\documentclass[reqno]{amsart}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2002(2002), No. 50, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2002 Southwest Texas State University.}
\vspace{1cm}}

\begin{document}

\title[\hfilneg EJDE--2002/50\hfil Metastability in the shadow system]
{Metastability in the shadow system for Gierer-Meinhardt's equations}

\author[Pieter de Groen \& Georgi Karadzhov\hfil EJDE--2002/50\hfilneg]
{Pieter de Groen \& Georgi  Karadzhov}

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

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

\date{}
\thanks{Submitted April 17, 2002. Published June 2, 2002.}

\thanks{ Partially supported by NATO Collaborative Research Grant No. 973981,
by the Research \hfil\break\indent
Community ``Advanced Numerical Methods for Mathematical Modeling'' of
the Fund for \hfil\break\indent
Scientific Research-Flanders, and by a FWO-BAS grant}

\subjclass[2000]{35B25, 35K60}
\keywords{Spike solution, singular perturbations,
 reaction-diffusion equations, \hfil\break\indent
 Gierer-Meinhardt equations}


\begin{abstract}
  In this paper we study the stability of the single internal 
  spike solution of the shadow system for the
  Gierer-Meinhardt equations in one space dimension.
  It is well-known, that the linearization around this spike
  consists of a differential operator plus a non-local term. 
  For parameter values in certain subsets of the 3D 
  $(p,q,r)$--parameter space we
  prove that the non-local term moves the negative $O(1)$ eigenvalue
  of the differential operator to the positive (stable) half plane
  and that an exponentially small
  eigenvalue remains in the negative half plane, indicating a
  marginal instability (dubbed ``metastability''). We also
  show, that for parameters $(p,q,r)$ in another region, 
  the $O(1)$ eigenvalue  remains in the negative half plane. 
  In all asymptotic approximations we compute
  rigorous bounds for the order of the error.
\end{abstract}

\maketitle


\def\C{\mathbb{C}}
\def\R{\mathbb{R}} 
\def\eps{\varepsilon}
\def\epsilon{\varepsilon}
\def\phi{\varphi}
\def\theta{\vartheta}
\def\rho{\varrho}
\def\D{\mathcal{D}}
\def\br{|}

\def\pref#1{{\rm(\ref{#1})}}
\newtheorem{tm}{Theorem}[section]
\newtheorem{rem}[tm]{Remark}
\newtheorem{lema}[tm]{Lemma}
\newtheorem{prop}[tm]{Proposition}
\newtheorem{defin}[tm]{Definition}
\numberwithin{equation}{section}

\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{array}{lll}
 U_t=\eps^2\Delta U-U+U^pH^{-q} & x\in\Omega,&t>0,\\
\tau H_t=D\Delta H-\mu H+U^rH^{-s}\quad& x\in\Omega,&t>0,
\vrule height 1.4em depth .8em width 0pt\\
\partial_nU=0=\partial_nH & x\in\partial\Omega,& t \ge 0,
\end{array}
\end{equation}
where $U$ and $H$ represent activator and inhibitor
concentrations, $\eps$ 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 large and $\eps$ and $\tau$
small (positive). $\Omega$ is a bounded domain; we shall restrict
our analysis to one space dimension and choose $\Omega:=[-1,1]$.
Since in our analysis the factor $\mu$ is not important, we set
$\mu:=1$; since the parameters $q$ and $s$ only appear together in
the quotient $q/(s{+}1)$ and $s\ge 0$, we may choose $s\!=\!0$
without loss of generality. The exponents $\{p>1, ~q>0,~r>1\}$
satisfy the inequality
\begin{equation} \label{eq1s1}
\gamma_r:=\frac{qr}{p-1}>1.
\end{equation}

The numerical study in \cite{gierer} revealed strongly localized
pulse solutions in the activator $U$ against a nearly constant background of
the inhibitor concentration  for large values of $D$ and $1/\eps$.
Irons \& Ward \cite{ward1} analyze by formal asymptotic expansions
metastable spike solutions for the reduced ``shadow''-system,
derived from \pref{gm} in the limit for $D\to \infty$ and $\tau\to 0$.
Doelman, Gardner \& Kaper \cite{doelman} consider the same problem, using ``geometric
singular perturbation theory'' to derive criteria for stability of
a solution with one spike.
The reader is advised to consult those papers
for a more general description of facts and results in this type of
problems.
As in \cite{ni}, \cite{lin} and \cite{ward1} we derive the simplified shadow
system from \pref{gm} as follows. The average $h$ of $H$ satisfies
the average of  (\ref{gm}b), $\tau h_t=-
h+\frac12 \int_{-1}^{1}U^rdx.$ If $\tau$ is small and $D$ is large,
$H(x)\approx h$ is approximately constant and $\tau h_t$
negligible, such that the averaged equation reduces to the
constraint
\begin{equation} \label{eq2s1}
 h=\frac12 \int_{-1}^{1}U^rdx.
\end{equation}
So we find from  (\ref{gm}a) the shadow system for
$(x,t)\in (-1,1)\times(0,\infty)$,
\begin{equation} \label{eq3s1}
\begin{gathered}
U_t=\eps^2 U_{xx}-U+h^{-q}U^p,\quad U_x(-1,t)=0=U_x(1,t).
\end{gathered}
\end{equation}
We can eliminate from this equation the direct dependence on $h$ by \pref{eq2s1}.

The aim of this paper is a study of metastability of the single
internal spike solution via the spectrum of the linearized operator.
We consider the shadow system of
\pref{gm} in one space dimension, construct a (stationary) solution to it
along the lines of \cite{takagi} and we show how the interesting
spectral properties of the variation around this solution can be
derived. We extend the results obtained by Wei in \cite{wei}. The
differential operator associated with this variation consists of a
selfadjoint differential operator $L_\eps$ and a non-local term (due to
the constant inhibitor background). $L_\eps$ has one large $(O(1))$
negative (i.e.\ unstable) eigenvalue associated to an even
eigenfunction and one exponentially small negative eigenvalue
associated to an odd eigenfunction. Using
several perturbational techniques, we derive conditions under
which the non-local term pushes the spectrum to the right (i.e.\ it
increases the stability) and maps the large negative eigenvalue
into the positive half plane, see figure~\ref{evc}.
Since the non-local term is zero on
the odd functions, it does not affect the exponentially small
eigenvalue. We remark, that Doelman et al.\ in \cite{doelman} assume
that the spatial domain is unbounded and consider this only a
minor simplification. Although this simplification looks quite
innocently, our study shows that it removes the asymptotically
small terms in the expansions of the eigenvalues of the shadow
equation, that are decisive for the (very weak) stability or
instability.


\begin{figure}[t] 
\begin{center}
\includegraphics[width=7cm]{eigen.eps}
\end{center}
\caption{\label{evc}
Plot of the four eigenvalues with  smallest real part
of the operator, restricted to the subspace of {\bf even} functions,
as a function of $q$ if $p=2$ and $r=2$.
As soon as $q$ crosses the point \mbox{$q={p-1 \over r}=\frac12 $}, condition
\pref{eq1s1} is met and the smallest eigenvalue crosses the imaginary axis.
Near $q=0.66$ the two smallest eigenvalues meet and go off into the complex
plane.}
\end{figure}



The boundary conditions on the finite domain are extremely important. With Neumann
boundary conditions, the only exact stationary spike solution, having exactly one interior
maximum and no boundary maxima, is the symmetric
spike with maximum at the center. If we want an off-center spike,
we have to change the Neumann condition at $x=1$ or at $x=-1$ into a mixed condition;
either $U_x(1)+\eps\delta U(1)=0$ for a spike on $(0,1)$ or
$U_x(1)-\eps\delta U(1)=0$ for a spike on $(-1,0)$ for some $\delta>0$.
Only using the exact spike, we can linearize around it in such a way
that the quadratic terms do not contain secular terms due to the approximation
of the spike. Also this aspect disappears, when we study the problem on
the whole real axis.


In section~\ref{spike}
we study stationary spike-solutions of \pref{eq3s1}, their asymptotics for small
$\eps$ and the linearization of this equation around such a solution.
In section~\ref{reference} we study the spectrum of the differential operator $L_\eps$
using the (new) explicit solution \ref{eq5s1} (see also \cite{gk1}) of the shadow system.
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 $p$, $q$ and $r$ using perturbational methods. Finally,
we study in section~\ref{contrac} the ``metastability''
of (\ref{eq2s1}--\ref{eq3s1}) along the lines of \cite{gk}.

While linearizing \pref{gm} or \pref{eq3s1} around a spike we should always keep
in mind that the non-linear system is well-defined only for positive solutions
and hence that only those variations are acceptable, that do not
compromise the positivity. For the study of the spectrum of the
linearized equation this does not matter. However, for a study of
metastability, we have to restrict the perturbations to functions that are not larger
than some (fixed positive) factor $\rho<1$ times the spike.

Essentially, the
methods are applicable too in the case where $D$ is large but not pushed to
the limit $D\to\infty$, see \cite{ward2}.

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

\subsection{A stationary one-spike solutions and its asymptotics}
\label{subspike}
A typical (stationary) spike solution of (\ref{eq2s1}--\ref{eq3s1}) should be
large on a small support around some interior point $x_{\rm o}$ and  negligible
elsewhere.  Assuming that such a spike solution $S$
 has the form $S(x_{\rm o}{+}\eps\xi,t)=h^\gamma u(\xi)$, where
$\xi$ is the stretched variable and $\gamma=q/(p{-}1{-}rq)$,
we find for $u$ the equation
\begin{equation} \label{eq4s1}
u''-u+u^p=0, \quad
u'((\pm 1-x_{\rm o})/\eps)=0.
\end{equation}
For $p>1$ the system of equations for $(u,u')$ in the phase plane
has a saddle point at
$(0,0)$ and a center at $(1,0)$. A unique homoclinic solution around the center
connects the saddle to itself, see figure~\ref{phase}.

\begin{figure}[t] 
\begin{center}
\includegraphics[width=7cm]{phase.eps}
\end{center}
\caption{\label{phase}Phasediagram in the $(u,u')$-plane for
the solution of $u''-u+u^p=0$ from \pref{eq4s1} with $p=3$.}
\end{figure}

This homoclinic solution happens to have for all $p>1$ the closed
form\footnote{It is remarkable
that we did not find this fact in any of the references consulted.}
\begin{equation} \label{eq5s1}
w_p(\xi):=\big({p+1\over 2}\big)^{1/(p-1)}
(\cosh({p-1\over 2}\xi))^{-2/(p-1)},
\end{equation}
and for large $\br\xi\br$ it has the asymptotic behaviour
\begin{equation} \label{eq6s1}
w_p(\xi)=\alpha e^{-\br\xi\br}(1+O(e^{-(p-1)\br\xi\br})),\quad
\alpha:=(2p+2)^{1/(p-1)}.
\end{equation}

Inside the homoclic orbit we find positive periodic orbits; from the phase
portrait it is clear that these are the only ones that
may solve the Neumann boundary conditions in \pref{eq4s1}. Necessary and sufficient is that
an integral multiple of its half  period is equal to the interval length $2/\eps$.
Let $\phi$ be such a periodic solution that has  minimum $a$ and maximum $b$,
then apparently
\begin{equation} \label{eqn1s2}
0<a<1<b<t_p:=\bigl({ p+1\over 2}\bigr)^{1/(p-1)}.
\end{equation}
Multiplying the equation for $\phi$ by $\dot\phi$ and integrating we find
\begin{equation} \label{eqn2s2}
\dot\phi^2=F(\phi)-F(a),\quad F(a)=F(b)\quad\mbox{and}\quad
F(\phi):=\phi^2-{{2\over p+1}}\phi^{p+1}.
\end{equation}
Minimum and maximum of the solution $\phi$ are related by the equation
$F(a)=F(b)$ and its period  is given
by the integral
\begin{equation} \label{eqn3s2}
T_a:=2\int_a^b{d\phi\over\sqrt{F(\phi)-F(a)}}.
\end{equation}

From this picture it is clear that for every $\eps>0$ there is a unique
stationary solution with a single spike in the interior
(and without boundary spikes). We shall denote it in the sequel by $\phi_\eps$;
its minimum we denote by $a_\eps$ and its maximum by $b_\eps$. If $\eps\to 0$, then
$\phi_\eps\to w_p$, $a_\eps\to 0$ and $b_\eps\to t_p $ and
\begin{equation} \label{eq4as1}
a_\eps=\sqrt{(p-1)t_p (t_p-b_\eps)}(1 +O(t_p-b_\eps)^\sigma),\quad \sigma:=\min(1,p-1).
\end{equation}
Because the period has to fit in the interval, we can derive the asymptotics
of $a_\eps$ and $b_\eps$ for $\eps\to 0$ from the equation $T_a=2/\eps$:

\begin{lema}
\begin{equation} \label{eq20s1}
a_\eps=2\alpha e^{-1/\eps} (1+O(e^{-\sigma/\eps}))\quad \mbox{and}\quad t_p-b_\epsilon=O(e^{-2/\epsilon}).
\end{equation}
\end{lema}

\noindent{\sc proof.}
We estimate the integral \pref{eqn3s2} by the corresponding integral for $w_p$,
for which we already know the asymptotics, \pref{eq6s1} and we show that the difference
tends to the constant $\log 2$ for $a\to 0,$
\begin{equation} \label{eq20as1}
\int_a^b\Bigl[{1\over\sqrt{F(t)-F(a)}}-{1\over\sqrt{F(t)}} \Bigr]dt
=\log 2+O(a^\sigma),
\end{equation}
To do so, we split the interval of integration into three parts,
$[a,c]$, $[c,d]$, and $[d,b]$, where $a<c<d<b$ and where $c~\mbox{and} ~d$ are constants,
not depending on
$a$ and bounded away from the zeros of $F$, $a$ and $b$.
Obviously, the integral over the middle segment is of order $F(a)=O(a^2).$

In the third part on $[d,b]$ we may estimate $F$ from below by
$$
F(t)-F(b)\ge \delta^2(b-t),\quad F(t)\ge\delta^2(t_p-t),\quad \mbox{if}\quad d\le t\le b
$$
for some $\delta>0$. Hence we estimate the integral from above as follows,
\begin{align*}
\int_d^b\Bigl[{1\over\sqrt{F(t)-F(b)}}&-{1\over\sqrt{F(t)}} \Bigr]dt\\
=&~\int_d^b{F(a)\,dt\over\sqrt{F(t)-F(b)}\sqrt{F(t)}
\Big(\sqrt{F(t)-F(b)}+\sqrt{F(t)}\Big)}\\
\le&~ \int_d^b {F(a)dt\over\delta^3\sqrt{t-b}\sqrt{t-t_p}
\bigl(\sqrt{t-b}+\sqrt{t-t_p}\bigr)}\\
=&~{\delta^{-3}F(a)\over t_p-b}
\int_d^b\Bigl[ {1\over\sqrt{t-b}}-{1\over\sqrt{t-t_p}}\Bigr]dt
\vrule height 2em depth 1.6 em width 0pt\\
=&~O(\sqrt{t_p-b})=O(a).
\end{align*}
In the first part on $[a,c]$ we substitute $s^2=F(t)=t^2-{2\over p+1}t^{p+1}$,
assuming $c<1$. This implies
$$
s=t+O(t^{p})\quad \mbox{or}\quad t=s+O(s^{p})\quad \mbox{and}\quad
dt={sds\over t-t^p}=ds\bigr(1+O(s^{p-1})\bigl)
$$
Writing $\widetilde a^2:= F(a)$ and $\widetilde c^2:= F(c)$ we find
$$
\int_a^c\Bigl[{1\over\sqrt{F(t)-F(b)}}-{1\over\sqrt{F(t)}} \Bigr]dt=
\int_{{\widetilde a}}^{{\widetilde c}}
\Bigl[{1\over\sqrt{s^2-\widetilde a^2}}-{1\over\sqrt{s^2}} \Bigr]\Bigr(1+O(s^{p-1})\Bigl)ds.
$$
The main term evaluates to
$$\int_{{\widetilde a}}^{{\widetilde c}}
\Bigl[{1\over\sqrt{s^2-\widetilde a^2}}-{1\over\sqrt{s^2}} \Bigr]ds
=\log\bigl(1+\sqrt{1-{{\widetilde a}^2}/{{\widetilde c^2}}}\bigr)
=\log 2+O(a^2).
$$
The remainder is bounded by a constant times the following integral,
in which we substitute $s=\widetilde a\sigma$,
$$
\int_{{\widetilde a}}^{{\widetilde c}}
\Bigl[{1\over\sqrt{s^2-\widetilde a^2}}-{1\over\sqrt{s^2}} \Bigr]s^{p-1}ds
=\int_1^{{\widetilde c}/{\widetilde a}}
{\widetilde a^{p-1}\sigma^{p-2}d\sigma\over
\sqrt{\sigma^2-1}(\sigma+\sqrt{\sigma^2-1})}.
$$
The integrand in the right-hand side has a square root singularity at $\sigma =1$ and behaves
like $\sigma^{p-4}$ for large $\sigma$, hence the integral is of the order
$O(a^{\min( 2, p-1)})$.
Finally, the integral for $w_p$ extends to $t_p$, hence we have to
subtract from \pref{eq20as1} the contribution
$$
\int_b^{t_p}{dt\over\sqrt{F(t)}} =O\Big(\int_b^{t_p}{dt\over\sqrt{t_p-t}}\Big)=O(\sqrt{t_p-b})=O(a).
$$
Summing up, we have proved
\begin{equation} \label{eq20bs1}
\int_a^b{dt\over\sqrt{F(t)-F(a)}}=\int_a^{t_p}{dt\over\sqrt{F(t)}} +\log 2+O(a^\sigma).
\end{equation}
Since the integral in the right-hand side is the inverse function of $w_p$,
its asymptotics \pref{eq6s1} imply \pref{eq20s1}.
\qed

As we expect we can use the function $w_p$ as an approximation to
$\phi_\eps$.

\begin{lema}\label{lem2}  For each $\eps>0$ equation \pref{eq4s1} has a unique spike solution
$\phi_\eps$ with a single
internal maximum, located at the center of the interval. It approximates $w_p$,
\begin{equation} \label{eq2cbs1}
|\phi_\eps(\xi)-w_p(\xi)|\leq c e^{-1/\epsilon},
\end{equation}
uniformly for $|\xi|\leq 1/\epsilon, 0<\epsilon<\epsilon_{\rm o}$ and it
satisfies on $[-1/\eps,1/\eps]$ the equivalence
$\phi_\eps \asymp w_p$.
\end{lema}

\noindent{\sc Proof:}
The equivalence is an immediate consequence of \pref{eq2cbs1} and the fact that both
$\phi_\eps$ and $w_p$ are bounded from below by $\alpha\approx e^{-1/\eps}$.
To prove \pref{eq2cbs1} we start with the evident property
$\phi_{\rm o}(\xi)=w_p(\xi)$.
For any $1<b<t_p$ we can define the
positive solution $\phi(\xi,b)$ with global maximum at zero by the
integral:
$$
\int_{\phi(\xi,b)}^b \frac{dt}{\sqrt{F(t)-F(b)}}=|\xi|\quad\mbox{or}
\quad \phi_{\eps}(\xi)=\phi(\xi,b_\eps).
$$
Thus $\phi(\xi,b)$ is defined for $|\xi|\leq \xi(b)$, where
$$
\xi(b):=\int_{a(b)}^b \frac{dt}{\sqrt{F(t)-F(b)}}\quad
\mbox{and}\quad F(a(b))=F(b),\; a(b)<1<b.
$$
In particular,
$\xi(b)\to \infty$ as $b\to t_p$.
We have
$$
\psi(\xi,b):=\phi(\xi,b)-w_p(\xi)=
\phi(\xi,b)-\phi(\xi,t_p)=\int_{t_p}^{b}\frac
{\partial \phi}{\partial b}(\xi,b)db
$$
and the function
$h(\xi,b):= \frac{\partial \phi}{\partial b}(\xi,b)$ satisfies the
problem (${~}'=d/d\xi$)
$$
h''=(1-p\phi^{p-1})h,\quad  h(0,b)=1,\quad  h'(0,b)=0.
$$
In particular $h(\xi,b)$ is bounded on any $\eps$-independent rectangle
$[-\eta_1,\eta_1] \times [t_p-\delta,t_p].$ Therefore
$$
\br\psi(\xi,b_\epsilon)\br\leq c e^{-2/\epsilon},\quad \mbox{if}\quad
\br\xi\br\leq \eta_1, 0<\epsilon<\epsilon_{\rm o}.
$$


Since $\psi$ is an even function it remains to consider the interval
$[\eta_1, 1/\epsilon].$ Now we shall use the maximum principle for
the boundary problem:
\begin{gather*}
\psi''=q_\eps\psi , \quad \quad
q(\xi,\epsilon):=\frac{f(\phi_\epsilon)-f(w_p)}
{\phi_\epsilon-w_p},\\
\psi(\eta_1,b_\epsilon)=O(e^{-1/\epsilon}),\quad
\psi(1/\epsilon,b_\epsilon)=O(e^{-1/\epsilon}),
\end{gather*}
where
$
f(t)=t-t^p.
$
The function $f$
is decreasing on $[0,t_{\rm o})~ t_{\rm o}:=p^{-1/(p-1)}<1$, hence $q$ is non-negative if
$\phi_\eps\le t_{\rm o}$ and $w_p\le t_{\rm o}$. If we choose $\eta_1$ such that those conditions
are satisfied by $\phi_\eps$ and $w_p$ in this point, they are true on all of
$[\eta_1, 1/\epsilon]$
because of the monotony of both functions.
 Then the maximum
principle implies
$$
|\psi(\xi,b_\epsilon)|\leq c e^{-1/\epsilon}
$$
in the interval $[\eta_1,1/\epsilon].$\qed

Thus we have only one steady state solution $S$ which has a single
internal spike at zero:
\begin{equation} \label{eq3as1}
S(x,\epsilon)=H(\Phi) \Phi(x,\epsilon),\quad ~H(\Phi):=
\Bigl[{\textstyle\frac{1}{2}}\int_{-1}^1 \Phi^r (\xi,
\epsilon) d\xi\Bigr]^{-\alpha_r},
\end{equation}
where
$\Phi(x,\epsilon):=\phi_\epsilon(x/\epsilon)$ and $\alpha_r:=\frac{q}{rq-p+1}$. For all
$x\in[-1,1]$ it satisfies the estimates
\begin{equation} \label{eq6as1}
|\Phi(x,\epsilon)-w_p(x/\epsilon)|\leq c
e^{-1/\epsilon},\quad
|\Phi'(x,\epsilon)-w_p'(x/\epsilon)|\leq c
e^{-1/\epsilon}.
\end{equation}
Using this internal spike solution $\Phi$ we can define two steady
state solutions with one boundary spike. Namely,
\begin{equation} \label{eq6bs1}\textstyle
\Phi_{+}(x,\epsilon):=\Phi(\frac{x}{2}-\frac12 ,\frac{\epsilon}{2}),\quad \mbox{and}\quad
\Phi_{-}(x,\epsilon):=\Phi(\frac{x}{2}+\frac12 ,\frac{\epsilon}{2}),\quad
0<\epsilon <\epsilon_{\rm o}
\end{equation}
have spikes at $x=1$ and at $x=-1$ respectively.
Both functions are smooth periodic
functions with period $4$.

\subsection{Linearization around the one-spike solution}
\label{linear}
To study stability of the spike solution, we consider the
first variation of the shadow system \pref{eq3s1}--\pref{eq2s1} around
this solution. It is convenient to rewrite this system as one
equation:
\begin{equation} \label{eq11s1}
\begin{gathered}
 u_t =\epsilon^2 u_{xx} -u +g(u),\\
u_x(\pm 1,t)=0,\; u(x,0)=u_{\rm o}(x),\; u_{\rm o}'(\pm 1)=0,
\end{gathered}
\end{equation}
where
\begin{equation} \label{eq12s1} g(u)=
[{\textstyle\frac{1}{2}}\int_{-1}^1 u^r (\xi,t) d\xi]^{-q}u^p.
\end{equation}
Let $v$ be the variation around $S$; set $u(x,t)=S(x,\epsilon)+v(x,t),$
then $v$ satisfies
\begin{gather*}
v_t =\epsilon^2 v_{xx} -v +g(S+v)-g(S),\\
v_x(\pm 1,t)=0,\quad v(x,0)=v_{\rm o}(x):=u_{\rm o}(x)-S(x,\epsilon),
\end{gather*}
or written in operator form
\begin{equation} \label{eq13s1}
v_t + A v =f[v],\quad v(x,0)=v_{\rm o}(x),
\end{equation}
where $f$ is the quadratic term
\begin{equation} \label{eq14s1}
f[v]:=\int_{0}^1 (1-\sigma)\partial_\sigma^2g(S+\sigma v)\,
d\sigma
\end{equation}
where $\partial_\sigma^2$ denotes the
second derivative of $\sigma \mapsto g(S+\sigma v)$ w.r.t.~$\sigma$ and
where $A$ is the spatial linear
operator,
\begin{equation} \label{eq15s1}
Av:=-\epsilon^2 v'' +v-p\Phi^{p-1} v +
qr \Phi^p\bigl(\int_{-1}^{1} \Phi^r(\xi,\epsilon)d\xi\bigr)^{-1}
\int_{-1}^{1} \Phi^{r-1}(\xi,\epsilon) v (\xi)d\xi,
\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 $x=\eps\xi$ and to define the
operators $A_\eps =L_\eps +B_\eps$ on the stretched interval
$[-1/\eps,1/\eps],$ where $(\dot u:=du/d\xi)$
\begin{equation} \label{eq9s1}
L_\eps u:=-\ddot u+u-p\phi_\eps^{p-1}u,\quad
\D(L_\eps):=\{u \in
H^2({\textstyle [- {1\over\eps}\,,\, {1 \over\eps}\,]})\,
\vrule height .8em depth .3em width .5pt \,
\dot u(\pm{\textstyle{1\over \eps}}\,)=0\}.
%\dot u(\mbox{$\pm 1/\eps$})=0\}.
\end{equation}
Its limit
$L_{\rm o}$ for $\eps\to 0$ has domain $H^2(\R)$. The non-local
operator $B_\eps$ (of rank one) is defined by
\begin{equation} \label{eq10s1}
B_\eps =d_{r,\eps} \langle \cdot,\phi_\eps^{r-1}\rangle \phi_\eps^p,
\end{equation}
where
$$
d_{r,\eps}=\frac{\gamma_r(p-1)}{\beta_{r,\eps}}=\frac{qr}{\beta_{r,\eps}},
\quad
\beta_{r,\eps}=\int_{-1/\eps}^{1/\eps} \phi_\eps^r (\xi,\eps)d\xi
$$
and
$\gamma_r>1$ is given by $\gamma_r =qr/(p-1)$, see \pref{eq1s1}.

The derivative of $\phi_\eps$ and powers of $\phi_\eps$ are to be used in
estimating eigenvalues etc.
Below we give some useful relations. For all $m>0$ we have
\begin{equation} \label{eq16s1}
L_\eps \phi_\eps^m =(1-m^2)\phi_\eps^m +\kappa_m
\phi^{m+p-1}+m(m{-}1)F(a_\eps)\phi_\eps^{m-2},
\end{equation}
where $\kappa_m:=m(2m +p-1)/(p+1) -p$.
The third term is uniformly of the order $O(e^{-\min(m,2)/\eps})$.
In  particular, the remainder is zero for~$m=1$:
\begin{equation} \label{eq17s1}
L_\eps \phi_\eps =(1-p)\phi_\eps^p.
\end{equation}
It is also easy to
check that
\begin{equation} \label{eq18s1}
L_\eps \bigl(\frac{\phi_\eps}{p-1}+\frac{\xi
\dot\phi_\eps}{2}\bigr)=-\phi_\eps\quad \mbox{and}\quad L_\eps \dot\phi_\eps=0.
\end{equation}
Finally, we can evaluate the integral
$\beta_{m,0}:= \int_{-\infty}^{\infty} w_p^m(\xi)d\xi $ in terms of the Gamma
function,
\begin{equation} \label{eq7as1}
\beta_{m,0}= {(\frac{p+1}{2})}^{m/(p-1)}{2\over p-1}
{\sqrt{\pi}\Gamma(\frac{m}{p-1})\over\Gamma(\frac{m}{p-1}+\frac12 )}.
\end{equation}


\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 $I\subset\R$ satisfy the
 minimax property,
see \cite[Theorem XIII.1, p.~76]{reed}.
If $L$ has isolated eigenvalues
$\lambda_{\rm o}\le\lambda_1\le\lambda_2\le\cdots$, 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} \quad \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 $L_\eps$ of our study (with ``potential''
$Q:=1-p\phi^{p-1})$ is a selfadjoint differential operator
bounded from below and it has a discrete spectrum consisting of
eigenvalues of multiplicity one for each $\eps>0$:
$\lambda_{\rm o}(\eps)<\lambda_1(\eps)<\lambda_2(\eps)<\cdots$
with corresponding eigenfunctions $\psi_{\rm o}(\cdot,\eps),
\psi_1(\cdot,\eps), \psi_2(\cdot,\eps),\cdot\cdot\cdot .$
 Its
spectrum converges for $\eps\to 0$ (and for all selfadjoint
boundary conditions) to the spectrum of $L_{\rm o}$,
see e.g.~\cite[ch.~9]{codd}.
We shall calculate the rate of convergence.

The ``limiting'' operator $L_{\rm o}$ (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}=-\,{(p-1)(p+3)\over 4}\,\psi_{\rm o},& p>1,\cr
\psi_1:=\dot w_p\,, & L_{\rm o}\psi_1=0, & p>1,
\vrule height 1.7em depth 1em width 0pt\cr
\psi_2:=w_p^{3-p\over 2}-\frac12 {{p+3\over p+1}}w_p^{p+1\over2},
&L_{\rm o}\psi_2 ={(p-1)(5-p)\over 4}\,\psi_2,
& 1<p<3.
\end{array}
\end{equation}
Since $\psi_{\rm o},~\psi_1$ and $\psi_2$ have zero, one and two zeros respectively,
and since the zeros of the eigenfunctions of second order ordinary
differential operators interlace, $\lambda_{\rm o}:=-\frac14  (p-1)(p+3)$,
$\lambda_1:=0$ and (if $p<3$) $\lambda_2:=\frac14  (p-1)(5-p)$
are the three smallest eigenvalues of $L_{\rm o}$.
In order to show that $L_{\rm o}$ does not have a second isolated eigenvalue
for $p>3$, we substitute $\psi(\xi)=\theta({p-1\over 2}\xi) $
in the eigenvalue equation $L_{\rm o}\psi=\lambda\psi$ using the explicit
form of $w_p$ from \pref{eq5s1}. This yields the equation
\begin{equation} \label{eq2s2}
M_p\theta:=-\ddot\theta-2p(p+1)(p-1)^{-2}\cosh^{-2}(\eta)\,\theta
=({{2\over p-1}})^2(\lambda-1)\theta=\mu\theta.
\end{equation}
Since the ``potential'' in $M_p$ is an increasing function of $p$,
its eigenvalues are increasing functions of $p$ by the minimax theorem
\pref{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 $L_{\rm o}$ has only two eigenvalues below 1 if $p\ge 3$.

In order to compute the exponentially small rate of convergence of the smallest
eigenvalues $\lambda_{\rm o}(\eps)$ and $\lambda_1(\eps)$ (and $\lambda_2(\eps)$ if $p<3$)
of $L_\eps$, 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.

The asymptotic expansion of $\lambda_k(\eps)$ will be calculated explicitly for $k=1$;
the expansions for the other eigenvalues are not needed explicitly
and are completely analogous.
We use the same technique as in \cite{pdg} and \cite{gk}.
We compute an approximate eigenfunction $w$ of
unit norm $\|w\|=1$ of the operator
$L_\eps$ and we show that
\begin{equation} \label{ev9}
\langle L_\eps w,w\rangle =\nu_\eps(1+O(\widetilde R_\eps))\quad
\mbox{and}\quad
\|L_\eps w\|^2=O(\widetilde R_\eps R_\eps),
\end{equation}
where both $\widetilde R_\eps =o(1)$ and
$R_\eps=o(1)$ as $\eps\to 0$.

The generalized Fourier expansion of $w$ in
the true eigenfunctions $\{\psi_k\br k=0,1,\cdots\}$ of $L_\eps$ is
\begin{equation} \label{ev10}
w=\sum_{k=0}^\infty c_k \psi_k\quad \mbox{with}\quad
\sum_{k=0}^\infty \br c_k|^2=\|w\|^2=1.
\end{equation}
Since all eigenvalues of $L_\eps$ except
$\lambda_1(\eps)$ are uniformly bounded  away from $\lambda_1(0)=0$
by a distance $d>0$, we find from \pref{ev9}
$$
1-\br c_1|^2=\sum_{k=0,~k\ne1}^\infty \br c_k|^2~\le~ d^{-2}
\sum_{k=0,~k\ne1}^\infty \lambda_k^2\br c_k|^2~\le~
d^{-2}\|L_\eps w\|^2=O(\widetilde R_\eps R_\eps),
$$
implying that $\br c_1|^2=1+O(\widetilde R_\eps R_\eps)$.
The estimate for the inner product in \pref{ev9} now implies that
$$
\langle L_\eps w,w\rangle -\nu_\eps=\br c_1|^2\lambda_1(\eps)-
\nu_\eps+\sum_{k=0,~k\ne1}^\infty \lambda_k\br c_k|^2
=O(\widetilde R_\eps(\nu_\eps + R_\eps))
$$
and hence that
\begin{equation} \label{ev11}
\lambda_1(\eps)=\nu_\eps+O(\widetilde R_\eps(\nu_\eps+ R_\eps )).
\end{equation}


Let $\psi_1(\cdot,\eps)$ be the true eigenfunction of $L_\eps$
corresponding to $\lambda_1(\eps)$. We look for an approximate eigenfunction
of the form $\psi_1(\cdot,\eps)\approx\dot\phi_\eps+$
boundary layer corrections. Within the interval $[-1/\eps,1/\eps]$
the tails of $\dot\phi_\eps$ are exponentially small by \pref{eq20s1} and
\pref{eq2cbs1},
\begin{equation} \label{eq4s2}\textstyle
\dot\phi_\eps({\pm { 1\over\eps}})=0
\quad \mbox{and}\quad
\ddot\phi_\eps(\pm {{ 1\over\eps}})=
2\alpha e^{-{ 1/\eps}}
(1+O(e^{-{\sigma/\eps}})).
\end{equation}
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{array}{l@{\,}l}
h(\xi)&:={}-\ddot\phi_\eps({{1\over\eps}})
\rho(\eps\xi)
\exp\big(\xi-{1\over\eps}\big),\\
k(\xi)&:=\ddot\phi_\eps({-{1\over\eps}})\rho(-\eps\xi)
\exp\big(-\xi-{1\over\eps}\big),\vrule height 1.7em depth 1em width 0pt\\
\widetilde\psi_1&:=\dot\phi_\eps+h+k
\end{array}
\end{equation}
where $\rho$ is a monotonous $\mathcal{C}^\infty$ cut-off function satisfying
$\rho(x)=1$ if $x\ge 1/2$ and $\rho(x)=0$ if
$x\le 1/4$.
From the definition it is clear that
$\widetilde\psi_1$ satisfies the boundary conditions at
$\xi={\pm 1/\eps}$  and
\begin{equation} \label{eq5s2}\textstyle
L_\eps h=-p \phi_\eps
^{p-1}h+\ddot\phi_\eps({{1\over\eps}})
\left(\eps^2\rho''+
2\eps\rho'\right)\exp\big(\xi-{1\over\eps}\big).
\end{equation}
In the first term the decay of $\phi_\eps$ towards the boundary compensates
(partially if $p<2$) the increase of the boundary layer term $h$ giving the order
$O(e^{-(1+\sigma)/\eps})$ with $\sigma:=\min(1,p-1)$.
 Since the support of $\rho'$ is
in $[1/4,\frac12 ]$, the second term in the r.h.s.\ is of the order
$O(e^{-{3/ 2\eps}})$. For $L_\eps k$ we
have an analogous estimate. Hence,
\begin{equation} \label{eq6s2}
\|L_\eps\widetilde\psi_1\|^2=O\left(
e^{-3/\eps} +e^{-2(1+\sigma)/\eps}\right) .
\end{equation}
Since $L_\eps \widetilde\psi_1=L_\eps (h+k)$ and
$\dot{\widetilde\psi_1}(\pm 1/\eps)=0$, we can
evaluate the quadratic form \pref{ev9}, using integration by parts:
\begin{equation} \label{eq7s2}
\begin{aligned}
\langle L_\eps \widetilde\psi_1, \widetilde\psi_1\rangle=&
\langle L_\eps (h+k), \widetilde\psi_1\rangle=
\left[-( \dot h+\dot k)\widetilde\psi_1
\right]^{1/\eps}_{-1/\eps}
+\langle h,L_\eps h\rangle+\langle k,L_\eps k\rangle\\
=&-8\alpha^2 e^{- 2/\eps} \big(1+O(e^{-\sigma/\eps})\big)
\end{aligned}
\end{equation}
By \pref{ev11} we may conclude, see \cite[eq.~(43)]{ward1}
\begin{equation} \label{eq7as2}
\lambda_1(\eps)=-8\alpha^2\zeta e^{- 2/\eps}\big(1 +O(e^{-\sigma/\eps})\big),
\quad \zeta^{-1}:=\int_{-1/\eps}^{1/\eps} \dot \phi_\eps^2(\xi)d\xi.
\end{equation}
The factor 8 is due to the factor 2 in \pref{eq20s1}, which makes the
approximate eigenfunction two times as large as the function $w_p$
used in \cite{ward1}.

The asymptotics of $\lambda_{\rm o}(\eps)$ and $\lambda_2(\eps)$
is derived in an analogous way. By analogy to \pref{eq1s2} and
\pref{eq5as2} suitable approximate eigenfunction corresponding to
$\lambda_{\rm o}(\eps)$ and $\lambda_2(\eps)$ are
\begin{equation} \label{eq7bs2}\textstyle
\widetilde\psi_{\rm o}:=\phi_\eps^{p+1\over 2}\quad \mbox{and}\quad
\widetilde\psi_2:=\phi_\eps^{3-p\over 2}
-\frac12\, {{p+3\over p+1}}\,\phi_\eps^{p+1\over 2}.
\end{equation}
Their asset is, that they satisfy the boundary conditions exactly; however, $L_\eps\psi -\lambda\psi$
is small but nonzero, due to \pref{eq16s1}. As an alternative, we can use the functions
given in \pref{eq1s2} plus boundary layer corrections, which produce  error terms likewise.
A lower bound on $\lambda_2$ for $p>3$ is produced likewise as before, using the fact that
$\phi_\eps$ is well approximated by $w_p$ according to lemma \ref{lem2}.
So we have proved the following statement.

\begin{tm}\label{eigenv}
The three smallest eigenvalues of $L_\eps$ satisfy the asymptotic formulae
for $p>1$ and for $\eps\to 0$
\begin{equation} \label{eq8s2}
\begin{array}{l@{\,}l}
\lambda_{\rm o}(\eps)&=-\frac14  (p-1)(p+3)+O\big(e^{-2/\eps}\big)
\\
\lambda_1(\eps)&=-8\alpha^2\zeta e^{- 2/\eps}
\big(1 +O(e^{-\sigma/\eps}) \big) 
\vrule height 1.5em depth .8em width 0pt\\
\lambda_2(\eps)&
\begin{cases} = \frac14  (p-1)(5-p) +O\left(e^{-(3-p)/\eps}\right)
 &\mbox{if }~ p<3\vrule height 1em depth .8em width 0pt\\
\ge 1+O(e^{-2/\eps}) &\mbox{if }~ p\ge 3.
\end{cases}
\end{array}
\end{equation}
where $\sigma:=\min(1,p-1)$, see \pref{eq4as1}.
\end{tm}

\begin{rem}\label{atboundary}\rm
When we linearize around the single boundary
spike steady state solutions (cf. \pref{eq6bs1}),
we can do the same analysis as before
by considering the respective operator
$L_\eps$ on $H^2(0,2/\eps)$ (using the translational
invariance).
 In comparison to the previous case this means that we have
to consider  on the interval $[-{2/\eps}, {2/\eps}]$ the
subset of even functions only and hence, that the odd-indexed
eigenvalues are not present. Hence there is no exponentially small
negative eigenvalue in this case: $\lambda_{\rm o}(\eps)$ persists
and the next eigenvalue is near $\frac14  (p-1)(5-p)$ if $p<3$ or
above $1$ otherwise.
\end{rem}

\begin{rem}\label{elsewhere}\rm
Likewise we can linearize the problem around $w_p$ at any off-central location
 instead of $\phi_\eps$, like Ward \& all do,
\cite{ward1,ward2,wei}; we need  not use an exact stationary solution of
\pref{eq3s1}. For the eigenvalue analysis this makes little difference.
All above results can be translated to this case, if we replace $1/\eps$ in the estimates
by the distance from the center of the (approximate) spike to the boundary.
\end{rem}

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

In this section we consider how the non-local operator $B_\eps$,
defined in \pref{eq10s1}, perturbs the eigenvalue of $L_\eps$ with smallest real
part.
Both the operators $L_\eps$ and $B_\eps$ are invariant under
change of sign $\xi\mapsto{-}\xi$ and hence leave the subspaces of even
and odd functions invariant. Since $B_\eps$ maps odd functions to
zero, the part of the spectrum of $L_\eps$ associated with odd
eigenfunctions remains unchanged. {\sl Hence we restrict our
analysis exclusively to even functions in this section.}
In particular, this removes the  negative exponentially small eigenvalue $\lambda_1(\eps)$
with asymptotics given by \pref{eq8s2}
from the spectrum under study, because it is associated with an odd eigenfunction.
Except for $\lambda_{\rm o}$, all eigenvalues of (the {\sl even} part) of $L_\eps$ are
positive and bounded from below by $\mu:=\min(\lambda_2,1)$.
The addition of the non-local perturbation $B_\eps$ changes the eigenvalues;
as seen from fig.~\ref{evc} they may go off into the complex plane.
Our goal
is to find conditions on the parameters $p,q,r$
so that the spectrum of $A_\eps$ (on even functions) lies in the
right half -plane. Condition \pref{eq1s1} is necessary. In \cite{wei} and \cite{ward2}
it is shown, that positivity is true if $r=p{+}1$ or if $r=2$ and $1<p<5$.
We show that positivity is true also if $r=(p+3)/2$ and, by perturbational techniques,
in a wide area around these cases. Finally we consider a negative result, that
the smallest eigenvalue remains in the negative half plane if $r$ is
slightly larger than $(p-1)/q$ and $q>2$. For consistency of the exposition we
repeat some arguments from the papers cited above.
For ease of
notation we use the spectral resolution of $L_\eps$ (restricted to
the subspace of {\sl even} functions).
\begin{equation} \label{s30}
L_\eps u=\lambda_{\rm o}\langle u,\psi_{\rm o}\rangle
\psi_{\rm o}+\int_{\mu_\eps}^\infty \nu dE_\nu u,
\end{equation}
where $\psi_{\rm o}$ is
the normalized ground state of $L_\eps$ and $\mu_\eps$ is the next
eigenvalue $\mu_\eps=\lambda_2(\eps)$ or
the bottom of the continuous spectrum (if $\eps=0~\mbox{and}~ p\ge 3$).

\subsection{Preliminaries}\label{prelim}
First we show that $\lambda_{\rm o}(\eps)$ is not in
the spectrum of $A_\eps$. Suppose the contrary, that $\lambda_{\rm o}$ is an eigenvalue
with eigenfunction $\psi$, $A_\eps \psi=\lambda_{\rm o}(\eps)\psi$.
Then $\langle\psi,\phi_\eps^{r-1}\rangle\neq 0$,
because
otherwise $B_\eps \psi=0$ and $L_\eps \psi =\lambda_{\rm o}(\eps)
\psi$, such that $\psi$ is a multiple of $\psi_{\rm o}$, which is a positive
and cannot have zero inner product with the positive function $\phi_\eps$.
Because $\lambda_{\rm o}$ is an eigenvalue of $L_\eps$, selfadjointness implies
$$
0=\langle(L_\eps -\lambda_{\rm o}(\eps))\psi, \psi_{\rm o}\rangle=-\langle B_\eps
\psi,\psi_{\rm o}\rangle=-d_{r,\eps}\langle\psi,\phi^{r-1}\rangle
\langle\psi_{\rm o},\phi^p\rangle\neq 0.
$$
This contradiction shows that $\lambda_{\rm o}(\eps)$ is not an
eigenvalue for $A_\eps$. In order to prove, in addition,
that $\lambda_{\rm o}$ is in the resolvent set
we must show that the inverse
of $A_\eps-\lambda_{\rm o}(\eps)$ is bounded.
Since this is a closed operator defined on the whole Hilbert space
$L^2 (-1/\eps,1/\eps)$, it suffices to show that we can solve the equation
$(A_\eps-\lambda_{\rm o}(\eps))\psi =g$ for arbitrary $g$. To this end
we apply Fourier's method and write $g$, $\phi_\eps$ and $\psi$ as a multiple of $\psi_{\rm o}$
and a rest orthogonal to it,
$$
\begin{array}{lll}
g=c\psi_{\rm o}+g_{\rm o},&&
\langle g_{\rm o},\psi_{\rm o}\rangle=0,\\
\psi=l\psi_{\rm o}+h,&\quad \mbox{where}\quad\vrule height 1.2em depth .6em width 0pt&
\langle h,\psi_{\rm o}\rangle=0,\\
\phi_\eps^p=c_{\rm o}\psi_{\rm o}+h_{\rm o}, & &
\langle h_{\rm o},\psi_{\rm o}\rangle=0,
\end{array}
$$
with $c_{\rm o}\neq 0$. This splits the equation into an equation for $h$
and  a scalar one for $l$:
$$
(L_\eps -\lambda_{\rm o}(\eps))h=g_{\rm o}-\frac{c}{c_{\rm o}} h_{\rm o},\quad
\quad lc_{\rm o} \langle\psi_{\rm o},\phi^{r-1}\rangle
+\langle h,\phi^{r-1}\rangle c_{\rm o} =c.
$$
Since both have a unique solution,
we conclude that $\lambda_{\rm o}(\eps)$ is not in the spectrum of $A_\eps$.

In this way the problem is reduced to investigate the spectrum
of $A_\eps$ on the resolvent set of $L_\eps$. There we can use the
formula of Sherman-Morrison for computing the inverse,
\begin{equation} \label{s3beq1}
\begin{array}{l@{\,}l}
(A_\eps-\lambda)^{-1}&=\left(1+(L_\eps-\lambda)^{-1}B_\eps\right)^{-1}(L_\eps
-\lambda)^{-1}\cr\vrule height 1.8em depth 0em width 0pt
& \displaystyle=\Big(1-{(L_\eps-\lambda)^{-1}B_\eps \over
h_\eps(\lambda)}\Big)(L_\eps-\lambda)^{-1},
\end{array}
\end{equation}
where the function $h_\eps(\lambda)$ in the  denominator by \pref{eq17s1}
and \pref{s30} satisfies
\begin{eqnarray}
h_\eps(\lambda)&\!\!\!:=\!\!\!&1+d_{r,\eps}\langle (L_\eps
-\lambda)^{-1}\phi_\eps^p,\phi_\eps^{r-1}\rangle\nonumber\\
&\!\!\!=\!\!\!&1-{d_{r,\eps}\over p-1}\langle (L_\eps
-\lambda)^{-1}L_\eps\phi_\eps,\phi_\eps^{r-1}\rangle\label{s32}\\
&\!\!\!=\!\!\!& 1-{qr\over p-1}-{d_{r,\eps}\over p-1}\langle(L_\eps
-\lambda)^{-1}\phi_\eps,\phi_\eps^{r-1}\rangle\nonumber\\
&\!\!\!=\!\!\!& 1-{qr\over p-1}-{qr\lambda\over\beta_{r,\eps} (p-1)}
\left\{ {\langle \phi_\eps,\psi_{\rm o}\rangle
\langle \psi_{\rm o},\phi_\eps^{r-1}\rangle\over \lambda_{\rm o}-\lambda}+
\int_{\mu_\eps}^\infty { \langle dE_\nu\phi_\eps,
\phi_\eps^{r-1}\rangle\over \nu-\lambda}\right\}.\nonumber
\end{eqnarray}
This formula shows that
$h_\eps$ is analytic on the resolvent set of $L_\eps$ and that any   zero of
$h_\eps$ outside the spectrum of $L_\eps$ is an eigenvalue of $A_\eps$
with associated eigenfunction $\psi:=(L_\eps -\lambda)^{-1} \phi_\eps^p$.
Its value at $\lambda=0$ is given by
$h_\eps(0)=1-{qr\over p-1}=1-\gamma_r<0$.
Since $h(\lambda)\searrow-\infty$ if $\lambda$ is real and
$\lambda\searrow\lambda_{\rm o}$,
$h(\lambda)$ has a negative real zero if $\gamma_r<1$, i.e.
if condition \pref{eq1s1} is not satisfied.
Moreover, this formula implies
\begin{equation} \label{s32a}
\br h_\eps(\lambda)-1+{qr\over p-1}\br\le
{d_{r,\eps}\|\phi_\eps\|\|\phi_\eps^{r-1}\|\over p-1}{\br\lambda\br\over
\min(\br\lambda-\mu_\eps\br ,\br\lambda_{\rm o}-\lambda\br)},
\end{equation}
such that a circle around 0 of radius $O(1{-}{p-1\over qr})$
(for $qr{+}p{-}1{\searrow} 0)$ must be contained
in the resolvent set~of~$A_\eps$.

\subsection{The selfadjoint case $r=p+1$}
\label{selfadjcase}
Since $L_\eps$ is a selfadjoint operator and
$B_\eps u= d_{r,\eps} \langle u,\phi_\eps^{r-1}\rangle \phi_\eps^p~$, the operator
$A_\eps$ is selfadjoint if $r=p+1$ and has real eigenvalues. For real $\lambda$ the derivative
of $h$ satisfies
$$
h_\eps'(\lambda)=d_{r,\eps}\|(L_\eps
-\lambda)^{-1}\phi_\eps^p\|^2\ge 0,\quad \lambda\in\R.
$$
 Hence $h$ is monotonically increasing and can not have negative zeros,
 because $h_\eps(0)<0$.

\begin{figure}[ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=5.9cm]{sab2.eps}\quad
\includegraphics[width=5.9cm]{sab3.eps}\\
\includegraphics[width=5.9cm]{sab4.eps}\quad
\includegraphics[width=5.9cm]{sab5.eps}
\end{tabular}
\end{center}
\caption{ \label{selfadj}Upper left: Lower and upper bounds for the smallest eigenvalue
             of $A_{\rm o}$ for $q{=}0.5,~q{=}1~\mbox{and}~ q{=}2$.
The lower bound is a monotone function of $q$ (as is the eigenvalue itself).
The dashed curve is $\mu_{\rm o}$ (the second eigenvalue or the bottom of the
continuous spectrum),
which provides an upper bound for all $q$. The dotted line provides a sharper upper
bound for $q{=}0.5$. The other pictures give the corresponding domains in $p{\times}r$-plane
around the selfadjoint line (dash-dotted), where the perturbations are smaller
than the lower bound and the real part of the smallest eigenvalue remains positive.
The width of this domain is (more or less) inversely proportional to $q$.
The dashed line represents condition~\pref{eq1s1}.}
\end{figure}


To find the magnitude of perturbations around this case that conserve
the positivity, we derive a lower bound for the smallest eigenvalue.
Because the resolution of the identity of
$L_\eps$ depends smoothly on $\eps\in[0,\eps_{\rm o}]$ for all $\lambda$
in a compact subset of the resolvent of $L_{\rm o}$, we may compute a lower estimate
for $\eps=0$ and extend the result by continuity to $[0,\eps_{\rm o}(p,q,r)]$.
Let $\alpha$ be the
smallest eigenvalue of $A_{\rm o}$. Evidently $\alpha<\mu_{\rm o}.$ We estimate
$\alpha$ from below by an upper estimate of the function
$h_{\rm o}$ given in \pref{s32}.
Using \pref{eq17s1}, we see that for $r=p{+}1$ the integral in its right-hand side may be written
as
\begin{equation} \label{s4eq1}
-\int_{\mu_{\rm o}}^\infty { \langle dE_\nu\phi_{\rm o},
\phi_{\rm o}^{r-1}\rangle\over \nu-\lambda}=
\int_{\mu_{\rm o}}^\infty {\nu\over p-1}{ \langle dE_\nu\phi_{\rm o},
\phi_{\rm o}\rangle\over \nu-\lambda}=
\int_{\mu_{\rm o}}^\infty {p-1\over \nu}{ \langle dE_\nu\phi_{\rm o}^p,
\phi_{\rm o}^{p}\rangle\over \nu-\lambda}
\end{equation}
and that both variants are positive for real $\lambda<\mu_{\rm o}$ and can be estimated from above by
$$
{M\over \mu_{\rm o}-\lambda}:=
\min\Big(
{(p-1)(\|\phi_{\rm o}^p\|^2-\langle \psi_{\rm o},\phi_{\rm o}^p\rangle^2)\over
\mu_{\rm o}(\mu_{\rm o}-\lambda)},
{\mu_{\rm o}(\|\phi_{\rm o}\|^2-\langle \psi_{\rm o},\phi_{\rm o}\rangle^2)\over
(p-1)(\mu_{\rm o}-\lambda)}
\Big)
$$
Inserting this in \pref{s32} we find a simple upper estimate for $h_{\rm o}(\lambda)$
which is monotone on $(\lambda_{\rm o},\mu_{\rm o})$; hence it has a zero
in this interval, that can be computed from  a simple quadratic equation.
For some values of $q$ this lower bound is plotted in figure~\ref{selfadj}.
Since the integral \pref{s4eq1} is positive, we obtain from \pref{s32} an
upper bound for $\alpha$ by omitting the integral.

Having the above control on the smallest positive eigenvalue of the
selfadjoint operator $A_\eps$ in the case $ r=p+1$, we
can consider it for neighbouring values of $r$ as a perturbation of the
selfadjoit one. Namely,
if $A=L+B,$ where $L$ is selfadjoint and $B$ has
relatively small norm, then from the formulae
\begin{equation} \label{eq0s3}
\begin{array}{l}
(A-\lambda)^{-1}=(1+(L-\lambda)^{-1}B)^{-1} (L-\lambda)^{-1},
\vrule height 1em depth .8em width 0pt
\cr \|(L-\lambda)^{-1}\|\leq 1/\mathop{\rm dist}(\lambda,\sigma(L)),
\end{array}
\end{equation}
where $\sigma(L)$ is the spectrum of $L$,
it follows that if $\lambda$ is in the spectrum of $A$ then the
distance from $\lambda$ to the spectrum of $L$ is at most the norm
of $B$. In particular, $\Re \lambda \geq \alpha -\|B\|,$ where
$\alpha$ is the smallest real part of the spectrum of $L$. In our
particular case
$$
B=u\mapsto\langle u,d_{r,\eps}\phi_\eps^{r-1}-d_{p+1,\eps}
\phi_\eps^p\rangle \phi_\eps^p
$$
and we can estimate the real part of any
eigenvalue $\lambda$ of $A_\eps$:
$$
\Re \lambda \geq \alpha_\eps
-\|\phi_\eps^p\|\|d_{r,\eps}\phi_\eps^{r-1}-d_{p+1,\eps} \phi_\eps^p\|,
$$
hence all eigenvalues of $A_\eps$ will have positive real part for
all $0<\eps<\eps_{\rm o}(p,q,r)$ if $(p,q,r)$ satisfy
\begin{equation} \label{selfa10}
\|w_p^p\|\,\|d_{r,\rm o} w_p^{r-1}-d_{p+1,\rm o} w_p^p\|<\alpha_{\rm o}.
\end{equation}


\subsection{The case $p=2r{-}3$}
\label{subgersh}
Using Gershgorin's theorem we prove that spectrum of
$A_\eps$ (restricted to the subspace of {\sl even}
functions)  is in the positive half plane for a large domain in parameter space $(p,q,r)$
around the plane $p=2r{-}3$. As before, we consider first the case $\eps=0$.


\begin{figure}[th] 
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=38mm]{figa1.eps}\quad
\includegraphics[width=38mm]{figa2.eps}\quad
\includegraphics[width=38mm]{figa3.eps}\\
\includegraphics[width=38mm]{figa4.eps}\quad
\includegraphics[width=38mm]{figa5.eps}\quad
\includegraphics[width=38mm]{figa6.eps}
\end{tabular}
\end{center}
\caption{\label{gersh}Domains in the $p{\times}r$-plane, where lemma \ref{gerhp}
implies positivity
of the real part of any eigenvalue, pictured for $q=0.1,~0.5,~1,~2,~4~\mbox{and}~ 8$.
The dashed line represents condition~\pref{eq1s1}; below it the smallest real part is negative,
see.
The straight line, emanating from the point $p=1~\mbox{and}~ r=2$ is the line $2r=p+3$, at
which positivity is proved in remark \ref{rangeBstar}, provided \pref{eq1s1} is satisfied.
}
\end{figure}


Let $(\lambda,\psi)$ with $\Re\lambda<\mu_{\rm o}$ be an eigenpair of $A_{\rm o}$ and let
$\psi=\xi\psi_{\rm o}+\eta g$ be the orthogonal decomposition with
$\langle\psi_{\rm o},g\rangle= 0 ~\mbox{and}~ \|g\|=1$, then $\lambda$
is eigenvalue and $(\xi,\eta)\in\R^2$ the corresponding eigenvector of the matrix
\begin{equation} \label{gersh1}
M_g=
\begin{pmatrix} m_{11}&m_{12}
\vrule height 0em depth .8em width 0em \cr m_{21}&m_{22}\end{pmatrix}
:=\begin{pmatrix}
\langle L_{\rm o}\psi_{\rm o},\psi_{\rm o}\rangle+
\langle B_\eps\psi_{\rm o},\psi_{\rm o}\rangle&
\langle B_\eps g,\psi_{\rm o}\rangle
\vrule height 0em depth .8em width 0em \\
\langle B_\eps\psi_{\rm o},g\rangle&
\langle L_{\rm o}g,g\rangle+\langle B_\eps g,g\rangle \end{pmatrix}.
\end{equation}
With constants $c_1$, $c_2$, $d_1$ and $d_2$ defined by
$$
c_1:=\langle \phi_{\rm o}^{p},\psi_{\rm o}\rangle
,\quad
c_2:=\langle \psi_{\rm o},\phi_{\rm o}^{r-1}\rangle,\quad
d_1^2:=\|\phi_{\rm o}^{p} \|^2-c_1^2 \quad \mbox{and}\quad
d_2^2:=\|\phi_{\rm o}^{r-1} \|^2-c_2^2
$$
the elements of $M_g$ are given by
$$
\begin{array}{ll}
 m_{11}=\lambda_{\rm o}+{d_{r,\eps}}c_1c_2,
&
m_{12}={d_{r,\eps}}c_1 \langle g,\phi_{\rm o}^{r-1}\rangle,
\vrule height 0em depth .8em width 0em \cr
m_{21}={d_{r,\eps}}c_2\langle \phi_{\rm o}^{p},g\rangle,
&
m_{22}=\langle L_{\rm o}g,g\rangle- d_{r,\eps}
\end{array}
$$
and satisfy
for any (normalized) function $g$ orthogonal to $\psi_{\rm o}$ the inequalities
\begin{equation} \label{gersh2}
|m_{12}| \le{d_{r,\eps}}c_1d_2,\quad
|m_{21}|\le{d_{r,\eps}}c_2d_1,\quad
\Re m_{22}\ge\mu_{\rm o}- {d_{r,\eps}}d_1d_2
\end{equation}

\begin{lema}\label{gerhp}
If both $m_{11}$ and $\mu_{\rm o}- {d_{r,\eps}}d_1d_2$ are positive,
then all eigenvalues
of $A_{\rm o}$ have positive real part if
\begin{equation} \label{gersh3}
m_{11}(\mu_{\rm o}- {d_{r,\eps}}d_1d_2)>
\left({d_{r,\eps}}\right)^2c_1c_2 d_1d_2
\end{equation}
\end{lema}

\noindent{\sc Proof.}
From  Gershgorin's circle theorem (see \cite{golub}, theorem 7.2.1) we
know that the eigenvalues of $M_g$ are in the union of the disks around
$m_{11}$ and $m_{22}$ with radii $\br tm_{12}\br$ and $\br m_{21}/t\br$ respectively
for any $t\in\R$. If both diagonal elements have positive real part, the Gershgorin circles
around them are contained in the positive half plane if their radii are smaller
than those real parts. The optimal choice for $t$,
valid for all $g$, leads to condition \pref{gersh3}.\qed

\begin{rem}\label{rangeBstar}\rm
If $2r=p+3$, the non-local term $B$ is zero on the orthogonal complement
of $\psi_{\rm o}$. This implies that $m_{12}=0$ and $m_{22}\ge\mu_{\rm o}$.
Hence, positivity of the spectrum reduces to the condition
$m_{11}>0$. Using relation \pref{eq17s1} we find
\begin{align*}
m_{11}=\,&\lambda_{\rm o} + {d_{r,\eps}}\langle\phi_{\rm o}^p,\psi_{\rm o}\rangle
\langle\psi_{\rm o},\phi_{\rm o}^{(p+1)/2}\rangle\\
=\,&\lambda_{\rm o}- {qr\lambda_{\rm o} \langle\phi_{\rm o},\psi_{\rm o}\rangle
\|\phi_{\rm o}^{(p+1)/ 2}\|\over\beta_{r,\eps}(p-1)}\\
=\,&\lambda_{\rm o}(1-{qr\over p-1})>0,
\end{align*}
implying positivity in the region satisfying \pref{eq1s1}.\qed
\end{rem}
Since the significance of condition \pref{gersh3} is difficult to grasp, we have
computed
the region in $p{\times}r$-plane where it is satisfied and  plotted its boundary for
several values of $q$ in figure~\ref{gersh}. Clearly, the line $2r=p+3$ is always inside the
computed domain. Moreover, we see that $r$ may take large values if $q$ is small.
If $q$ and/or $r$ are large, the non-local term is large too. Its contribution to $m_{11}$ in
\pref{gersh2} is always positive. However, in the lower estimate, that we have to use for $m_{22}$,
it gives a negative contribution. It can be made small only if the components of $\phi_\eps^p$
and $\phi_\eps^{r-1}$ in the orthogonal complement of $\psi_{\rm o}$ are small.

For $\eps>0$ and small we may conclude as
before, that all eigenvalues of $A_\eps$ will have positive real part
for  all $0<\eps<\eps_{\rm o}(p,q,r)$ if \pref{gersh3} is satisfied.

\subsection{A negative result}
\label{subneg}

\begin{figure}[t] 
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=58mm]{neg25.eps}\quad
\includegraphics[width=58mm]{neg3.eps}\\
\includegraphics[width=58mm]{neg5.eps}\quad
\includegraphics[width=58mm]{neg10.eps}
\end{tabular}
\end{center}
\caption{\label{negdom}
Region above the line $qr=p-1$ with $q>3$, where the operator $A_\epsilon$
(restricted to the subspace of even functions) has two real negative eigenvalues.
Because the region is quite small, we rescale the domain and plot
vertically $r-{p-1\over q}$ versus horizontally $p$ for the values
$q=2.5,~3,~5$, and $10$.
}
\end{figure}
\begin{figure}[t] 
\begin{center}
\includegraphics[width=6cm]{eigneg.eps}
\end{center}
\caption{\label{negeig}
Plot of (the real parts of) the four eigenvalues with  smallest real part as a function of $q$
 if $p=2.5$ and $r=1.201$ near the critical value of q, where $p-1=2r$.
}
\end{figure}

As we pointed out in the beginning of this section, condition \pref{eq1s1}
is necessary for positivity of the real parts of all eigenvalues. It is however not
sufficient; we show that $h_\eps$ has a negative zero near the line $qr=p-1$ and $q>3$.
As before, we do the analysis for $\eps=0$ and extend the result to $\eps>0$ by continuity.
We can compute the next term in the expansion
\pref{s32} of $h_{\rm o}$, writing
$$
h_{\rm o}(\lambda)=1-{qr\over p-1} -{d_{r,{\rm o}}\lambda g(\lambda)\over p-1}
,\quad \quad  g(\lambda):=\langle (L_{\rm o} -\lambda)^{-1}\phi,
\phi^{r-1}\rangle.
$$
By \pref{eq18s1} we can compute the value of $g$ at zero
\begin{equation} \label{neg1}
g(0)=\langle L_{\rm o} ^{-1}\phi,
\phi^{r-1}\rangle=\langle \frac{\phi}{p-1}+\frac{\xi \phi'}{2},
\phi^{r-1}\rangle=(\frac{1}{2r}-\frac{1}{p-1})\beta_{r,{\rm o}}.
\end{equation}
and for the remainder $R$ with $g(\lambda)=g(0)+\lambda R(\lambda)$ we have
\begin{equation} \label{neg2}
R(\lambda)=\langle [(L_{\rm o} -\lambda)^{-1}-L_{\rm o}^{-1}]\phi,
\phi^{r-1}\rangle/\lambda=\langle(L_{\rm o} -\lambda)^{-1} L_{\rm o}^{-1}\phi,
\phi^{r-1}\rangle.
\end{equation}
From \pref{neg1} we see that $g(0)>0$ if  $p>2r+1$ and, hence, that $h$ has a negative slope
at $\lambda=0$; since $h(0)<0$ this may imply that $h(\lambda)$ may be positive
for some $\lambda<0$ and hence that $h$ has two negative zeros,
if the remainder $R$ is not too bad.
Using the spectral resolution \pref{s30} of $L_{\rm o}$ (with $\mu_{\rm o}=1$ because
$p>2r{+}1\ge 3$) we get for real $\lambda$ in the interval $[-1,0]$ %$(\lambda_{\rm o},1)$
$$
R(\lambda)={\langle \phi_{\rm o},\psi_{\rm o}\rangle\langle \psi_{\rm o},\phi_{\rm o}^{r-1}\rangle\over
\lambda_{\rm o}(\lambda_{\rm o}-\lambda)}
+\int_1^\infty {\langle dE_\nu \phi_{\rm o},\phi_{\rm o}^{r-1}\rangle\over\nu(\nu-\lambda)}
\le
\frac{c_1c_2}{|\lambda_{\rm o}|(|\lambda_{\rm o}|-1)}+
{d_1d_2}=:a_{\rm o},
$$
where $c_j~\mbox{and}~ d_j$ are given in \pref{gersh2}. Hence, we can estimate $h_{\rm o}$
from below for $\lambda\in[-1,0]$ by the quadratic
\begin{equation} \label{neg3}
h_{\rm o}(\lambda)\ge1-{qr\over p-1} -\frac{q\lambda(p-1-2r)}{2(p-1)^2}
-{d_{r,\rm o}a_{\rm o}\lambda^2\over p-1},
\end{equation}
for which we can easily check, whether it has positive values on $[-1,0]$.
In figure~\ref{negdom} we have plotted the small strip above the line
 $qr=p-1$ with $q>3$, where this estimate proves the existence of two negative
eigenvalues. An obvious drawback of the method is, that it detects a negative result
only if there are two real negative eigenvalues and that the method fails if both
eigenvalues coalesce and go off into a complex pair with a negative real part
as can be seen in figure~\ref{negeig}, where the (real parts of) the four smallest
eigenvalues (of a numerical discretisation) are plotted in the neighbourhood
of the critical point, where $rq=p-1$.


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

In this section we study metastability of the spike solution $S$ of
(\ref{eq2s1}--\ref{eq3s1})
as given in \pref{eq3as1}. We assume that the parameters $(p,q,r)$ are such that all
eigenvalues of $A_\eps$ are located in the
right half plane, except for the exponentially small negative eigenvalue $\lambda_1(\eps)$
associated with an odd (or antisymmetric) eigenfunction.

\noindent{\sc Norms.} Besides the standard $L^2$-norm for functions on the interval $[-1,1]$
denoted by $\|\cdot\|$, we use in this section the
\mbox{``energy norm''}  $\|\cdot\|_1^{}$,
and the ``operator norm''  $\|\cdot\|_2^{}$,
which are associated naturally to a problem with a small parameter like \pref{eq4s1}
and are defined by:
$$ \|u\|_1^{2}:=\|u\|^2+\|\eps u'\|^2 \quad\mbox{and}\quad
 \|u\|_2^{2}:=\|u\|^2+\|\eps^2u''\|^2.
$$
For fixed positive $a$ and  uniformly for all $\eps\in(0,\eps_{\rm o}]$ these norms satisfy
the equivalences
\begin{equation} \label{eq0s4}
\langle(A+a)u,u\rangle^{1/2} \asymp \|u\|_1^{}\quad \mbox{and}\quad \|(A+a)u\|
 \asymp\|Au\|+a\|u\| \asymp
\|u\|_2^{} \quad (a>0)
\end{equation}
We denote by $\|\cdot\|_\infty$ the supremum norm (on the continuous functions);
 it satisfies the Sobolev inequality
\begin{equation} \label{eq0s4a}
\|u\|_\infty^{}\le  \sqrt{2/\eps}\|u\|_1^{}.
\end{equation}
Finally, $C$ denotes a generic positive constant that may differ at each
occurrence.

Metastability is generally used in a vague way in association with small
eigenvalues of some linearized operator.
The idea is, that a solution $S$ is to be called metastable, if a small
perturbation of the initial condition yields a solution that
remains in the vicinity of $S$ during a long time interval, whose length
grows beyond bound, if $\eps\to0$. To be precise:

\begin{defin}\label{metastab} \rm
 The spike solution
$S(\cdot,\eps),~ 0<\eps<\eps_{\rm o}$, is called metastable in the norm
$|\cdot|$ if there exist (monotonous) order functions
$\delta(\eps)\to 0,\; \delta_{\rm o}(\eps)\to 0$ and
$T(\eps)\to \infty$ as $\eps \to 0$, such that if a solution $U$ of \pref{eq3s1}
initially satisfies
$\br U(\cdot,0)-S\br<\delta_{\rm o}(\eps)$, then
it satisfies $\br U(\cdot,t)-S\br<\delta(\eps)$ for
all $0<t<T(\eps),\;0<\eps<\eps_{\rm o}.$
\end{defin}

Obviously, we are not satisfied with pure existence of such order functions and we
want to find explicit estimates on their decay or growth.


We study perturbations around the steady state spike solution $S$.
As in \cite{gk} we use a contraction method to show, that a solution starting near $S$
stays confined to a small tube around $S$ for an (exponentially) long time lapse.
The perturbation satisfies  eq.~\pref{eq13s1}, which reads:
$$
v_t + A v =f[v],\quad v(x,0)=v_{\rm o}(x),
$$
where the quadratic term $f$ is given by \pref{eq14s1}
and the (linear) operator $A$ is defined by \pref{eq15s1}. Obviously, this
operator $A$ has the same spectral properties as its (stretched) cousin $A_\eps$
has in sections \ref{reference} and \ref{perturb}.
Under the positivity condition, stated above, $A$ is a sectorial
operator, see \cite{kato}:
there exists an angle $\chi\in (0,\pi/2)$, such that the
resolvent set of $A$ contains the sector
$$
\Lambda:=\{\lambda\in \C \,\vrule height 1em depth .4em width .5pt\,
 \chi \leq |\arg (\lambda-\lambda_1
(\epsilon))|\leq \pi,\; \lambda\neq \lambda_1(\epsilon)\}.
$$
In this sector the resolvent satisfies for some constant $M$, not depending on
$\eps$ for all small $\eps$, the estimate
$$
\|(A-\lambda)^{-1}\|\leq \frac{M}{|\lambda-\lambda_1(\epsilon)|}\;\quad
\mbox{for all}\; \lambda\in\Lambda.
$$
Associated to $A$ is the semigroup
\begin{equation} \label{eq1s4}
e^{-At} :=\frac{1}{2\pi i} \int_{\Gamma} (A-\lambda)^{-1}e^{-\lambda t}
d\lambda,\quad  t>0,
\end{equation}
where $\Gamma$ is a suitable contour in the resolvent set $\Lambda$.

\begin{lema}\label{l2bounds}
For all $t>0$, all $\eps\in(0,\eps_0]$
and for some constant $c>0$ not depending on
$t$ and $\epsilon$ this semigroup satisfies:
\begin{eqnarray}
\label{eq3s4}\displaystyle \|e^{-At}\| &\leq& C 
e^{-\lambda_1(\epsilon) t}\,,\\
\label{eq4s4}\displaystyle
\|Ae^{-At}\| &\leq& C\,(|\lambda_1|+t^{-1})
e^{-\lambda_1(\epsilon) t}\,,\vrule height 1.4em depth .5em width 0pt\\
\label{eq3s4a}\displaystyle \|e^{-At}\,u\|_1^{}& \leq& C 
e^{-\lambda_1(\epsilon) t}\,
\left\{
\begin{array}{l}
\|u\|_1^{}\,,\cr
(1+t^{-1/2})\,\|u\|\,,\vrule height 1.3em depth 0em width 0pt
\end{array}\right.
\\
\label{eq4s4a}\displaystyle
\|e^{-At}\,u\|_2^{} &\leq& C\,
e^{-\lambda_1(\epsilon) t}\,\|u\|_2^{}\,.\vrule height 1.4em depth .1em width 0pt
\end{eqnarray}
\end{lema}

\noindent {\sc Proof}.
We choose in \pref{eq1s4} the contour $\Gamma:=\{-\delta+\lambda_1+\rho e^{\pm i\chi}\br\rho\in\R^+\}$
for some $\delta>0$ to be determined later on.
Because $\cos\chi>0$, we may estimate the resolvent for any point in this set by
$$
\|(A+\delta-\lambda_1+\rho e^{\pm i\chi})^{-1}\|\leq
\frac{M}{|\delta+\rho e^{\pm i\chi}|}
\le {M\over \sqrt{(1-\cos\chi)(\delta^2+\rho^2})}.
$$
Hence,
$$
\|e^{-At}\|\leq {Me^{\delta t-\lambda_1t}\over\pi\sqrt{1-\cos\chi} }
\int_{0}^\infty \frac{e^{-\rho t \cos\chi}}{\sqrt{\rho^2+\delta^2}}
d\rho.
$$
Clearly, the choice $\delta=1/t$ yields the desired constant for a proof of
\pref{eq3s4}. To check \pref{eq4s4} we use the same contour and $\delta$ in
the $t$-derivative of \pref{eq1s4}:
\begin{align*}
\|A e^{-At}\| =&\|\frac{1}{2\pi i} \int_{\Gamma}
( A-\lambda)^{-1}e^{-\lambda t} \lambda d\lambda\| \\
\le & {Me^{\delta t-\lambda_1t}\over\pi\sqrt{1-\cos\chi} }
\int_{0}^\infty \frac{e^{-\rho t \cos\chi}}{\sqrt{\rho^2+\delta^2}}
(|\delta|+|\lambda_1|+\rho)d\rho.
\end{align*}
To prove \pref{eq4s4a} we use the equivalence \pref{eq0s4}
$$
\|e^{-At}u\|_2^{} \asymp \|e^{-At}u\|+\|e^{-At}Au\|
\leq C e^{-\lambda_1(\epsilon) t}(\|u\|+\|Au\|)\le
C e^{-\lambda_1(\epsilon) t}\|u\|_2^{}.
$$
The inequalities \pref{eq3s4a} results from interpolation between
\pref{eq4s4a} and \pref{eq3s4}.\qed

\begin{rem}\label{Asub}\rm
Let $\overline A$ be the restriction of $A$ to the orthogonal complement of the
(true) eigenfunction $\psi_1(x/\eps,\eps)$ associated with $\lambda_1$
and let $2\mu_1$ be equal to the real part of the eigenvalue of $\overline A$
with smallest real part, then the semigroup generated by this restriction
satisfies the estimates
\begin{gather}
\label{eq3as4} %\label{eq3s4b}
 \|e^{-\overline At}\|_j^{} \leq C e^{-\mu_1 t}\|u\|_j^{},\quad ~j=0,~1~\mbox{or}~2,\\
\label{eq4s4b} %\label{eq4as4}
\|\overline Ae^{-\overline At}\| \leq Ct^{-1} e^{-\mu_1 t},
\end{gather}
proved in analogous way.
\end{rem}


Now we estimate the nonlinear term $f[v]$ in
\pref{eq13s1}, using $\langle\cdot,\cdot\rangle$ for the inner product in $L^2(-1,1)$.
From $g(u)=2^{q}\langle u^r,1\rangle^{-q}u^p$
we find the explicit formula
\begin{equation} \label{eq5s4}
\begin{aligned}
f[v]=\,& 2^{q}\int_{0}^1(1-\sigma)
\Big\{ p(p-1)\langle (S+\sigma v)^r,1\rangle^{-q}(S+\sigma v)^{p-2} v^2+\\
&{}-2pqr\langle (S+\sigma v)^r,1\rangle^{-q-1}
\langle (S+\sigma v)^{r-1},v \rangle(S+\sigma v)^{p-1}v+
\vrule height 1.2em depth .6em width 0pt\\
&{}+ q(q+1)\langle (S+\sigma v)^r,1\rangle^{-q-2}
\langle (S+\sigma v)^{r-1},v \rangle^2(S+\sigma v)^{p}+\\
&{}-qr(r-1)\langle (S+\sigma v)^r,1\rangle^{-q-1}
\langle (S+\sigma v)^{r-2},v ^2\rangle(S+\sigma v)^p
\Big\}d\sigma,\vrule height 1.2em depth 0em width 0pt
\end{aligned}
\end{equation}
provided $(S+\sigma v)$ is strictly positive.
Using \pref{eq2cbs1}, \pref{eq3as1} and \pref{eq7as1}
$$
\int_{-1}^1 \Phi^r(\xi,\epsilon)d\xi \asymp
\int_{-1}^1 w_p^r(\xi/\epsilon)d\xi \asymp \epsilon
$$
implying
$$
\int_{-1}^1 S^r d\xi \asymp \epsilon^{1-r\alpha_r}=\epsilon^{(1-p)/(qr+1-p)},
$$
and assuming that $v(\cdot,t)\in H^1(-1,1)$ is bounded in modulus by
$\rho S(\cdot,\eps)$
for some $\rho\in(0,\frac12 )$, we find the estimate
\begin{equation} \label{eq7s4a}
\br f[v](x,\cdot)\br\le C\rho^2\eps^{q(p-1)/(qr-p+1)}S^p(x,\eps)
\end{equation}
if $|v(x,\cdot)|\le\rho S(x,\eps)$ and $\rho\le1/2$.
To estimate the difference $f[v]-f[w]$ we write
$$
f[v]-f[w]=\int_{0}^1 (1-\sigma) \partial_\sigma^2g(S+w+\sigma (v-w))
d\sigma +\int_{0}^1 \partial_\sigma[\partial_\tau
g(S+\sigma w+ \tau (v-w))_{|\tau=0}] d\sigma
$$
Assuming that $v(\cdot,t)~\mbox{and}~ w(\cdot,t)\in H^1(-1,1)$ are bounded
in modulus by $\rho S(\cdot,\eps)$
and that the difference satisfies
$\br v(\cdot,t)-w(\cdot,t)\br\le \tau S(\cdot,\eps)$ we find by analogy to
\pref{eq5s4}
\begin{equation} \label{eq7s4b}
\br f[v]-f[w]\br\le C\tau\rho\eps^{q(p-1)/(qr-p+1)}S^p(x,\eps)
\end{equation}
Now we have the problem, that bounds weighted by $S$ are not compatible with
the $L^2$-bounds in lemma \ref{l2bounds}. We choose here for a rough way out
by using the maximum and the minimum of $S$ for estimates from above and below:
\begin{equation} \label{eq5s4a}
0<c_1\eps^{-\alpha_r}e^{-1/\eps}\le S(x,\eps)\le c_2 \eps^{-\alpha_r}
\end{equation}
for some positive constants $c_1$ and $c_2$.
This certainly will influence the magnitude of the region of
contraction but not the existence of such a region.
Using \pref{eq0s4a}, we get the estimates
\begin{equation} \label{eq7s4}
\|f[v]\| \leq c R(\epsilon) \|v\|_1^2,
\quad \mbox{and}\quad \|f[v]-f[w]\| \leq c R(\epsilon) (\|v\|_1+\|w\|_1)\|v-w\|_1,
\end{equation}
where
\begin{equation} \label{eq8s4}
R(\epsilon):=\epsilon^{\alpha_r-1}
e^{2/\epsilon}.
\end{equation}
To apply the contraction method we consider the integral equation,
equivalent to \pref{eq13s1},
$$
v=Gv,\quad \mbox{where}\quad  Gv(\cdot,t)=e^{-At}v_{\rm o} +
\int_{0}^t e^{-A(t-s)}f[v(\cdot,s)]ds.
$$
On the time interval $[0,T]$ we have for all $t$ the estimates:
$$
\|Gv(\cdot,t)\|_1 \leq ce^{|\lambda_1| T} \|v_{\rm o}\|_1+
cR(\eps)(1+T)e^{|\lambda_1| T}
\sup_{0<s<T} \|v(\cdot,s)\|_1^2 ,
$$
and
\begin{align*}
 \|Gv&(\cdot,t)-Gw(\cdot,t)\|_1 \leq ce^{|\lambda_1| T}\\
 &\times \Big\{ \|v_{\rm o}-w_{\rm o}\|_1+R(\eps)(1+T)
\sup_{0<s<T} ( \|v(\cdot,s)\|_1+w\|(\cdot,s)\|_1)\|v(\cdot,s)-(\cdot,s)\|_1
\Big\}.
\end{align*}
Hence, the operator $G$ maps the cylinder of functions $v$ on $(-1,1)\times [0,T]$, that
satisfy \mbox{$\|v(\cdot,0)\|_1\le \rho$,} into itself and is a contraction there,
provided $\rho R(\eps)(1+T)\le\gamma $ for some sufficiently small
positive constant $\gamma$. This proves
the metastability of the single
internal spike solution $S(x,\eps)$ in the Sobolev norm
$\|\cdot\|_1$ in the sense of definition \ref{metastab}:

\begin{tm}[Metastability of the single internal spike]
 There exist positive constants $c,~\gamma$ and $ \eps_{\rm o}$,
depending on $p,$ $q$ and $r$ only, such that
the solution $U$ of the shadow
equation \pref{eq3s1} exists for all times $0<t<1/|\lambda_1(\eps)|$ and
satisfies
$$
\|U(\cdot,t)-S\|_1 \leq \rho,\quad \quad
0<t<1/|\lambda_1(\eps)|,\quad 0<\eps<\eps_{\rm o}\,,\quad 0<\rho<\rho_\eps\,.
$$
for all initial conditions $U_{\rm o}\in
H^1 (-1,1)$ in the vicinity of $S$, that satisfy the compatibility
conditions \mbox{$U_{\rm o}'(-1)=U_{\rm o}'(1)=0$} and satisfy the bound
$$
\|U_{\rm o}-S\|_1 \leq c \rho,\quad \quad 
\rho_\eps:=\gamma|\lambda_1|/R(\eps),\quad  0<\eps<\eps_{\rm o}.
$$
\end{tm}


\begin{rem}[Local stability of the single boundary spike] \rm
We already know from remark
\ref{atboundary} that the  operator $A$ will not have small negative
eigenvalue,
if we linearize around a single boundary spike steady
state solution then. Thus in the region of parameters $(p,q,r)$,
where the large negative eigenvalue is shifted to the positive halfplane
by the non-local term, the real part of the spectrum
of $A$ is positive and we have a positive lower bound
$\Re \lambda > \alpha(p,q,r)>0.$ Therefore we have local stability:
\\
There exist positive constants $c,~\gamma$ and $ \eps_{\rm o}$,
depending on $p,$ $q$ and $r$ only, such that
the solution $U$ of the shadow
equation \pref{eq3s1} exists for all times $t>0$ and
satisfies
$$
\|U(\cdot,t)-S\|_1 \leq \rho e^{-\alpha(p,q,r)t},\quad
0<\eps<\eps_{\rm o}\,,\quad 0<\rho<\rho_\eps\,.
$$
for all
initial conditions $U_{\rm o}\in
H^1 (-1,1)$ in the vicinity of $S$, that satisfy the compatibility
conditions \mbox{$U_{\rm o}'(-1)=U_{\rm o}'(1)=0$} and satisfy the bound
$$
\|U_{\rm o}-S\|_1 \leq  c\rho,\quad \rho_\eps:=\gamma/R(\eps),\quad
0<\eps<\eps_{\rm o}.
$$
\end{rem}

\begin{rem} \rm
Since the exponentially small negative eigenvalue has a one dimensional eigenspace
consisting of the derivative of the spike profile and since $A$ is ``stable'' on the
orthogonal complement, we can show the existence of an 1-D unstable manifold
tangent to this eigenspace at the origin, like in
\cite{gk} or in \cite{henry} (page 113).
Any small perturbation is attracted to this unstable manifold exponentially fast
and hence develops into (exponentially) slow motion of the spike.
\end{rem}


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\end{document}