11}
|g_1(u_1,u_2)|&\leq&\displaystyle
C_\delta|u|^{\sigma_p}|u_1-u_2|\leq C_\delta
\rho^{\sigma_p}\omega^{\sigma_p}
|u_1-u_2|\leq C_\delta \rho^{\sigma_p}|u_1-u_2|\,\widetilde\omega/\omega\,,\hspace*{3em}
\\
|g_2(u_1,u_2)|&\leq&\displaystyle\vsep{1.9em}
C_\delta\widetilde\phi^{p-1}(\rho+\int_{-1/\varepsilon}^{1/\varepsilon}|u|^r d\xi)|u_1-u_2|
\nonumber\\  \label{est21}
&&\hspace*{3em}\leq\displaystyle\vsep{1.5em}C \rho |u_1-u_2| \widetilde\phi^{p-1}\leq
C_\delta \rho |u_1-u_2|\,\widetilde\omega/\omega,
\\
|g_3(u_1,u_2)|&\leq&\displaystyle\vsep{1.9em}
C_\delta(\widetilde\phi^{p}+|u|^p)
\int_{-1/\varepsilon}^{1/\varepsilon}|u|^{\sigma_r}|u_1-u_2| d\xi
\leq \nonumber\\  \label{est31}
&&\hspace*{3em}\leq\displaystyle\vsep{1.5em}C_\delta \rho^{\sigma_r}
\omega (\widetilde\phi^{p}+\omega^p)\|u_1-u_2\|
\leq C_\delta \rho^{\sigma_r}\|u_1-u_2\|\,\widetilde\omega,
\\
|g_4(u_1,u_2)|&\leq&\displaystyle\vsep{1.9em}
C_\delta  |u|(\widetilde\phi^{p-1}+|u|^{p-1})|\langle\,
\widetilde\phi^{r-1},
u_1-u_2\,\rangle|
\nonumber\\  \label{est41}
&&\hspace*{3em}\leq C_\delta \rho
\omega (\widetilde\phi^{p-1}+\omega^{p-1})\|u_1-u_2\|\leq
\displaystyle\vsep{1.5em}C_\delta \rho \|u_1-u_2\|\,\widetilde\omega,
\\  \label{est4a1}
|g_5(u_1,u_2)|&\leq&\displaystyle\vsep{1.9em}
C_\delta \rho \widetilde\phi^{p}|\langle\,\widetilde\phi^{r-1},
u_1-u_2\,\rangle|
\leq C_\delta \rho \|u_1-u_2\|\,\widetilde\omega.
\end{eqnarray}
Hence if $u\in X_\varepsilon$ we find the uniform estimate
\begin{equation} \label{contr5}
\|f(u_1)-f(u_2)\|\leq C_\delta \varrho \|u_1-u_2\|,\quad  0<\varepsilon<\varepsilon_0(\delta).
\end{equation}
Further we have
$$
g_\varepsilon(u_1)-g_\varepsilon(u_2)=\int_0^\varepsilon\int_0^1 \partial_\sigma\partial_\nu Q_\nu
[\widetilde\varphi+u_2+
\sigma(u_1-u_2)]d\sigma d\nu \,;
$$
defining
$$
Q_\nu[\varphi]:=|\varphi|^p q_\nu(\varphi)\,,\quad
 q_\nu(\varphi):=\Bigl(\int_{-1/\varepsilon}^{1/\varepsilon}
G_\delta(\nu\xi,\nu\eta)|\varphi(\eta)|^r d\eta\Bigr)^{-q}\,,$$
we find
\begin{gather*}
g_\varepsilon(u_1)-g_\varepsilon(u_2)=h_1(u_1,u_2)+h_2(u_1,u_2),\\
h_1(u_1,u_2)=|\widetilde\varphi+u_1|^p \int_0^\varepsilon[\partial_\nu
q_\nu(\widetilde\varphi+u_1)-\partial_\nu q_\nu(\widetilde\varphi+u_2)]d\nu,\\
h_2(u_1,u_2)=[|\widetilde\varphi+u_1|^p-|\widetilde\varphi+u_2|^p]
\int_0^\varepsilon \partial_\nu q_\nu(\widetilde\varphi+u_2) d\nu.
\end{gather*}
We have
$$
\partial_\nu q_\nu(\varphi)=-q\Bigl(\int_{-1/\varepsilon}^{1/\varepsilon}
G_\delta(\nu\xi,\nu\eta)|\varphi(\eta)|^r d\eta\Bigr)^{-q-1}
\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta(\nu\xi,\nu\eta)|\varphi(\eta)|^r d\eta.
$$
Hence,
\begin{align*}
&\partial_\nu q_\nu(\widetilde\varphi+u_1)-
\partial_\nu q_\nu(\widetilde\varphi+u_2)
={}-q\Bigl(\int_{-1/\varepsilon}^{1/\varepsilon}
G_\delta(\nu\xi,\nu\eta)|\widetilde\varphi(\eta)+u_1(\eta)|^r d\eta\Bigr)^{-q-1}\\
&{}\hspace*{8em}\quad \hfill\times \int_{-1/\varepsilon}^{1/\varepsilon}
\partial_\nu G_\delta(\nu\xi,\nu\eta)\bigl[|\widetilde\varphi(\eta)+u_1(\eta)|^r-
|\widetilde\varphi(\eta)+u_2(\eta)|^r\bigr]d\eta \\
&\quad -q\int_{-1/\varepsilon}^{1/\varepsilon}\!
\partial_\nu G_\delta(\nu\xi,\nu\eta)|\widetilde\varphi(\eta){+}u_2(\eta)|^r
d\eta
\Big[\Bigl(\int_{-1/\varepsilon}^{1/\varepsilon}
G_\delta(\nu\xi,\nu\eta)|\widetilde\varphi(\eta){+}u_1(\eta)|^r d\eta\Bigr)^{-q{-}1}
\\
&{}\hspace*{15em}\quad\times \Bigl(\int_{-1/\varepsilon}^{1/\varepsilon}
 G_\delta(\nu\xi,\nu\eta)|\widetilde\varphi(\eta){+}u_2(\eta)|^r d\eta\Bigr)
^{-q{-}1}\Big]
\end{align*} 
If $u_j\in X_\varepsilon$ then using \eqref{est2a}, \eqref{est0a} we get
\begin{align*}
&|\partial_\nu q_\nu(\widetilde\varphi+u_1)-
\partial_\nu q_\nu(\widetilde\varphi+u_2)|\leq\\
&\quad\quad\leq C_\delta \int_{-1/\varepsilon}^{1/\varepsilon}
(|\xi|+|\eta|)|u_1(\eta)-u_2(\eta)\bigl(\widetilde\varphi(\eta)^{r-1}
+|u_1(\eta)|^{r-1}+ |u_2(\eta)|^{r-1}\bigr)d\eta\\
&\quad\quad\quad +C_\delta \int_{-1/\varepsilon}^{1/\varepsilon}
(|\xi|+|\eta|)|u_1(\eta)-u_2(\eta)\bigl(\widetilde\varphi(\eta)^{r}
+|u_1(\eta)|^{r}+ |u_2(\eta)|^{r}\bigr)d\eta\,.
\end{align*}
Hence
\begin{align*} 
|h_1(u_1,u_2)|
&\leq C_\delta \varepsilon  |\widetilde\varphi+u_1|^p
(1+|\xi|) \int_{-1/\varepsilon}^{1/\varepsilon}
|u_1(\eta)-u_2(\eta)|(1+|\eta|)\\
&\hspace*{10em}\times \Bigl(\widetilde\varphi(\eta)^{r-1}+|u_1(\eta)|^{r-1}+
|u_2(\eta)|^{r-1}\Bigr)d\eta \\
&\leq C_\delta\varrho^{1+\gamma}(1+|\xi|)(\widetilde\varphi^p+\omega^p)\|u_1-u_2\|
\leq C_\delta\varrho^{\gamma}\|u_1-u_2\|\widetilde\omega\,.
\end{align*}
Analogously,
\begin{align*} 
|h_2(u_1,u_2)|
&\leq C_\delta \varepsilon  |u_1-u_2|
(1+|\xi|)\big(\widetilde\varphi^{p-1}+|u_1|^{p-1}+|u_2|^{p-1}\big)\\
&\hspace*{10em}\times \int_{-1/\varepsilon}^{1/\varepsilon}
(1+|\eta|)(\widetilde\varphi(\eta)^{r}+|u_2(\eta)|^{r})d\eta\\
&\leq  C_\delta\varrho^{1+\gamma}(1+|\xi|)(\widetilde\varphi^{p-1}
+\omega^{p-1})|u_1-u_2| 
\leq  C_\delta\varrho^{\gamma}|u_1-u_2|\widetilde\omega/\omega\,.
\end{align*}
Thus for $u_j\in X_\varepsilon$ we have the uniform (pointwise) estimate
\begin{equation} \label{contr6a}
|g_\varepsilon(u_1)-g_\varepsilon(u_2)|
\leq C_\delta\varrho^{\gamma}\|u_1-u_2\|\widetilde\omega+
\leq C_\delta\varrho^{\gamma}|u_1-u_2|\widetilde\omega/\omega\,.
\end{equation}
In particular,
\begin{equation} \label{contr7}
\|g_\varepsilon(u_1)-g_\varepsilon(u_2)\|
\leq C_\delta\varrho^{\gamma}\|u_1-u_2\|,\quad \mbox{if } 0<\varepsilon<\varepsilon_0(\delta).
\end{equation}
Hence \eqref{contr5} and \eqref{contr7} imply the uniform estimate
\begin{equation} \label{contr7a}
\|M(u_1)-M(u_2)\|_2
\leq C_\delta\varrho^{\gamma}\|u_1-u_2\|,\quad \mbox{if }0<\varepsilon<\varepsilon_0(\delta).
\end{equation}
Next we prove that for $u_j\in X_\varepsilon$ the uniform (pointwise) estimate holds
\begin{equation} \label{contr1b}
|Mu_1-Mu_2|\leq C_\delta \varrho^\gamma \omega
\|u_1-u_2\|_\omega,\quad \mbox{if } 0<\varepsilon<\varepsilon_0(\delta).
\end{equation}
To this end we use the equation \eqref{int1c} and write
$$
(-\partial^2+1)(Mu_1-Mu_2)=F(u_1,u_2),
$$
where
$$
 F(u_1,u_2):=p\psi_\varepsilon^{p{-}1}(M_1{-}Mu_2)
 {+}B_\varepsilon(Mu_1{-}Mu_2){+}f(u_1){-}f(u_2){+}
 g_\varepsilon(u_1){-}g_\varepsilon(u_2).
$$
We claim that
\begin{equation} \label{est1b}
|F(u_1,u_2)|\leq C_\delta \varrho^\gamma \widetilde\omega
\|u_1-u_2\|_\omega.
\end{equation}
Indeed, if $|u_j|\leq \varrho\omega$ then
$$
 \psi_\varepsilon^{p-1}|Mu_1-Mu_2|\leq C_\delta\varrho^\gamma \|u_1-u_2\| \widetilde\omega
$$
and
\begin{gather*}
 |f(u_1)-f(u_2)|\leq \sum_{j=1}^5 |g_j(u_1,u_2)|, \\
 |B_\varepsilon( Mu_1-Mu_2)|\leq C_\delta\varrho^\gamma \|u_1-u_2\| \widetilde\varphi^p
\leq C_\delta\varrho^\gamma \|u_1-u_2\| \widetilde\omega.
\end{gather*}
Thus estimate \eqref{est1b} follows from \eqref{est11}--\eqref{est41}, \eqref{est4a1},
\eqref{contr6a}.
Then using again the formula
$$
Mu_1(\xi)-Mu_2(\xi)=\varepsilon^{-1}\int_{-1/\varepsilon}^{1/\varepsilon} {\widetilde G}_{1/\varepsilon}
(\varepsilon,\xi,\varepsilon\eta) F(u_1,u_2)(\eta)d\eta
$$
we get \eqref{contr1b}
Finally, from \eqref{contr7a}, \eqref{contr1b} we find the uniform estimate for
$u_j\in X_\varepsilon$:
$$
\|Mu_1 -Mu_2\|_\omega \leq\frac12 \|u_1-u_2\|_\omega,\quad 0<\varepsilon<\varepsilon_0(\delta).
$$
Thus the problem \eqref{spike1} has unique solution in $X_\varepsilon$. Moreover,
this solution is positive because it also solves the integral equation
\begin{equation} \label{contr9}
\varphi=\frac{1}{\varepsilon}\int_{-1/\varepsilon}^{1/\varepsilon}{\widetilde G}_{1/\varepsilon} (\varepsilon\xi,
\varepsilon\eta) Q_\varepsilon[\varphi] d\eta.
\end{equation}
Below we need the asymptotic behaviour of the spike solution at the boundary:
\begin{equation} \label{asympt1}
\varphi(1/\varepsilon)=2\alpha\Bigl(\int_{-\infty}^\infty g_\delta
w_p^r(\eta)d\eta\Bigr)^{-\alpha_r} e^{-1/\varepsilon}(1+O(\varepsilon^\gamma)),\quad
 0<\varepsilon<\varepsilon_0(\delta).
\end{equation}
To prove this,  we notice first that the shadow spike $\widetilde\varphi$
has the same asymptotic behaviour. This follows from \eqref{shad2}, \eqref{shad2a} and since
(see  \cite{gk1})
$$
\psi_\varepsilon(1/\varepsilon)=2\alpha e^{-1/\varepsilon}(1+O(e^{-(p-1)/\varepsilon})),\quad p> 1,
\; \alpha:=(2p+2)^{\frac{1}{p-1}}.
$$
Hence it is sufficient to prove the estimate
\begin{equation} \label{est5}
|\varphi(1/\varepsilon)-\widetilde\varphi(1/\varepsilon)|\leq C_\delta \varepsilon^\gamma e^{-1/\varepsilon},\quad
0<\varepsilon<\varepsilon_0(\delta).
\end{equation}
We can estimate this difference using the integral equation \eqref{contr9}, where
$Q_\nu[\varphi]$ is defined by \eqref{modspike0}. We have (using Taylor's formula,
\eqref{positiv1}, \eqref{est2a}),
\begin{equation} \label{contr10}
Q_\nu[\varphi](\eta)=
Q_0[\varphi](\eta)+ (1+|\eta|)\varphi^p(\eta)\,O(\nu),\quad |\eta\|\leq 1/\varepsilon,
\end{equation}
where
$$
Q_0[\varphi](\eta)=\varphi^p(\eta) \Bigl(
\int_{-1/\varepsilon}^{1/\varepsilon} g_\delta  \varphi^r(\xi)d\xi\Bigr)^{-q}.
$$
Using also the estimate $|\varphi(\xi)-\widetilde\varphi(\xi)|\leq \varrho\omega(\xi)$,
$0<\varepsilon<\varepsilon_0(\delta)$, we find
$$
Q_0[\varphi](\eta)=
Q_0[\widetilde\varphi](\eta)+ |\varphi^p(\eta)-\widetilde\varphi^p(\eta)|\,O(1)
+ \varphi^p(\eta)\varrho\,.
$$
Hence
\begin{equation} \label{contr11}
\varphi(\xi)=\widetilde\varphi(\xi){ +}\frac{1}{\varepsilon}
\int_{-1/\varepsilon}^{1/\varepsilon}{\widetilde G}_{1/\varepsilon} (\varepsilon\xi, \varepsilon\eta)
\Bigl[|\varphi^{p}(\eta){-}\widetilde\varphi^{p}(\eta)|\,O(1)
+(1+|\eta|)\varphi^p(\eta)\,O(\varrho)\Bigr]d\eta\,.
\end{equation}
To estimate this integral,
 we need a better estimate of $\varphi$, namely
\begin{equation} \label{est8}
\varphi(\xi)=O_\delta(e^{-|\xi|}),\quad |\xi|\leq 1/\varepsilon,\;
0<\varepsilon<\varepsilon_0(\delta).
\end{equation}
Indeed $\varphi$ satisfies
the equation
$$
\varphi''=q \varphi,\; q=1-\varphi^{p-1}
\Bigl(\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta(\varepsilon\xi,\varepsilon\eta)\varphi^r(\eta)d\eta\Bigr)^{-q}.
$$
Since
$\varphi\leq \widetilde\varphi +\varrho\omega$,
it follows that
$$q(\xi)=1-O(e^{-\gamma|\xi|}),$$
hence applying the classical asymptotic theory we get \eqref{est8}.


On the other hand, to estimate the difference $\varphi^p -\widetilde\varphi^p$,
we use the estimates
$|\varphi-\widetilde\varphi|\leq \varrho\omega$ and
$|\varphi|,\;|\widetilde\varphi|\leq C_\delta e^{-|\eta|}$. Thus
$$
|\varphi^p(\eta)-\widetilde\varphi^p(\eta)|\leq C_\delta |\varphi(\eta)-\widetilde\varphi(\eta)|
e^{-(p-1)\eta} \leq C_\delta
|\varphi(\eta)-\widetilde\varphi(\eta)|^b
e^{-(p-b)\eta}
$$
if $0<b\leq 1$. We choose $b$ as follows: 
\begin{quote}$0<b<\frac{p-1}{2-p}$ if $1<p\leq 3/2$
and $b=1$ if $p>3/2$.\end{quote}
 Then
\begin{equation} \label{est6}
|\varphi^p(\eta)-\widetilde\varphi^p(\eta)|\leq C_\delta \varrho^b \omega^b
e^{-(p-b)\eta}.
\end{equation}
Returning now to \eqref{contr11} we estimate the integral by \eqref{est6}:
\begin{align*}
\frac{1}{\varepsilon}\int_{-1/\varepsilon}^{1/\varepsilon}&{\widetilde G}_{1/\varepsilon} (1,
\varepsilon\eta)|\varphi^p(\eta)-\widetilde\varphi^p(\eta)|d\eta\leq
\\
&\leq \frac{C_\delta\varrho^b}{\sinh 2/\varepsilon}\int_{-1/\varepsilon}^{1/\varepsilon}
\cosh(1/\varepsilon+\eta) e^{-|\eta|(p-b+b(p-1))} d\eta 
\leq C_\delta \varrho^b e^{-1/\varepsilon}
\end{align*}
if $1<p<2$. Analogous estimate is valid for $p\geq 2$.
Thus \eqref{contr11} becomes
$$
\varphi(1/\varepsilon)=\widetilde\varphi(1/\varepsilon)+O(\varrho^b e^{-1/\varepsilon}),
\quad 0<\varepsilon<\varepsilon_0(\delta),
$$
whence estimate \eqref{est5} follows.



\subsection{Linearization around the one-spike solution\label{linear}}
In order to study stability of the spike solution, we consider the
first variation of the system \eqref{simplgm} around
this solution. It is convenient to rewrite this system as one
equation, solving first the second equation
\begin{equation} \label{eq11s1}
\begin{gathered}
h(x,t)=\frac{1}{\varepsilon}\int_{-1}^1 G_\delta(x,y) u^r(y,t)dy,\\
 u_t =\varepsilon^2 u_{xx} -u +g(u), \\
u_x(\pm 1,t)=0,\quad u(x,0)=u_{0}(x),\quad u_{0}^{\prime}(\pm 1)=0\,,
\end{gathered}
\end{equation}
where
$$
g(u)=u^p \Bigl(\frac{1}{\varepsilon}\int_{-1}^1 G_\delta(x,y)u^r (y,t)dy\Bigr)^{-q}.
$$
Let $v$ be the variation around $S$; set $u(x,t)=S(x,\varepsilon)+v(x,t)$,
then $v$ satisfies the
equations
\begin{gather*}
v_t =\varepsilon^2 v_{xx} -v +g(S+v)-g(S),\\
v_x(\pm 1,t)=0,\quad v(x,0)=v_{0}(x):=u_{0}(x)-S(x,\varepsilon),
\end{gather*}
or written in operator form
\begin{equation} \label{eq13s1}
v_t + A v =f[v]\,,\quad v(x,0)=v_{0}(x)\,,
\end{equation}
where $f$ is the quadratic term
\begin{equation} \label{eq14s1}
f[v]:=\int_{0}^1 (1-\sigma)\,\partial_\sigma^2\,g(S+\sigma v)
\,d\sigma
\end{equation}
and $\partial_\sigma^2$ denotes the
second derivative of $\sigma \mapsto g(S+\sigma v)$ w.r.t. $\sigma$ and
where $A=L+B$ is the spatial linear
operator,
\begin{equation} \label{eq15s1}
\begin{gathered}
Lv:=-\varepsilon^2 v''+v-pvS^{p-1}\Big(\frac{1}{\varepsilon}\int_{-1}^1
G_\delta(x,y) S^r(y)dy\Big)^{-q},\\
Bv=qr \Bigl(\frac{1}{\varepsilon}\int_{-1}^1 G_\delta(x,y)
S^r(y)dy\Bigr)^{-q-1}
\Bigl(\frac{1}{\varepsilon}\int_{-1}^1 G_\delta(x,y)
S^{r-1}(y)v(y)dy\Bigr) S^p
\end{gathered}
\end{equation}
defined on the Sobolev space $H^2 (-1,1)$ with boundary conditions
$v'(\pm 1)=0$.


For the study of the spectrum and the study of stability using this spectrum
it is convenient to stretch the spatial variable by $x=\varepsilon\,\xi$ and to define the
operator $A_\varepsilon =L_\varepsilon +B_\varepsilon$ on the stretched interval
$[-1/\varepsilon,1/\varepsilon]$, where $(\dot u:=du/d\xi)$,
\begin{gather*}
L_\varepsilon u:=-\ddot u+u-pu\varphi_\varepsilon^{p-1}\Bigl(\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta
(\varepsilon\xi,\varepsilon\eta) \varphi_\varepsilon^r(\eta)d\eta\Big)^{-q},\\
\mathcal{D}(L_\varepsilon):=\{u \in H^2([- 1/\varepsilon, 1/\varepsilon]):
\dot u(\pm 1/\varepsilon)=0\}
\end{gather*}
Here and later on we write for simplicity $\varphi_\varepsilon$ instead of
$\varphi_{\varepsilon,\delta}$.

The non-local operator $B_\varepsilon$  is defined by
$$
B_\varepsilon v =qr\varphi_\varepsilon^p
\Bigl(\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta(\varepsilon\xi,\varepsilon\eta)
\varphi_\varepsilon^r(\eta)d\eta\Bigl)^{-q-1}
\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta(\varepsilon\xi,\varepsilon\eta)
\varphi_\varepsilon^{r-1}(\eta)v(\eta)d\eta.
$$
We can calculate the ``limiting'' operators
$$
L_0\,u:=-\ddot u+u-pw_p^{p-1} u,\quad
B_0 u=\frac{qr}{\int_{-\infty}^{\infty} w_p^r(\xi)d\xi}\langle  u, w_p^{r-1}
\rangle w_p^p.
$$
Finally, we can evaluate the integral $\beta_m:= \int_{-
\infty}^{\infty} w_p^m(\xi)\,d\xi $ in terms of the Gamma
function,
$$
\beta_m= {(\frac{p+1}{2})}^{\frac{m}{p-1}}
{2\over p-1} {\sqrt{\pi}\Gamma(\frac{m}{p-1})\over\Gamma(\frac{m}{p-1}+\frac12)}\,.
$$
In section \ref{reference} we shall study the spectrum of $L_\varepsilon$.
 In section \ref{perturb} we study the way in which the spectrum of $L_\varepsilon$
is shifted by adding $B_\varepsilon$.

\section{The spectrum of the differential operator
 \label{reference}}

The eigenvalues of a selfadjoint differential operator
$L:=-d^2/dx^2+Q(x)$ with domain $\mathcal{D}(L)$ of functions on a
bounded or unbounded interval $I\subset\mathbb{R}$ satisfy the
 minimax property,
see \cite[theorem~XIII.1,~p.~76]{reed}.
If $L$ has isolated eigenvalues
$\lambda_{0}\le\lambda_1\le\lambda_2\le\dots$, ordered in increasing
sense and counted according their multiplicity (and below the
continuous spectrum if present), these satisfy
\begin{equation} \label{minmax}
\lambda_k=\inf_{E\subset \mathcal{C},\; \dim(E)\ge k+1} \max_{u\in
E,\; \|u\|=1} \langle  L u, u\rangle\,,
\end{equation}
where
$\langle \cdot,\cdot\rangle$ denotes the inner product and
where $\mathcal{C}$ is the domain of the operator.


The operator $L_\varepsilon$ of our study
with ``potential''
$$
Q:=1-p\,\varphi_\varepsilon^{p-1}\Bigl(\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta(\varepsilon\xi,\varepsilon\eta)
\varphi_\varepsilon^r(\eta)d\eta\Bigr)^{-q}
$$
is a selfadjoint differential operator
bounded from below and it has a discrete spectrum consisting of
eigenvalues of multiplicity one for each $\varepsilon>0$\,:
$\lambda_{0}(\varepsilon)<\lambda_1(\varepsilon)<\lambda_2(\varepsilon)<\dots$
with corresponding eigenfunctions $\psi_{0}(\cdot,\varepsilon)$,
$\psi_1(\cdot,\varepsilon)$, $\psi_2(\cdot,\varepsilon)$, \dots.
 Its spectrum converges for $\varepsilon\to 0$ (and for all selfadjoint
boundary conditions) to the spectrum of $L_{0}$, see
e.g.~\cite[ch.~9]{codd}.
We shall calculate the rate of convergence.


The ``limiting'' operator $L_{0}$ (on the whole real axis) has
the continuous spectrum $[ 1,\infty)$ and may have discrete
eigenvalues below this interval (see \cite[p.~140]{henry}).
Simple calculations show:
\begin{equation} \label{eq1s2}
\begin{array}{lll}
\psi_{\rm o}:=w_p^{p+1\over 2}&L_{\rm o}\,\psi_{\rm o}=
-\frac14(p-1)(p+3)\,\psi_{\rm o}\,,\hspace{2em}&p>1\,,\\
\psi_1:=\dot w_p&L_{\rm o}\,\psi_1=0\,&p>1\,,\vsep{1.7em}\\
\psi_2:=w_p^{3-p\over 2}-\frac12{\textstyle{p+3\over p+1}}\,w_p^{p+1\over 2}\hspace{1em}&
L_{\rm o}\,\psi_2
=\frac14 (p-1)(5-p)\,\psi_2\,,~~&1<p<3\vsep{1.7em}\,.
\end{array}
\end{equation}
Since $\psi_{0},~\psi_1$ and $\psi_2$ have zero, one and two zeros respectively,
and since the zeros of the eigenfunctions of second order ordinary
differential operators interlace, $\lambda_{0}:=-\frac14 (p-1)(p+3)$,
$\lambda_1:=0$ and (if $p<3$) $\lambda_2:=\frac14 (p-1)(5-p)$
are the three smallest eigenvalues of $L_{0}$.
In order to show that $L_{0}$ does not have a second isolated eigenvalue
for $p>3$, we substitute $\psi(\xi)=\vartheta({p-1\over 2}\xi) $
in the eigenvalue equation $L_{0}\psi=\lambda\psi$ using the explicit
form of $w_p$ from \eqref{eq5s1}. This yields the equation
$$
M_p \vartheta:=-\ddot\vartheta-2p(p+1)(p-1)^{-2}\cosh^{-2}(\eta)\vartheta
=({\textstyle{2\over p-1}})^2\,(\lambda-1)\vartheta=\mu \vartheta\,.
$$
Since the ``potential'' in $M_p$ is an increasing function of $p$,
its eigenvalues are increasing functions of $p$ by the minimax theorem
\eqref{minmax}. Since $\lambda_2\to1$ if $p\to 3$ from below, the second
eigenvalue of $M_p$ tends to zero for $p\nearrow3$
and gets absorbed into the continuous spectrum if $p\ge 3$.
So $L_{0}$ has only two eigenvalues below 1 if $p\ge 3$.

In order to compute the  rate of convergence of the smallest
eigenvalues $\lambda_{0}(\varepsilon)$ and $\lambda_1(\varepsilon)$
(and $\lambda_2(\varepsilon)$ if $p<3$)
of $L_\varepsilon$, we can use the technique of \cite{pdg}
and \cite{gk}. We compute (formally) approximate eigenfunctions
and project them onto the true eigenfunctions; the residuals
yields estimates for the eigenvalues.

Let $\widetilde L_\varepsilon, \widetilde B_\varepsilon$ be the corresponding operators
resulting in linearization around the shadow spike solution
$\widetilde\varphi_\varepsilon$.
(We use the  notation $\widetilde\varphi_\varepsilon$ for $\widetilde\varphi$.)
More precisely (see \eqref{shad1a}) we have,
\[
\widetilde L_\varepsilon\,u:=-\ddot u+u-p\,\widetilde\varphi_\varepsilon^{p-1}\,
\Big(\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta(0,0)
\widetilde\varphi_\varepsilon^r(\eta)d\eta\Big)^{-q}u =
-\ddot u+u-\psi_\varepsilon^{p-1} u,
\]
\begin{align*}
\widetilde B_\varepsilon v &:=qr \widetilde\varphi_\varepsilon^p
\Big(\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta(0,0)
\widetilde\varphi_\varepsilon^r(\eta)d\eta\Big)^{-q-1}
\Big(\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta(0,0)
\widetilde\varphi_\varepsilon^{r-1}(\eta)v(\eta)d\eta\Big)\\
&= qr\psi_\varepsilon^p \frac{\langle  u,\psi_\varepsilon^{r-1}\rangle}
{\langle \psi_\varepsilon^r,1\rangle}.
\end{align*}
Thus the operator $\widetilde A_\varepsilon =\widetilde L_\varepsilon +\widetilde B_\varepsilon$
does not depend on $\delta$ and coincides with the shadow operator
from \cite{gk1}.
Since
$$
L_\varepsilon =\widetilde L_\varepsilon +O\bigl(|\varphi_\varepsilon^{p-1}-\widetilde\varphi_\varepsilon^{p-1}|+
\bigl[\varepsilon(1+|\xi|)+|\langle \varphi_\varepsilon^{r}-
\widetilde\varphi_\varepsilon^r,g_\delta\rangle|\bigr]
\widetilde \varphi_\varepsilon^{p-1}\bigr)$$
it follows the uniform estimate
\begin{equation} \label{oper3}
\|L_\varepsilon-\widetilde L_\varepsilon\| =O(\varepsilon^\gamma )\,,\quad 0<\varepsilon<\varepsilon_0(\delta).
\end{equation}
Here and later on the positive quantity $O(1)$ depends on $\delta$ and is
equivalent to $1$ on any compact interval $[0,\delta_0]$. All estimates
will be uniform in the same sense.
Analogously,
\begin{equation} \label{oper3a}
\|B_\varepsilon-\widetilde B_\varepsilon\| =O(\varepsilon^\gamma)\,,\quad 0<\varepsilon<\varepsilon_0(\delta).
\end{equation}
In particular, using the asymptotic behaviour  of the eigenvalues of $\widetilde L_\varepsilon$
\cite{gk1}, we find
$$
\lambda_0(\varepsilon)=\lambda_0 +O(\varepsilon^\gamma) \,,\quad
\lambda_2(\varepsilon)=\mu_0 +O(\varepsilon^\gamma )\,,
$$
where $\mu_0:=\lambda_2$ if $p<3$ and $\mu_0:=1$ if $p\geq 3$.


To find the asymptotic behaviour of the small eigenvalue $\lambda_1(\varepsilon)$ we shall
use $\dot\varphi_\varepsilon$ as an approximate eigenfunction. Differentiating
\eqref{spike1}, we get
$$
L_\varepsilon \dot\varphi_\varepsilon=-q\varepsilon\varphi_\varepsilon^p
\Big(\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta(\varepsilon\xi,\varepsilon\eta)
\varphi_\varepsilon^r(\eta)d\eta\Big)^{-q-1}
\int_{-1/\varepsilon}^{1/\varepsilon}\partial_x G_\delta(\varepsilon\xi,\varepsilon\eta)
\varphi_\varepsilon^r(\eta)d\eta.
$$
We evaluate this expression as follows. We have
$$
\partial_x G_\delta(\varepsilon\xi,\varepsilon\eta)=\pm \delta^2/2\mu
+\delta^2 \varepsilon(|\xi|+|\eta|)\,O(1),\quad \xi\neq \eta,
$$
where ``+" corresponds to the region $\xi<\eta$ and "--" corresponds to
$\xi>\eta$;
\begin{equation} \label{est10}
\Big(\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta(\varepsilon\xi,\varepsilon\eta)
\varphi_\varepsilon^r(\eta)d\eta\Big)^{-q-1}=
\Big(\int_{-1/\varepsilon}^{1/\varepsilon} g_\delta
\varphi_\varepsilon^r(\eta)d\eta\Big)^{-q-1}+O(\delta^2 \varepsilon (1+|\xi|))\,.
\end{equation}
Therefore,
$$
L_\varepsilon \dot\varphi_\varepsilon=-\frac{q\,\varepsilon\,\delta^2\varphi_\varepsilon^p }{2\mu}\,
\Big(\int_\xi^{1/\varepsilon} \varphi_\varepsilon^r(\eta)d\eta-
\int_{-1/\varepsilon}^\xi \varphi_\varepsilon^r(\eta)d\eta\Big)
+O(\delta^2 \varepsilon^2(1+|\xi|)
\varphi_\varepsilon^p) \,.
$$
In particular,
$$
\langle  L_\varepsilon \dot\varphi_\varepsilon,\dot\varphi_\varepsilon \rangle=
-\frac{q\,\varepsilon\,\delta^2}{2\mu\,(p+1)}
\int_{-1/\varepsilon}^{1/\varepsilon} \varphi_\varepsilon^{p+r+1}(\eta)\,d\eta\,
\Big(\int_{-1/\varepsilon}^{1/\varepsilon} g_\delta
\varphi_\varepsilon^r(\eta)d\eta\Big)^{-q-1}+O(\delta^2 \varepsilon^2 ).
$$
The asymptotic expansion of $\lambda_1(\varepsilon)$ will be calculated
using the same technique as in \cite{pdg} and \cite{gk}.
We compute an approximate eigenfunction $w$, $\|w\|=1$
 of the operator
$L_\varepsilon$ and we show that
\begin{equation} \label{ev9}
\langle  L_\varepsilon w,w\rangle =\nu_\varepsilon(1+O(\widetilde R_\varepsilon))\quad
\mbox{and}\quad \|L_\varepsilon w\|^2=O(\widetilde R_\varepsilon\,R_\varepsilon)\,,
\end{equation}
where $R_\varepsilon=o(1)$ and $\widetilde R_\varepsilon=o(1)$ for $\varepsilon\to 0$.

The generalized Fourier expansion of $w$ in
the true eigenfunctions $\{\psi_k: k=0,1,\dots\}$ of $L_\varepsilon$ is
$$
w=\sum_{k=0}^\infty c_k \psi_k\quad \mbox{with}\quad
\sum_{k=0}^\infty| c_k|^2=\|w\|^2=1\,.
$$
Since all eigenvalues of $L_\varepsilon$ except
$\lambda_1(\varepsilon)$ are uniformly bounded  away from $\lambda_1(0)=0$
by a distance $d>0$, we find from \eqref{ev9}
$$
1-|c_1|^2=\sum_{k=0\,,~k\ne1}^\infty |c_k|^2~\le~ d^{-2}\,
\sum_{k=0\,,~k\ne1}^\infty \lambda_k^2\,|c_k|^2~\le~
d^{-2}\,\|L_\varepsilon w\|^2=O(\widetilde R_\varepsilon\,R_\varepsilon)\,,
$$
implying that $|c_1|^2=1+O(\widetilde R_\varepsilon\,R_\varepsilon)$.
The estimate for the inner product in \eqref{ev9} now implies that
$$
\langle  L_\varepsilon w,w\rangle -\nu_\varepsilon=|c_1|^2\lambda_1(\varepsilon)-
\nu_\varepsilon+\sum_{k=0,\; k\ne1}^\infty \lambda_k |c_k|^2
=O\bigl(\widetilde R_\varepsilon(\nu_\varepsilon + R_\varepsilon)\bigr)
$$
and hence that
$$
\lambda_1(\varepsilon)=\nu_\varepsilon+O\bigl(\widetilde R_\varepsilon(\nu_\varepsilon+ R_\varepsilon )\bigr).
$$
Let $\psi_1(\cdot,\varepsilon)$ be the true eigenfunction of $L_\varepsilon$
corresponding to $\lambda_1(\varepsilon)$. We look for an approximate eigenfunction
of the form $\psi_1(\cdot,\varepsilon)\approx\dot\varphi_\varepsilon+$
boundary layer corrections. Within the interval $[-1/\varepsilon,1/\varepsilon]$
the tails of $\dot\varphi_\varepsilon$ are exponentially small by \eqref{asympt1} and
\eqref{spike1}, \begin{equation} \label{eq4s2}
\dot\varphi_\varepsilon({\pm 1\over\varepsilon})=0 \quad \mbox{and}\quad
\ddot\varphi_\varepsilon(\pm {1\over\varepsilon})=
a e^{-{1/\varepsilon}}(1+O(\varepsilon^\gamma)),\quad 0<\varepsilon<\varepsilon_0(\delta),\;\gamma>0,
\end{equation}
where $a:=2\alpha(\int_{-\infty}^\infty g_\delta w_p^r(\eta)d\eta)^{-\alpha_r}$.
\vrule height 0pt depth 10pt width 0pt.


We construct boundary layer terms at both endpoints
by standard matched asymptotic
expansions. Suitable boundary layer corrections at
the right and left endpoints are
\begin{equation} \label{eq5as2}
\begin{gathered}
h(\xi):={}-\ddot\varphi_\varepsilon({\textstyle{1\over\varepsilon}}) \varrho(\varepsilon\xi)
\,\exp\big(\xi-{1\over\varepsilon})\big)\,, \\
k(\xi):= \ddot\varphi_\varepsilon({-\textstyle{1\over\varepsilon}})\,\varrho(-\varepsilon\xi)
\,\exp\big(-\xi-{1\over\varepsilon}) \big)\,, \\
\widetilde\psi_1 :=\dot\varphi_\varepsilon+h+k
\end{gathered}
\end{equation}
where $\varrho$ is a monotonic $\mathcal{C}^\infty$ cut-off function satisfying
$\varrho(x)=1$ if $x\ge 3/4$ and $\varrho(x)=0$ if
$x\le 1/2$.
 From the definition it is clear that
$\widetilde\psi_1$ satisfies the boundary conditions at $\xi={\pm 1/\varepsilon}$  and
$$
L_\varepsilon h={}-p \,\varphi_\varepsilon^{p-1}h
\Big(\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta (\varepsilon\xi,\varepsilon\eta)\varphi_\varepsilon^r
(\eta)d\eta\Big)^{-q}+\ddot\varphi_\varepsilon({\textstyle{1\over\varepsilon}})
\left(\varepsilon^2\varrho''+2\varepsilon\varrho'\right)\exp(\xi-
\textstyle{1\over\varepsilon})\,.
$$
For $p>1$ we have
\begin{equation} \label{eq6s2}
\|L_\varepsilon \widetilde\psi_1\|^2=\delta^4 \varepsilon^2+
O(e^{-(2+\gamma)/\varepsilon})\,, \quad 0<\varepsilon<\varepsilon_0(\delta).
\end{equation}
Further,
$$
\langle  L_\varepsilon \widetilde\psi_1,\widetilde\psi_1\rangle =
\langle  L_\varepsilon \dot\varphi_\varepsilon,\dot\varphi_\varepsilon \rangle -
\langle  L_\varepsilon (h+k), h+k \rangle -
[(\dot h +\dot k)\widetilde\psi_1]_{-1/\varepsilon}^{1/\varepsilon}.
$$
We can calculate the last two terms, hence
\begin{equation} \label{eq6bs2}
\langle  L_\varepsilon \widetilde\psi_1,\widetilde\psi_1\rangle =
\langle  L_\varepsilon \dot\varphi_\varepsilon,\dot\varphi_\varepsilon \rangle -
2[\ddot\varphi_\varepsilon(1/\varepsilon)]^2+e^{-(2+\gamma)/\varepsilon} O(1).
\end{equation}
On the other hand,
$$
\|\dot\varphi_\varepsilon\|^2=\Big(\int_{-\infty}^\infty g_\delta
w_p^r(\eta)d\eta\Big)^{-2\alpha_r}\int_{-\infty}^\infty
(\dot w_p)^2d\eta\,(1 +\varepsilon^\gamma\, O(1)).
$$
 Therefore, the above estimates show that
\begin{equation} \label{eq7as2}
\lambda_1(\varepsilon)=-a(\delta) \varepsilon -a_1 e^{-2/\varepsilon} +
(\delta^2 \varepsilon^{1+\gamma} + e^{-(2+\gamma)/\varepsilon})\,O(1),\quad
 0<\varepsilon<\varepsilon_0(\delta),
\end{equation}
where $a(\delta)>0,\; a(\delta)=\delta^2\, O(1)$ and
\begin{equation} \label{est12}
a_1=8\alpha^2 \Big(\int_{-\infty}^\infty (\dot w_p)^2d\eta\Big)^{-1}.
\end{equation}
In particular, for any fixed $\delta>0$ we have the asymptotic
$$
\lambda_1(\varepsilon)=-a(\delta) \varepsilon  +
 \varepsilon^{1+\gamma} \,O(1),\quad  0<\varepsilon<\varepsilon_0(\delta).
$$
Thus the small eigenvalue $\lambda_1(\varepsilon)$ of the differential operator
$L_\varepsilon$ is always negative. In contrast, in the next section we shall prove
that the small eigenvalue $\lambda_\varepsilon$ of the perturbed operator $A_\varepsilon$
is positive for any fixed $\delta>0$ if $0<\varepsilon<\varepsilon_0(\delta)$. If we allow
dependence of $\delta$ on $\varepsilon$, then $\lambda_\varepsilon$ is positive for all
$\delta>\delta(\varepsilon)$, where $\delta(\varepsilon)$ is exponentially small w.r.t.
$\varepsilon\in (0,\varepsilon_0)$. To prove these facts, we need two type of estimates:
for any fixed $\delta>0$ or for all small $\delta$.

\section{Perturbation of the spectrum by the non-local term
 \label{perturb}}

In this section we consider how the nonlocal operator $B_\varepsilon$ perturbs the
eigenvalues of $L_\varepsilon$.
 Since $\|B_\varepsilon\|=O(1)$ it follows that the spectrum of $A_\varepsilon$
lies in a strip around the real axis. Hence this is an operator with compact
resolvent and according to Kato, p. 237 \cite{kato}, its spectrum consists
of eigenvalues with finite multiplicity.


Our goal
is to find conditions on the parameters $p,q,r$
so that the spectrum of $A_\varepsilon$  lies in the
right half-plane. We shall prove that this is true under the same conditions
on the parameters $p,q,r$ as in the shadow case, cf. \cite{gk1}.


\subsection{ \label{perturb1}
Perturbation of the small eigenvalue by the non-local term}
In this subsection we consider how the non-local operator $B_\varepsilon$
 perturbs the small eigenvalue $\lambda_1 (\varepsilon)$ of
$L_\varepsilon$.
Both the operators $L_\varepsilon$ and $B_\varepsilon$ are invariant under
the change of  sign $\xi\mapsto{-}\xi$ and hence leave the subspaces of even
and odd functions invariant. Hence in this subsection we can consider the
operator $A_\varepsilon=L_\varepsilon +B_\varepsilon$ on the subspace of odd functions only. Then
$A_\varepsilon$ is a small perturbation of the selfadjoint operator $L_\varepsilon$.
Indeed, since
$$
B_\varepsilon = B_{0\varepsilon} +\delta^2 \varepsilon (1+|\xi|)\varphi_\varepsilon^p O(1),
$$
where
$$
B_{0\varepsilon} v =q\,r\,\varphi_\varepsilon^p \Big(\int_{-1/\varepsilon}^{1/\varepsilon} g_\delta
\varphi_\varepsilon^r(\eta)d\eta\Big)^{-q-1}
\int_{-1/\varepsilon}^{1/\varepsilon} g_\delta
\varphi_\varepsilon^{r-1}(\eta)v(\eta)d\eta\,,
$$
and
$ B_{0\varepsilon}=0$ on odd functions, it follows
$$
\|B_\varepsilon\|=\delta^2 \varepsilon O(1),\quad  0<\varepsilon<\varepsilon_0(\delta).
$$
Hence by  Kato \cite[p. 364]{kato} the spectrum of $A_\varepsilon$
(on odd functions) consists of one simple small eigenvalue $\lambda_\varepsilon$
(see \eqref{eq7as2}),
\begin{equation} \label{est14a}
\lambda_\varepsilon =(\delta^2 \varepsilon+e^{-2/\varepsilon})\,O(1),
\end{equation}
and eigenvalues close to the real axis and in the half plane $Re\lambda>1/2$.


Thus the problem is reduced to determine the sign of $Re\lambda_\varepsilon$. To
this end we shall find its asymptotic behaviour. This will be done in two steps. In
the first step we use the a priori estimate \eqref{est14a} and derive a better
estimate for $\lambda_\varepsilon$ (see \eqref{eigenv11a} below). To this end we use
the same technique as for the selfadjoint operator $L_\varepsilon$, exploiting the
fact that the non-selfadjoint operator $A_\varepsilon$ is a sufficiently small
perturbation of $L_\varepsilon$.

Let
$$
A_\varepsilon \psi_\varepsilon =\lambda_\varepsilon \psi_\varepsilon,\quad \|\psi_\varepsilon\|=1
$$
(the eigenfunction being odd one). As an approximate eigenfunction we use
the same function $\widetilde\psi$ as before: $\widetilde\psi=\dot\varphi_\varepsilon+h+k$.
Note that this is also an odd function.
Let
$$
\widetilde\psi=c\psi_\varepsilon +dg,\quad \|g\|=1,
\mbox{ with $g$ orthogonal to }\psi_\varepsilon .
$$
Then
\begin{equation} \label{eig1}
\|L_\varepsilon \widetilde\psi\|^2=|c|^2 \|L_\varepsilon \psi_\varepsilon\|^2 +|d|^2 \|L_\varepsilon g\|^2
\end{equation}
and $L_\varepsilon \psi_\varepsilon =\lambda_\varepsilon \psi_\varepsilon +\delta^2 \varepsilon \,O(1)$,
hence
$$\|L_\varepsilon \psi_\varepsilon\| =(\delta^2 \varepsilon+e^{-2/\varepsilon})\,O(1).
$$
On the other hand we already know that (see \eqref{eq6s2}),
\begin{equation} \label{eig2}
\|L_\varepsilon \widetilde\psi\| =(\delta^2 \varepsilon+e^{-(1+\gamma)/\varepsilon})\,O(1),\quad
p>1.
\end{equation}
Now we need the uniform estimate $\|L_\varepsilon g\|\geq C$. Suppose on the
contrary that $\|L_\varepsilon g\|=o(1)$ as $\varepsilon \to 0$. Let $L_\varepsilon \omega_1=
\lambda_1 \omega_1$, $\|\omega_1\|=1$. If $g=c_1\omega_1+d_1h_1$,
$|c_1|^2+|d_1|^2=1$ is the orthogonal decomposition, we find that
$g=\omega_1+o(1)$.
On the other hand, if $\psi_\varepsilon=c_2\omega_1+d_2h_2$, $|c_2|^2+|d_2|^2=1$
is the orthogonal decomposition of $\psi_\varepsilon$, then since
$\|L_\varepsilon \psi_\varepsilon\|=O(\lambda_1)$ we find that $\psi_\varepsilon=\omega_1
+O(\lambda_1)$. Then $\langle  g, \psi_\varepsilon \rangle=1+o(1)$, what contradicts
orthogonality of $g$ and $\psi_\varepsilon$.
Hence
\begin{equation} \label{est14}
|d| =(\delta^2 \varepsilon+e^{-(1+\gamma)/\varepsilon})\,O(1).
\end{equation}
Further,
$$
\langle  A_\varepsilon \widetilde\psi,\widetilde\psi\rangle =
|c|^2\lambda_\varepsilon +d{\bar c} \langle  A_\varepsilon g, \psi_\varepsilon \rangle +
|d|^2 \langle  A_\varepsilon g, g \rangle .
$$
Since
$$
\langle  A_\varepsilon g, \psi_\varepsilon \rangle =
\langle  L_\varepsilon g, \psi_\varepsilon \rangle +\delta^2 \varepsilon \,O(1)
$$
it follows
$$
\langle  A_\varepsilon g, \psi_\varepsilon \rangle = (\delta^2 \varepsilon +e^{-2/\varepsilon})\, O(1).
$$
On the other hand, \eqref{eig1}, \eqref{eig2} imply
$$
|d|\|A_\varepsilon g\| = (\delta^2 \varepsilon+e^{-(1+\gamma)/\varepsilon})O(1).
$$
Therefore,
$$
\langle  A_\varepsilon \widetilde\psi,\widetilde\psi\rangle
= \lambda_\varepsilon \|\widetilde\psi\|^2
+(\delta^4 \varepsilon^2 +e^{-(2+\gamma)/\varepsilon})\,O(1).
$$
To evaluate the quadratic form we write
$$
\langle  A_\varepsilon \widetilde\psi,\widetilde\psi\rangle =
\langle  A_\varepsilon \dot\varphi_\varepsilon,\dot\varphi_\varepsilon\rangle -[(\dot h+\dot k)\widetilde\psi]
_{-1/\varepsilon}^{1/\varepsilon} -
\langle  L_\varepsilon (h+k), h+k\rangle +
\langle  B_\varepsilon \dot\varphi_\varepsilon, h+k\rangle .
$$
Since
$$
\langle  B_\varepsilon \dot\varphi_\varepsilon, h+k\rangle =\delta^2 e^{-2/\varepsilon}\, O(1),
\quad p>1,
$$
we get from above estimates
\begin{equation} \label{rel2}
\lambda_\varepsilon =\frac{\langle  A_\varepsilon \dot\varphi_\varepsilon,\dot\varphi_\varepsilon\rangle}
{\|\dot\varphi_\varepsilon\|^2} -\frac{2[\ddot\varphi_\varepsilon(1/\varepsilon)]^2}
{\|\dot\varphi_\varepsilon\|^2} +
(\delta^4 \varepsilon^2+e^{-(2+\gamma)/\varepsilon}+\delta^2 e^{-2/\varepsilon})\,O(1),
\end{equation}
where $0<\varepsilon<\varepsilon_0(\delta),\; p>1$,
and it remains to evaluate
$\langle  A_\varepsilon \dot\varphi_\varepsilon,\dot\varphi_\varepsilon\rangle$.
We have
$$
L_\varepsilon \dot\varphi_\varepsilon={}-q\,\varepsilon\,\varphi_\varepsilon^p\,
\Big(\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta(\varepsilon\xi,\varepsilon\eta)
\varphi_\varepsilon^r(\eta)d\eta\Big)^{-q-1}
\int_{-1/\varepsilon}^{1/\varepsilon}\partial_x G_\delta(\varepsilon\xi,\varepsilon\eta)
\varphi_\varepsilon^r(\eta)d\eta
$$
and
$$
B_\varepsilon \dot\varphi_\varepsilon=q\,\varphi_\varepsilon^p\,
\Big(\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta(\varepsilon\xi,\varepsilon\eta)
\varphi_\varepsilon^r(\eta)d\eta\Big)^{-q-1}
\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta(\varepsilon\xi,\varepsilon\eta)
d\varphi_\varepsilon^r(\eta).
$$
Since
\begin{align*} 
&\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta(\varepsilon\xi,\varepsilon\eta)
d\varphi_\varepsilon^r(\eta)\,=\\
&\hspace{4em}=\,\big[ G_\delta(\varepsilon\xi,1)- G_\delta(\varepsilon\xi,-1)\big]\varphi_\varepsilon^r(1/\varepsilon)
-\varepsilon\int_{-1/\varepsilon}^{1/\varepsilon}\partial_y G_\delta(\varepsilon\xi,\varepsilon\eta)
\varphi_\varepsilon^r(\eta)d\eta,
\\
&\partial_y G_\delta(\varepsilon\xi,\varepsilon\eta)+
\partial_x G_\delta(\varepsilon\xi,\varepsilon\eta)=
\delta c_\delta \sinh\delta\varepsilon(\xi+\eta)\,,\quad \text{ where }
 c_\delta:=\frac{\delta}{\mu\sinh 2\delta}
\\
&G_\delta(\varepsilon\xi,1)-G_\delta(\varepsilon\xi,-1)=\delta^2 \,O(1)
\end{align*}
we find
\begin{align*}
A_\varepsilon \dot\varphi_\varepsilon&={}-q\delta c_\delta \varepsilon \varphi_\varepsilon^p
\int_{-1/\varepsilon}^{1/\varepsilon} \sinh \delta \varepsilon(\xi+\eta)
\varphi_\varepsilon^r(\eta) d\eta
\Big(\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta(\varepsilon\xi,\varepsilon\eta)
\varphi_\varepsilon^r(\eta)d\eta\Big)^{-q-1}
\\
&\quad{} +O\Bigl(\delta^2 \varphi_\varepsilon^r(1/\varepsilon) \varphi_\varepsilon^p\,
\Big(\int_{-1/\varepsilon}^{1/\varepsilon} G_\delta(\varepsilon\xi,\varepsilon\eta)
\varphi_\varepsilon^r(\eta)d\eta\Big)^{-q-1}\Bigr).
\end{align*}
We simplify this expression as follows. Since
$$
\sinh \delta\varepsilon(\xi+\eta)=\delta\varepsilon(\xi+\eta)+O(\delta^3\varepsilon^3(|\xi|^3+
|\eta|^3)),
$$
it follows
$$
\int_{-1/\varepsilon}^{1/\varepsilon} \sinh \delta \varepsilon(\xi+\eta)
\varphi_\varepsilon^r(\eta) d\eta = \delta\varepsilon\xi\int_{-1/\varepsilon}^{1/\varepsilon}
\varphi_\varepsilon^r(\eta) d\eta +O(\delta^3\varepsilon^3(1+|\xi|^3)).
$$
Using also \eqref{est10} we get
\begin{align*}
&\langle  A_\varepsilon \dot\varphi_\varepsilon,\dot\varphi_\varepsilon\rangle=\\
&= -q\delta^2 \varepsilon^2 c_\delta\int_{-1/\varepsilon}^{1/\varepsilon}\varphi_\varepsilon^r(\eta)d\eta
\int_{-1/\varepsilon}^{1/\varepsilon}\xi\varphi_\varepsilon^p(\eta)d\varphi_\varepsilon
\Bigl(\int_{-1/\varepsilon}^{1/\varepsilon} g_\delta
\varphi_\varepsilon^r(\eta)d\eta\Bigr)^{-q-1}+O(\delta^2\varepsilon^3)\\
&=\frac{q}{p+1}\delta^2 \varepsilon^2 c_\delta\int_{-1/\varepsilon}^{1/\varepsilon}
\varphi_\varepsilon^r(\eta)d\eta
\int_{-1/\varepsilon}^{1/\varepsilon}\varphi_\varepsilon^{p+1}(\eta)d\eta
\Bigl(\int_{-1/\varepsilon}^{1/\varepsilon} g_\delta
\varphi_\varepsilon^r(\eta)d\eta\Bigr)^{-q-1}+O(\delta^2\varepsilon^3 ).
\end{align*}
In this expression we can replace as before $\varphi_\varepsilon$ by
$\widetilde\varphi_\varepsilon$ and then by $w_p$. As a result we get
\begin{equation} \label{eigenv10}
\frac{\langle  A_\varepsilon \dot\varphi_\varepsilon,\dot\varphi_\varepsilon\rangle}
{\|\dot\varphi_\varepsilon\|^2}=
\frac{ q\,\delta^2 \varepsilon^2\int_{-\infty}^\infty w_p^{p+1}(\eta)d\eta}
{(p+1)\cosh^2 \delta\int_{-\infty}^\infty (\dot w_p)^2d\eta}+
O(\delta^2\varepsilon^{5/2} ).
\end{equation}
Finally, \eqref{rel2}, \eqref{eigenv10} and \eqref{eq4s2}
give
\begin{equation} \label{eigenv11}
\frac{ q\,\delta^2 \varepsilon^2\int_{-\infty}^\infty w_p^{p+1}(\eta)d\eta}
{(p+1)\cosh^2 \delta\int_{-\infty}^\infty (\dot w_p)^2 d\eta}-
\frac{8\alpha^2 e^{-2/\varepsilon}}
{\int_{-\infty}^\infty (\dot w_p)^2 d\eta}+
(\delta^2\varepsilon^{5/2}+\delta^4\varepsilon^2 +O(e^{-(2+\gamma)/\varepsilon}))
\end{equation}
if $0<\varepsilon<\varepsilon_0(\delta)$. In particular,
\begin{equation} \label{eigenv11a}
\lambda_\varepsilon=O(\delta^2 \varepsilon^2 +e^{-2/\varepsilon})\,,\hspace{2em}
 0<\varepsilon<\varepsilon_0(\delta).
\end{equation}
To find the asymptotic behaviour of $\lambda_\varepsilon$,
we notice that using \eqref{eigenv11a} we can improve the bound for $d$ (cf.
\eqref{est14}):
\begin{equation} \label{est14ab}
|d|=O(\delta^2\varepsilon^2+e^{-(1+\gamma)/\varepsilon})\,.
\end{equation}
Indeed, since $A_\varepsilon \widetilde\psi =c\lambda_\varepsilon \psi_\varepsilon+dA_\varepsilon g$ and
$A_\varepsilon g=L_\varepsilon g+\delta^2\varepsilon\, O(1)$, hence $\|A_\varepsilon g\|\geq C>0$, it follows
$|d|=(\|A_\varepsilon \widetilde\psi\|+\delta^2\varepsilon^2 +e^{-2/\varepsilon})$.
Using $\|A_\varepsilon \dot\varphi_\varepsilon\|
=\delta^2\varepsilon^2 \,O(1)$ and $\|A_\varepsilon h\|=e^{-(1+\gamma)/\varepsilon} \,O(1)$ we get
\eqref{est14ab}.


Now, having the better estimate \eqref{est14ab} we can repeat the above arguments
and show that instead of \eqref{eigenv11} we have
for $ 0<\varepsilon<\varepsilon_0(\delta)$\,:
\begin{equation} \label{eigenv11b}
\lambda_\varepsilon=
\frac{q\delta^2 \varepsilon^2\int_{-\infty}^\infty w_p^{p+1}(\eta)d\eta}
{(p+1)\cosh^2 \delta\int_{-\infty}^\infty (\dot w_p)^2 d\eta}-
\frac{8\alpha^2 e^{-2/\varepsilon}}{\int_{-\infty}^\infty (\dot w_p)^2 d\eta}+
(\delta^2\varepsilon^{5/2}+e^{-(2+\gamma)/\varepsilon})\,O(1)\,.
\end{equation}
In particular, for any fixed $\delta>0$ we have the asymptotic
$$
\lambda_\varepsilon=
\frac{q\delta^2 \varepsilon^2\int_{-\infty}^\infty w_p^{p+1}(\eta)d\eta}
{(p+1)\cosh^2 \delta\int_{-\infty}^\infty (\dot w_p)^2 d\eta}
+\varepsilon^{5/2} \,O(1),\; 0<\varepsilon<\varepsilon_0(\delta).
$$
If $\delta$ is not fixed and we allow $\delta\to 0$ as $\varepsilon\to 0$, then
$Re \lambda_\varepsilon$ changes sign around the point $\delta(\varepsilon)$ given
as a solution to the
equation
$$
\frac{q}{p+1}\,\frac{\mu}{8\alpha^2}\, \varepsilon^2 \delta^2 e^{2/\varepsilon}\int_{-\infty}
^\infty w_p^{p+1}(\eta)d\eta=1.
$$
 Note that the same expression
is obtained in \cite{ward2} using formal asymptotic
methods.



\subsection{ \label{perturb2}Perturbation of the non-small eigenvalues and uniform estimates
of the resolvent}
 According to \eqref{oper3}, \eqref{oper3a} and Kato \cite[p.\,364]{kato},
 the eigenvalues of
 $A_\varepsilon$ lie in a $O(\varepsilon^\gamma)$ neighbourhood of the eigenvalues of the
shadow operator $\widetilde A_\varepsilon$. Hence, under the same conditions
on the parameters $p,q,r$ as in \cite{gk1}, the spectrum of $A_\varepsilon$
 lies in
the right-half plane for all $0<\varepsilon<\varepsilon(D)$. In particular,
there exists an angle $\chi_D\in (0,\pi/2)$, such that the
resolvent set of $A_\varepsilon$ contains the sector
$$
\Lambda:=\{\lambda\in \mathbb{C} :  \chi_D \leq |\arg (\lambda-\mu_\varepsilon)
|\leq \pi \},
$$
where
$\mu_\varepsilon=\frac12 \mathop{\rm Re} \lambda_\varepsilon $.
Moreover, in this sector the resolvent satisfies for some constant $M_{\varepsilon,D}$,
for all $0<\varepsilon<\varepsilon(D)$,
 the estimate
\begin{equation} \label{resest1}
\|(A_\varepsilon-\lambda)^{-1}\|\leq \frac{M_{\varepsilon,D}}{|\lambda-\mu_\varepsilon|}\quad
\mbox{for all } \lambda\in\Lambda.
\end{equation}
To prove this estimate, we use the formula
\begin{gather*}
(A_\varepsilon-\lambda)^{-1}=(1+(L_\varepsilon-\lambda)^{-1}B_\varepsilon)^{-1}
(L_\varepsilon-\lambda)^{-1},\\
\|(L_\varepsilon-\lambda)^{-1}\|\leq 1/\mathop{\rm dist}(\lambda,\sigma(L_\varepsilon)).
\end{gather*}
 Since
$$
\|(L_\varepsilon-\lambda)^{-1}\|\leq \frac{C}{|\lambda|},\quad
\|(L_\varepsilon-\lambda)^{-1}B_\varepsilon\|\leq \frac{1}{2},
$$
uniformly for all $|\lambda|>N_D$, $\lambda\in \Lambda$, $0<\varepsilon<\varepsilon(D)$
for some $N_D$ large enough, it follows
$$
\|(A_\varepsilon-\lambda)^{-1}\|\leq \frac{C}{|\lambda-\mu_\varepsilon|},
$$
uniformly for all $|\lambda|>N_D$, $\lambda\in \Lambda$, and
$ 0<\varepsilon<\varepsilon(D)$.
If $|\lambda|\leq N_D$, $\lambda\in \Lambda$, then
$$
\|(\widetilde A_\varepsilon-\lambda)^{-1}\|\leq C_{\varepsilon,D}.
$$

\section{Contraction around the steady state $S(x,\varepsilon)$
 \label{contrac}}

In this section we study stability of the spike solution $S$ of \eqref{simplgm}
as given in \eqref{spike0}. We assume that the parameters $(p,q,r,\mu,\varepsilon)$ are such that all
eigenvalues of $A_\varepsilon$ are located in the
right half plane.

%{\sc Norms.}
Besides the standard $L^2$-norm for functions on the interval $[-1,1]$
denoted by $\|\cdot\|\,$, we use in this section the
``energy norm''  $\|\cdot\|_1$,
which is associated naturally to a problem with a small parameter
like \eqref{eq11s1} and is defined by:
$ \|u\|_1^{2}:=\|u\|^2+\|\varepsilon\,u'\|^2$.


For fixed positive $a$ large enough and  uniformly for all $\varepsilon\in(0,\varepsilon_{0}]$
this norm satisfies
the equivalences
$$
\langle (A+a)u,u\rangle^{1/2} \asymp \|u\|_1.
$$
We study perturbations around the steady state spike solution $S$, using
the contraction method as in \cite{gk1}
The perturbation satisfies equation \eqref{eq13s1}, which reads:
$$
v_t + A v =f[v]\,,\quad v(x,0)=v_{0}(x),
$$
where the quadratic term
$ f$ is given by \eqref{eq14s1}
and the (linear) operator $A$ is defined by \eqref{eq15s1}. Obviously, this
operator $A$ has the same spectral properties as its (stretched) cousin $A_\varepsilon$
has in sections \ref{reference} and \ref{perturb}.
Hence, under the positivity condition, stated above, $A$ is a sectorial
operator, see \cite{henry}, it satisfies the estimate \eqref{resest1}.


Associated to $A$ is the semigroup
$$
e^{-At} :=\frac{1}{2\pi i} \int_{\Gamma} (A-\lambda)^{-1}e^{-\lambda t}
d\lambda\,,\quad t>0\,,
$$
where $\Gamma$ is a suitable contour in the resolvent set $\Lambda$.
As in \cite{gk1} we prove the following statement.

\begin{lemma} \label{l2bounds}
For all $t>0$, all $\varepsilon\in(0,\varepsilon(D)]$
and for some constant $C_{\varepsilon,D}$, not depending on
$t$, this semigroup satisfies:
\begin{gather}
 \label{eq3s4} \|e^{-At}\| \leq C_{\varepsilon,D}
e^{-\mu_\varepsilon t}\,,\\
 \label{eq4s4} \|e^{-At}\|_1 \leq C_{\varepsilon,D}
e^{-\mu_\varepsilon t}\,
\begin{cases}
\|u\|_1^{}\,,\\
(1+t^{-1/2})\|u\|
\end{cases}
\end{gather}
\end{lemma}


Hence we can apply the contraction method as in \cite{henry}, \cite{gk1},
and prove the local stability of the single
internal spike solution $S(x,\varepsilon)$ in the Sobolev norm
$\|\cdot\|_1$.

\begin{theorem}[Local stability of the single internal spike for
$0<\varepsilon<\varepsilon(D)$]
 There exist positive constants $C(D), C_\varepsilon(D)$ and $ \varepsilon(D)$,
depending also on $p$, $q$, $r$ and $\mu$, and small $\varrho_\varepsilon(D)$
 such that the solution $(U,H)$ of the system
 \eqref{simplgm} exists for all times $t>0$ and
satisfies
\begin{gather*}
\|U(\cdot,t)-S\|_1 \leq \varrho e^{-\mu_\varepsilon t},\\
\|H(\cdot,t)-H\|_1 \leq C(D)\varrho \varepsilon^{-1}e^{-\mu_\varepsilon t},
\end{gather*}
for all $\varepsilon$ and $\varrho$ satisfying
\[
0<\varepsilon<\varepsilon(D),\quad  0<\varrho<\varrho_\varepsilon(D),
\]
for all initial conditions $U_{0}\in
H^1 (-1,1)$ in the vicinity of $S$, that satisfy the compatibility
conditions $U_{0}' (-1)=U_{0}' (1)=0$
and satisfy the bound
$$
\|U_{0}-S\|_1 \leq C_\varepsilon(D) \varrho.
$$
\end{theorem}

\subsection*{Acknowledgement}
We are grateful to the referee for a number of
useful suggestions. 
The first author (G. E. Karadzhov) was partially supported by the
Research Community ``Advanced Numerical Methods for Mathematical Modelling''
of the Fund for Scientific Research - Flanders, and by the University of
Sussex at Brighton.


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\end{document}