# From transient elastic linkages to friction:
a complete study of a penalized fourth order equation with delay

Vuk Milišić<EMAIL_ADDRESS>Univ. Brest, CNRS UMR 6205 Laboratoire
de Mathématiques de Bretagne Atlantique, 6, avenue Victor Le Gorgeu, 29200,
Brest, France  Philippe Souplet<EMAIL_ADDRESS>Université
Sorbonne Paris Nord, LAGA, CNRS UMR 7539, 99, avenue Jean-Baptiste Clément,
93430, Villetaneuse, France

###### Abstract

In this paper we consider a fourth order nonlinear parabolic delayed problem
modelling a quasi-instantaneous turn-over of linkages in the context of cell-
motility. The model depends on a small parameter $\varepsilon$ which
represents a typical time scale of the memory effect. We first prove global
existence and uniqueness of solutions for $\varepsilon$ fixed. This is
achieved by combining suitable fixed-point and energy arguments and by
uncovering a nonlocal in time, integral conserved quantity. After giving a
complete classification of steady states in terms of elliptic functions, we
next show that every solution converges to a steady state as $t\to\infty$.
When $\varepsilon\to 0$, we then establish convergence results on finite time
intervals, showing that the solution tends in a suitable sense towards the
solution of a parabolic problem without delay. Moreover, we establish the
convergence of energies as $\varepsilon\to 0$, which enables us to show that,
for $\varepsilon$ small enough, the $\varepsilon$-dependent problem inherits
part of the large time asymptotics of the limiting parabolic problem.

###### keywords:

integral equations, memory effects, cell motility, parabolic equations,
nonlinear penalized pointwise constraint, adhesion, Allen-Cahn equation, limit
sets, steady states, fourth order elliptic problems.

###### MSC:

35Q92

##  1 Introduction

This article is a contribution to the mathematical study of adhesion forces in
the context of cell motility, in continuation to the project [7, 8, 9, 10].
Cell adhesion and migration play a crucial role in many biological phenomena
such as embryonic development, inflammatory responses, wound healing and tumor
metastasis. The main motivation comes from the seminal papers [13, 11], where
the authors built a complex and realistic model of the Lamellipodium. It is a
cytoskeletal quasi-two-dimensional actin mesh, and the whole structure propels
the cell on the substrate.

In this setting, the authors of [13, 11] consider an axi-symmetric
idealization of the network. It is represented by two families of filaments
(clockwise and anti-clockwise) interacting with each other. Thanks to these
hypotheses, the problem reduces to a single equation whose solution is the
position $z(s,t)\in\mathbb{R}^{2}$ of a single filament evolving in time
$t\in{\mathbb{R}_{+}}$ and with respect to a reference configuration
$s\in(0,L)$. Using gradient flow techniques, the authors obtain a nonlinear
equation:

$\left\\{\begin{aligned}
&\underbrace{z_{0}^{\prime\prime\prime\prime}}_{\text{bending}}-\underbrace{(\lambda
z_{0}^{\prime})^{\prime}}_{\text{unextensibility}}+\underbrace{\mu^{A}D_{t}z_{0}}_{\text{adhesion
}}+\underbrace{\left(\left(\arccos\left(\left|z_{0}\right|^{\prime}\right)-\varphi_{0}\right)(z_{0}^{\prime})^{\bot}\right)^{\prime}}_{\text{twisting}}+\underbrace{D_{t}\varphi
z^{\bot}_{0}}_{\text{stretching}}=0\\\ &\left|\partial_{s}z_{0}\right|=1,\\\
\end{aligned}\right.$ (1.1)

where $\varphi=\arg z_{0}$ and $z^{\bot}_{0}=(-z_{0,2},z_{0,1})^{T}$ and ′
denotes the space derivative (with respect to $s$). The Lagrange multiplier
$\lambda(s,t)$ accounts for the inextensibility constraint and is an unknown
of the problem. They complement this equation with initial and natural
boundary conditions that we omit for sake of conciseness. The adhesion and
stretching friction terms appearing in this force balance equation were
obtained as formal limits of a microscopic description of adhesion mechanisms.
For instance the first friction term is obtained as the limit when
$\varepsilon$, a small dimensionless parameter, goes to zero in the
expression:

$\frac{1}{\varepsilon}\int_{0}^{\infty}\left(z_{\varepsilon}(s,t)-z_{\varepsilon}(s+\varepsilon
a,t-\varepsilon
a)\right)\varrho(s,a,t)da\to\mu_{1}(s,t)\left(\partial_{t}-\partial_{s}\right)z_{0}(s,t)=:\mu^{A}D_{t}z.$

Here $a$ represents the age of a linkage established with the previous
locations of the filament and the shift in the reference configuration
$s+\varepsilon a$ comes from the (de)polymerization of the filaments inside
the lamellipodium. The parameter $\varepsilon$ represents at the same time a
characteristic age of the linkages, and the inverse of their stiffness. The
elongation $(z(s,t)-z(s+\varepsilon a,t-\varepsilon a))/\varepsilon$, for a
fixed $a$, is the linear elastic force exerted on $z(s,t)$ by a linkage
established at $z(s+\varepsilon a,t-\varepsilon a)$ (see also [12, 4] for an
extension to the non-axi-symmetric case).

Our initial goal was to give a rigorous justification of these formal
computations but the nonlinear space-dependent feature of (1.1) made it out of
reach at that time. A first attempt to cope with the fully nonlinear space-
dependent problem was performed in [6]. In this latter article the first
author considered delayed harmonic maps ($z^{\prime\prime\prime\prime}$ is
replaced by $z^{\prime\prime}$ and the constraint $|z^{\prime}(s,t)|=1$ is
replaced by $|z(s,t)|=1$ for a.e. $s\in(0,L)$) and showed rigorously the
asymptotics with respect to $\varepsilon$. In that work a key feature is
compactness in time. When compared with the classical parabolic problem
without delay [14], compactness is far more difficult to obtain. Indeed, in
the standard case, using minimizing movements [1], a variational principle
provides this estimate first, and convergence with respect to the
discretisation step of the semi-discrete scheme in time is then immediate.
This provides in turn existence (and uniqueness) for the continuous problem
(1.1) for instance (this was exactly the method used in [11] when starting
from classical minimizing movements à la De Giorgi [1]).

The present work is the first attempt in order to handle the fourth order case
from [11], and this turns out to be quite challenging mathematically. Because
the geometric structure (related to the constraint) of the problem is
different, results from [6] do not apply. Here we consider a penalization of
the constraint replacing it by a non-convex potential $F(\xi)=(1-\xi^{2})^{2}$
in the variational principle. These hypotheses lead to study a delayed
parabolic model of Allen-Cahn type (but of fourth order in space) where the
penalization term applies to $z^{\prime}$, that we detail in the next
subsection.

###  1.1 The model

We define $I:=(0,L)$, with $L>0$, to be a segment of the real line. We now
assume that the filament at time $t$ is described by the graph of a planar
curve $I\ni s\mapsto z(s,t)\in\mathbb{R}$. First consider the time dependent
energy (depending on the past positions of the filament), defined by

$\displaystyle\tilde{\mathcal{E}}_{t}(w):=\frac{1}{2\varepsilon}$
$\displaystyle\int_{I}\int_{0}^{\infty}\bigl{(}w(s)-z_{\varepsilon}(s,t-\varepsilon
a)\bigr{)}^{2}\varrho(s,a)dads+\int_{I}\kappa^{2}(w(s))ds+\frac{1}{\delta}\int_{I}\tilde{F}(w^{\prime}(s))ds.$
(1.2)

Here $\varepsilon,\delta,\ell>0$ are parameters, the double well potential
reads $\tilde{F}(\xi):=(\xi^{2}-\ell^{2})^{2}$,

$\kappa(w):=\frac{w^{\prime\prime}}{(1+|w^{\prime}|^{2})^{3/2}}$

is the curvature, $\rho$ is a given kernel and
$z_{\varepsilon}(s,t)={z_{p}}(s,t)$ for $s\in I$ and for negative times, with
a given past data ${z_{p}}$. The $\tilde{F}$-term in (1.2) is a penalization
taking into account the limited extensibility of the filament, whereas the
$\kappa$-term in (1.2) accounts for the bending energy. In this paper we shall
actually consider the simplified case where $\kappa(w)$ is replaced by
$w^{\prime\prime}$. We will also assume $\delta=1$ and (with no loss of
generality up to proper normalization) $\ell=1$. This leads to the energy
functional

$\displaystyle{\mathcal{E}}_{t}(w):=\frac{1}{2\varepsilon}$
$\displaystyle\int_{I}\int_{0}^{\infty}\bigl{(}w(s)-z_{\varepsilon}(s,t-\varepsilon
a)\bigr{)}^{2}\varrho(s,a)dads+\frac{1}{2}\int_{I}|w^{\prime\prime}(s)|^{2}ds+\int_{I}F(w^{\prime}(s))ds,$
(1.3)

where $F(\xi):=(\xi^{2}-1)^{2}$. Although this will to some extent simplify
the treatment, this captures the essential mathematical features of the
problem under study, namely the conjunction of a fourth-order operator, a time
delay operator and an Allen-Cahn type nonlinearity acting on the space
derivative. In a future study, we plan to further develop the ideas of the
present work to treat the fully nonlinear case (1.2). Here we will focus on
the vanishing memory limit $\varepsilon\to 0$, leaving the inextensible limit
$\delta\to 0$ for future investigation. The, potentially difficult, case when
the filament is described by a curve which is not a graph will be also
addressed elsewhere.

Given the energy in (1.3), we start from the formal variational minimization
principle:

$z_{\varepsilon}(t):=\operatorname*{argmin}_{w\in
H^{2}(I)}{\mathcal{E}}_{t}(w),\quad t>0.$

Define the delay operator ${\mathcal{L}}_{\varepsilon}$ by

$\displaystyle{\mathcal{L}}_{\varepsilon}[z_{\varepsilon}](s,t)$
$\displaystyle=\frac{1}{\varepsilon}\int_{0}^{\infty}\bigl{(}z_{\varepsilon}(s,t)-z_{\varepsilon}(s,t-\varepsilon
a)\bigr{)}\varrho(s,a)da,$

which can be seen as a time derivative with memory effect and compared to a
fractional derivative of order greater or equal to one. The above minimization
procedure leads to the variational problem: find
$z_{\varepsilon}=z_{\varepsilon}(s,t)$, with $z_{\varepsilon}({\cdot},t)$
valued in $H^{2}(I)$, solving

$\left\\{\begin{aligned}
&\int_{I}{\mathcal{L}}_{\varepsilon}[z_{\varepsilon}](t)vds+\int_{I}\bigl{(}z_{\varepsilon}^{\prime\prime}(t)v^{\prime\prime}+F^{\prime}(z_{\varepsilon}^{\prime}(t))v^{\prime}\bigr{)}ds=0,&\forall
v\in H^{2}(I),\ t>0,\\\
&z_{\varepsilon}(t)=z_{p}(t),&t<0.\end{aligned}\right.$ (1.4)

Here and throughout the paper we omit the variable $s$ when no risk of
confusion arises. The strong form of the integral equation in the latter
problem reads:

${\mathcal{L}}_{\varepsilon}[z_{\varepsilon}]+z_{\varepsilon}^{\prime\prime\prime\prime}-\left(F^{\prime}(z_{\varepsilon}^{\prime})\right)^{\prime}=0,\quad
s\in I,\ t>0,$ (1.5)

complemented by the corresponding natural boundary conditions:

$z_{\varepsilon}^{\prime\prime\prime}-F^{\prime}(z_{\varepsilon}^{\prime})=z_{\varepsilon}^{\prime\prime}=0,\quad
s\in\partial I,\ t>0.$ (1.6)

###  1.2 Assumptions and notation

In our main results we will assume that the given kernel $\rho$ satisfies

$0\leq\,\rho(s,a)\in L_{a}^{\infty}(0,\infty;H^{2}(I))\cap
L_{a}^{1}(0,\infty;H^{2}(I)),$ (1.7) $\rho\in
W_{a}^{1,1}(0,\infty;L^{\infty}(I)),\quad\partial_{a}\rho\leq 0\ \ a.e.,$
(1.8) $0<\mu_{\min}\leq\mu(s):=\int_{\mathbb{R}}\rho(s,a)da,\quad s\in I,$
(1.9)
$\int_{I}\int_{a=0}^{\infty}a^{3/2}\|\rho(\cdot,a)\|_{2}da<\infty,\quad\int_{0}^{\infty}a\|\rho(\cdot,a)\|_{\infty}da<\infty.$
(1.10)

For some results, we will need the following mild coercivity assumption on the
kernel:

$\int_{I}\int_{0}^{\infty}\rho^{2}(s,a)|\partial_{a}\rho(s,a)|^{-1}dads<\infty.$
(1.11)

Note that assumption (1.10) is for instance satisfied if $0\leq\rho(s,a)\leq
K(1+a)^{-m}$ with some constants $m>\frac{5}{2}$, $K>0$, and that (1.11) then
holds if in addition $-\partial_{a}\rho(s,a)\geq c(1+a)^{-4}$ with $c>0$.

As for the past data, we will assume

$z_{p}\in W^{1,\infty}(-\infty,0;H^{2}(I)).$ (1.12)

In connection with the kernel, we define the useful functions

$\varphi(s,\tau)=\int_{\tau}^{\infty}\rho(s,a)da,\quad\mu_{1}(s)=\int_{0}^{\infty}a\rho(s,a)da=\int_{\tau=0}^{\infty}\varphi(s,\tau)d\tau$
(1.13)

and, for given past data $z_{p}$ and all $\varepsilon\geq 0$, we define the
following constants

$\kappa_{\varepsilon}=\int_{\tau=0}^{\infty}\int_{I}z_{p}(s,-\varepsilon\tau)\varphi(s,\tau)dsd\tau.$
(1.14)

Throughout the paper, the $L^{2}(I)$ scalar product and the $L^{p}(I)$ norm
($1\leq p\leq\infty)$ will be respectively denoted by $(\cdot,\cdot)$ and
$\|\cdot\|_{p}$. The Sobolev-Slobodecki spaces (over $(0,T)$ or $I$, possibly
fractional and vector valued) will be denoted by $W^{k,p}$ for
$k\in[0,\infty)$, $p\in[1,\infty]$ and we will set $H^{k}=W^{k,2}$. Spaces
like $L^{2}(I),H^{k}(I)$ will be often abreviated by $L^{2},H^{k}$. We will
sometimes omit the subscript $\varepsilon$ when no confusion may arise.
Throughout this paper, $C$ will denote a generic positive constant depending
only on $\rho,L$.

###  1.3 Steady states and limiting problem

First, in order to motivate our results and explain their significance, we
need to introduce the stationary problem associated with (1.5) and the
limiting problem as $\varepsilon\to 0$. The parameter $\varepsilon$ in (1.5)
represents a typical time scale of the memory effect, which gradually “fades
out” as $\varepsilon$ becomes smaller. At a formal level, the limiting problem
to (1.5) as $\varepsilon\to 0$ is given by the following parabolic problem
without delay:

$\left\\{\begin{aligned}
b(s)\partial_{t}z_{0}+z_{0}^{\prime\prime\prime\prime}-\bigl{(}F^{\prime}(z_{0}^{\prime})\bigr{)}^{\prime}&=0,&s\in
I,\ t>0,\\\
z_{0}^{\prime\prime\prime}-F^{\prime}(z_{0}^{\prime})=z_{0}^{\prime\prime}&=0,&s\in\partial
I,\ t>0,\\\ z_{0}(s,0)&=\phi(s),&s\in I,\end{aligned}\right.$ (1.15)

where $b=\mu_{1}$ is the first moment of $\rho$, defined in (1.13), and
$\phi=z_{p}(\cdot,0)$. We note that (1.5) and (1.15) have the same steady
states, namely the solutions of the stationary problem

$\left\\{\hskip 5.69054pt\begin{aligned}
Z^{\prime\prime\prime\prime}-\bigl{(}F^{\prime}(Z^{\prime})\bigr{)}^{\prime}=0,&\quad&s\in
I,\\\
F^{\prime}(Z^{\prime})-Z^{\prime\prime\prime}=Z^{\prime\prime}=0,&\quad&s\in\partial
I.\end{aligned}\right.$ (1.16)

As a preliminary to the rest of our analysis, we first give the complete
classification of these steady states (see Section 2 for precise definition
and proof).

######  Proposition 1.1.

  * (i)

Assume $L\leq\pi/2$. Then, up to additive constants, problem (1.16) admits
only the affine solutions $Z_{0}\equiv 0$ and $Z_{\pm 1}\equiv\pm x$.

  * (ii)

Assume $L>\pi/2$ and let $m=\lfloor 2L/\pi\rfloor\in N^{*}$. Beside the affine
solutions, up to additive constants, (1.16) admits exactly $2m$ nonaffine
solutions, namely $Z_{2},\cdots,Z_{m+1}$, and their opposites $Z_{-i}=-Z_{i}$.

As for the, more classical, parabolic problem without delay (1.15), we have
the following result (see Definition A.1 and Remark A.1 for the precise notion
of solution).

######  Proposition 1.2.

Let $b\in H^{2}(I)$ with $\inf_{I}b>0$ and $\phi\in H^{2}(I)$.

(i) Problem (1.15) has a unique, global solution

$z_{0}\in C_{b}([0,\infty);H^{2}(I))\cap H^{1}_{loc}([0,\infty);L^{2}(I))\cap
L^{2}_{loc}([0,\infty);H^{4}(I)).$ (1.17)

Moreover, for each $p\in(2,\infty)$, we have

$\sup_{t\geq 1}\
\left(\,\|z_{0}\|_{W^{1,p}(t,t+1;L^{2}(I))}+\|z_{0}\|_{L^{p}(t,t+1;H^{4}(I))}\,\right)<\infty.$
(1.18)

(ii) There exists a stationary solution $Z_{0}$, with $(Z_{0},b)=(\phi,b)$,
such that

$\lim_{t\to\infty}\|z_{0}(t)-Z_{0}\|_{H^{2}(I)}=0.$ (1.19)

Although Proposition 1.2 may be known, we have been unable to find a suitable
reference, especially in view of the needed time regularity, in presence of
nonlinear boundary conditions. We thus provide a proof in A, based on maximal
regularity estimates from [2] for inhomogeneous, linear higher order parabolic
problems.

###  1.4 Main results

The main goals of this work are to show that:

$\bullet$ For sufficiently small $\varepsilon>0$, the time-nonlocal problem
(1.5) is globally well posed and its solution converges to one of the steady
states as $t\to\infty$.

$\bullet$ When $\varepsilon\to 0$, the solution $z_{\varepsilon}$ of (1.5)
converges in a certain sense to the solution $z_{0}$ of problem (1.15).

$\bullet$ For sufficiently small $\varepsilon>0$, problem (1.5) inherits part
of the large time asymptotic properties of (1.15).

Although this may seem reasonable, this is by no means obvious, in view of the
nonlocal nature of (1.5), and will require rather involved arguments.

######  Theorem 1.

Assume (1.7)–(1.10) and (1.12). There exists $\varepsilon_{0}>0$ depending
only on $\rho,L$ and on $\|z_{p}\|_{L^{\infty}(-\infty,0;H^{2}(I))}$ such
that, for all $\varepsilon\in(0,\varepsilon_{0}]$, the following holds:

(i) Problem (1.4) admits a unique global solution $z_{\varepsilon}\in
L^{\infty}(\mathbb{R};H^{2}(I))$.

(ii) The curve $\varepsilon\mapsto z_{\varepsilon}(t)$ is locally $1/4$-Hölder
continuous from $(0,\varepsilon_{0}]$ into $H^{2}(I)$, uniformly for $t>0$ in
bounded intervals.

(iii) There exists a stationary solution $Z_{\varepsilon}$, with
$(Z_{\varepsilon},\mu_{1})=\kappa_{\varepsilon}$, such that,

$z_{\varepsilon}(t)\to Z_{\varepsilon}\hbox{ in $H^{2}(I)$, as $t\to\infty$.}$
(1.20)

(iv) The set of $\varepsilon$ such that $Z^{\prime}_{\varepsilon}\equiv 1$
(resp. $-1$) is relatively open in $(0,\varepsilon_{0}]$.

######  Theorem 2.

Assume (1.7)–(1.10), (1.12), and let $z_{0}$ be the solution of (1.15) with
$b=\mu_{1}$ and $\phi=z_{p}(s,0)$.

(i) For every fixed $T>0$, we have

$\lim_{\varepsilon\to 0}z_{\varepsilon}=z_{0},$ (1.21)

where the convergence is strong in $C([0,T];C^{1}(\bar{I}))$, weak in
$H^{1}(0,T;L^{2}(I))$ and weak-∗ in $L^{\infty}(0,T;H^{2}(I))$. If in addition
(1.11) holds, then the convergence in (1.21) is strong in
$L^{2}(0,T;H^{2}(I))$.

(ii) Assume that (1.11) holds and that $Z^{\prime}_{0}\equiv 1$ (resp. $-1$).
Then there exists $\bar{\varepsilon}_{0}\in(0,\varepsilon_{0}]$ such that, for
all $\varepsilon\in(0,\bar{\varepsilon}_{0}]$, $Z^{\prime}_{\varepsilon}\equiv
1$ (resp. $-1$).

######  Remark 1.1.

(i) By a simple energy argument (see Remark A.2(i) for details), the stable
steady states of (1.15) are exactly the nonconstant affine steady states
($Z\equiv\pm x+Const.$). More precisely, if $\phi\in H^{2}(I)$ and
$\|\phi^{\prime\prime}\|_{2}+\|\phi^{\prime}\pm 1\|_{\infty}$ is sufficiently
small, then $Z^{\prime}_{0}\equiv\mp 1$. Theorem 2(ii) guarantees that under
this assumption for $\phi=z_{p}(0)$, we have
$Z^{\prime}_{\varepsilon}\equiv-1$ (resp. $1$) for $\varepsilon>0$ small.

On the contrary, the result of Theorem 2(ii) is not expected to hold if
$Z^{\prime}_{0}\not\equiv\pm 1$, since the steady state is then unstable even
for the limiting problem (cf. Remark A.2(ii)).

(ii) In Theorem 1, by $z_{\varepsilon}\in L^{\infty}(\mathbb{R};H^{2}(I))$
being a solution we mean that the first (resp., second) part of (1.4) is
satisfied for a.e. $t>0$ (resp., $t<0$). Under the assumptions of Theorem 1,
the solution actually enjoys the additional regularity

$z_{\varepsilon}\in L^{\infty}(0,\infty;W^{4,\infty}(I))\cap
W^{1,\infty}(0,\infty;H^{2}(I)),$ (1.22)

so that the first part of (1.4) is satisfied for all $t>0$ and
$z_{\varepsilon}$ is in fact a strong solution. Moreover, it satisfies the
uniform global bound

$\|z_{\varepsilon}(t)\|_{H^{2}(I)}\leq C\bar{R},\quad t>0,\
\varepsilon\in(0,\varepsilon_{0}],$

and the uniform, time derivative global integrability property

$\int_{0}^{\infty}\int_{I}|\partial_{t}z_{\varepsilon}|^{2}dsdt\leq
C\bar{R},\quad\varepsilon\in(0,\varepsilon_{0}],$ (1.23)

where

$R:=1+\|z_{p}\|_{L^{\infty}(-\infty,0;H^{2}(I))},\quad\hat{R}:=\|z_{p}\|_{W^{1,\infty}(-\infty,0;L^{\infty}(I))},\quad\bar{R}=R^{4}+\hat{R}^{2}.$

Also we may take $\varepsilon_{0}=c_{0}\bar{R}^{-4}$, with
$c_{0}=c_{0}(\rho,L)>0$. See Propositions 4.1 and 6.1.

(iii) The solution $z_{\varepsilon}$ is continuous in $H^{2}(I)$ for $t>0$ and
has a limit as $t\to 0^{+}$ (owing to (1.22)), but it generally has a jump
discontinuity at $t=0$, unless a suitable compatibility condition is imposed
on $z_{p}$. A useful estimate of the jump, of independent interest, is given
in Proposition 5.3.

Local existence and uniqueness rely on a Banach fixed point. The proof is
based on fixed point arguments built on the resolution of an elliptic fourth-
order problem. It is to be noticed that this is not standard in the framework
of partial differential evolution equations where a parabolic problem is more
often used. Here we fix-point not only the nonlinearity but also the past of
the solution, involved as a source term in the fixed-point operator. We
analyze the fourth order elliptic problem in the Hilbert setting and for
classical solutions, giving by interpolation a general result in $L^{p}$
spaces for $p\geq 2$ which is of interest per se. The previous step guarantees
a local time of existence for $\varepsilon$ small enough. In order to extend
this time we show a stability result. Namely we derive (1.23). These estimates
are uniform with respect to $\varepsilon$ and to $t$. It is much more involved
than for (1.15). Indeed, for smooth solutions of classical gradient flow
models one has easily that:

$\frac{d}{dt}\left\\{\int_{0}^{L}F({z_{0}}^{\prime}(t))ds+\frac{1}{2}\|{z_{0}}^{\prime\prime}(t)\|_{2}^{2}\right\\}=-\int_{0}^{L}b(s)\left|\partial_{t}{z_{0}}(t)\right|^{2}ds,$
(1.24)

giving directly the result after integration in time. For delayed problems
this does not hold and an alternative approach has to be found. Thanks to the
monotonicity condition (1.8), one first ensures that

$\frac{d}{dt}{\mathcal{E}}_{t}(z_{\varepsilon})=\frac{1}{2\varepsilon^{2}}\int_{0}^{L}\int_{0}^{\infty}\bigl{(}z_{\varepsilon}(t)-z_{\varepsilon}(t-\varepsilon
a)\bigr{)}^{2}\partial_{a}\varrho\,dads<0,$ (1.25)

which guarantees the time integrability of the above right hand side. This
latter term is then used as a source term for a closed elliptic problem
satisfied by $\partial_{t}z_{\varepsilon}$. These estimates provide an
$L^{\infty}_{t}L^{2}_{s}$ bound for $z_{\varepsilon}^{\prime}$ and
$z_{\varepsilon}^{\prime\prime}$. In a second step we deduce the global
existence and $H^{2}$-boundedness for all $t>0$ by showing a uniform
$L^{\infty}$ bound in time and space, based on a new invariant
$\int_{0}^{L}\int_{0}^{\infty}z_{\varepsilon}(s,t-\varepsilon
a)\varphi(s,a)dads=\kappa_{\varepsilon}$, where $\varphi$ is defined in
(1.13), which appears here in as a key quantity. While $\varphi$ is somehow
reminiscent of the so-called left eigenvector in studies of hyperbolic
equations from population dynamics [15, Chap. 3], it is completely new in the
present context. This invariant is to be related to $(\mu_{1},{z_{0}})$ to
which it formally reduces when $\varepsilon$ goes to zero. As a matter of
fact, we show, cf. Theorem 1(iii), that actually for $\varepsilon>0$ fixed,
$z_{\varepsilon}(t)\to Z_{\varepsilon}$ when $t$ grows large and
$(Z_{\varepsilon},\mu_{1})=\int_{0}^{L}\int_{0}^{\infty}{z_{p}}(-\varepsilon
a)\varphi(s,a)dads$. The same result holds as well when $\varepsilon=0$ and is
more standard (cf. Proposition 1.2(ii)). Next, in Section 9, we show that
problem (1.1) is consistent with the weak solution of (1.15) when
$\varepsilon$ vanishes. This result relies mainly on compactness arguments
which make use of the a priori estimates on $\partial_{t}z_{\varepsilon}$
mentioned above. Then we show rigorously how (1.25) converges to (1.24) when
$\varepsilon$ vanishes. This result is new as compared with [8, 10, 6], to
which it applies as well (see Remark 10.1). Moreover it allows to show
stability of affine stationary states $Z^{\prime}=\pm 1$. This relies on three
main ingredients: (i) the mechanical energy
$\int_{0}^{L}(\frac{1}{2}|Z^{\prime\prime}|^{2}+F(Z^{\prime}))ds$ vanishes for
those, (ii) $L^{2}_{t}H^{2}_{s}$-convergence of $z_{\varepsilon}$ towards
$z_{0}$ when $\varepsilon$ goes to zero, (iii) decrease of energies in time.

The outline of the rest of the paper is as follows. In Section 2 we describe
the steady states of the problem. In Section 3, we study a linear auxiliary
problem and provide resolvent estimates in terms of $\varepsilon$ that will be
useful for the local existence-uniqueness. In Section 4 we prove the local
existence-uniqueness (Theorem 3) and time regularity (Proposition 4.1) of the
solution. In Section 5, we establish some key energy estimates. Section 6 is
devoted to uniform $L^{\infty}$ bounds and to the proof of global existence.
Continuity with respect to $\varepsilon$ is established in Section 7.
Convergence as $t\to\infty$ for fixed $\varepsilon$ is proved in Section 8.
Theorem 2 is finally proved in Sections 9-10. In the appendix we provide a
proof of Proposition 1.2 and of the energy balance associated to the limiting
problem (1.15).

##  2 Steady states

Before entering the analysis of the evolution problem (1.4), a basic and more
elementary task is to describe the steady states and to establish Proposition
1.1, which is needed in the proof of the convergence part of Theorem 1.

We say that $Z$ is a (classical) solution of the stationary problem (1.16) if
$Z\in C^{4}([0,1])$ and $Z$ satisfies (1.16) pointwise. It is not difficult to
see that weak and classical solutions of (1.16) are equivalent notions.
Indeed, if $Z\in H^{2}(I)$ is a weak solution, it solves:

$\left(Z^{\prime\prime},v^{\prime\prime}\right)+(F^{\prime}(Z^{\prime}),v^{\prime})=0,\quad\forall
v\in H^{2}(I)$

then it solves (1.16) in the distributional sense. As $Z^{\prime}\in
C(\overline{I})$ and $Z^{\prime\prime}\in L^{2}(I)$, this implies that
$Z^{\prime\prime\prime\prime}\in L^{2}(I)$ and thus $Z\in H^{4}(I)\subset
C^{3}(\overline{I})$ thanks to Sobolev’s embeddings. Again using
$\eqref{eqstatz}$ provides $Z^{\prime\prime\prime\prime}\in
C^{1}(\overline{I})$. The sufficient condition is obvious.

To prove Proposition 1.1 (and give additional information on the solutions),
we proceed as follows. Observe that $z$ is a solution of (1.16) if and only if
$w:=z^{\prime}$ solves

$w^{\prime\prime}=F^{\prime}(w)=-4w(1-w^{2})\ \ \hbox{ in $(0,L)$, \quad with
}w^{\prime}(0)=w^{\prime}(L)=0.$ (2.1)

To study (2.1), for all $a\in\mathbb{R}$, we consider the shooting problem

$w^{\prime\prime}=F^{\prime}(w)\quad\hbox{ with $w^{\prime}(0)=0$ and
$w(0)=a$.}$ (2.2)

Problem (2.2) has a unique solution $w_{a}$, defined on a maximal time
interval $[0,S_{a}^{*})$ for some $S_{a}^{*}\in(0,\infty]$. We clearly have
$w_{0}=0$, $w_{\pm 1}=\pm 1$ (constant solutions). Also by symmetry, we have
$w_{-a}=-w_{a}$, so that it thus suffices to consider $a>0$, $a\neq 1$. This
case is treated in the following proposition.

######  Proposition 2.1.

  * (i)

If $a>1$ then $w^{\prime}_{a}>0$ on $(0,S_{a}^{*})$. Moreover,
$S_{a}^{*}<\infty$ and $\lim_{s\to S_{a}^{*}}w^{\prime}_{a}(s)=\infty$.

  * (ii)

For $a\in(0,1)$, we have $S_{a}^{*}=\infty$ and $w_{a}$ is given by an anti-
periodic solution of $w^{\prime\prime}=F^{\prime}(w)$, for some half-period
$\Pi_{a}>0$ (which is in particular a periodic solution of period $2\Pi_{a}$)
and is explicit:

$w_{a}(s)=a\,{\mathrm{sn}}\left(\sqrt{2(2-a^{2})}s+K(k_{a}),k_{a}\right),\quad
k_{a}=\frac{a}{\sqrt{2-a^{2}}},$ (2.3)

where ${\mathrm{sn}}$ denotes the sine amplitude elliptic function and $K(k)$
the complete elliptic integral of the first kind. The number $K(k)$ is a
quarter period associated to ${\mathrm{sn}}(\cdot,k)$ and reads:

$K(k):=\int_{0}^{\frac{\pi}{2}}\frac{1}{\sqrt{1-k^{2}\sin^{2}\hskip-1.5pt\theta}}\,d\theta,\quad
k\in(0,1).$

Moreover we have $w^{\prime}_{a}<0$ on $(0,\Pi_{a})$.

  * (iii)

The half-period function $(0,1)\ni a\mapsto\Pi_{a}$ is continuous monotone
increasing with

$\lim_{a\to 0^{+}}\Pi_{a}=\pi/2,\qquad\lim_{a\to 1}\Pi_{a}=\infty$ (2.4)

and reads: $\Pi_{a}:=K(k_{a})/\sqrt{2(2-a^{2})}$.

As a consequence we can describe the solutions of (2.1) and (1.16) as follows.

######  Proposition 2.2.

  * (i)

Assume $L\leq\pi/2$. Then problem (2.1) admits only the contant solutions $0$
and $\pm 1$.

  * (ii)

Assume $L>\pi/2$ and let $m=\lfloor 2L/\pi\rfloor\in N^{*}$. Beside the
constant solutions, (2.1) admits an even finite number $2m$ of solutions $\pm
w_{1},\cdots,w_{m}$ where, for each $n\in\\{1,\dots,m\\}$, $w_{n}$ is the
unique antiperiodic solution of (2.2) with $a>0$ such that $\Pi_{a}=L/n$.

  * (iii)

The solutions of (1.16) are given by $z(s)=\int_{0}^{s}w(\tau)d\tau+c$ where
$w$ is any solution of (2.1) and $c\in\mathbb{R}$ is arbitrary.

###### Proof of Proposition 2.1.

We shall denote $w=w_{a}$ for simplicity.

(i) If $a>1$ then $w$ cannot be a solution of (2.1). Indeed, we have
$w^{\prime\prime}(0)>0$ and we easily deduce that $w^{\prime\prime}$ and
$w^{\prime}$ remain $>0$ for $s>0$ as long as $w$ exists. It follows that
$w^{\prime\prime}\geq cw^{3}$ for all $s\in(0,S^{*})$ with some $c>0$, and a
standard argument then implies $S^{*}<\infty$ and $\lim_{s\to
S^{*}}w^{\prime}(s)=\infty$.

(ii) Multiplying (2.2) by $w^{\prime}(s)$ gives
$\frac{1}{2}\left(w_{a}^{\prime}\right)^{2}{\phantom{}}^{\prime}=F(w_{a})^{\prime}$.
We integrate with respect to $s$. This leads to:

$\frac{1}{2}\left(w_{a}^{\prime}\right)^{2}(s)-F(w_{a}(s))=\frac{1}{2}\left(w_{a}^{\prime}\right)^{2}(0)-F(w_{a}(0)).$

Now using the boundary conditions in (2.2) provides:

$\frac{1}{2}\left(w_{a}^{\prime}(s)\right)^{2}=F(w_{a}(s))-F(a),$

which finally gives:

$\left(w_{a}^{\prime}(s)\right)^{2}=2(1-F(a))-4w_{a}(s)^{2}+2w_{a}(s)^{4}=:Aw_{a}^{4}+Bw_{a}^{2}+c.$

We compare this equation with ${\mathrm{sn}}(s,k)$ satisfying [16]:

$\left({\mathrm{sn}}^{\prime}(s,k)\right)^{2}=\alpha\,{\mathrm{sn}}^{4}(s,k)+\beta\,{\mathrm{sn}}^{2}(s,k)+\gamma,\quad\alpha:=k^{2},\quad\beta:=-(1+k^{2}),\quad\gamma=1,$

where $k$ is a given parameter in $(0,1)$. We look for $w_{a}$ to be of the
form:

$w(s)=\xi\,{\mathrm{sn}}(\chi\,(s-s_{0}),k),$

where $\xi$, $\chi$ and $k$ are constants to be found. This provides a system
of 3 unknowns s.t.

$2=\frac{\chi^{2}}{\xi^{2}}k^{2},\quad 4=\chi^{2}(1+k^{2}),\quad
2(1-F(a))=\xi^{2}\chi^{2}.$

After some computations and accounting that $k\in(0,1)$, one obtains the
following definition of the constants $\xi,\chi,k$ solving the previous
system:

$\xi_{a}=a,\quad\chi_{a}=\sqrt{2(2-a^{2})},\quad
k_{a}=\frac{a}{\sqrt{2-a^{2}}}.$

We next ensure that the solution satisfies the boundary conditions (2.2) at
the origin $s=0$. All this leads to the explicit form (2.3). It is a periodic
solution of period $4K(k_{a})/\sqrt{2(2-a^{2})}$.

(iii) The complete elliptic integral of the first kind $K(k)$ is a regular
monotone increasing function of $k\in(0,1)$ s.t. (cf. [16]), $\lim_{k\to
0}K(k)=\frac{\pi}{2}$, and $\lim_{k\to 1}=K(k)=+\infty$, so that
$\Pi_{a}=2\frac{K(k_{a})}{\chi_{a}}$ and one has

$\lim_{a\to 0}\Pi_{a}=\frac{\pi}{2},\quad\lim_{a\to 1}\Pi_{a}=+\infty.$

Moreover,

$\Pi_{a}^{\prime}=-\int_{0}^{\frac{\pi}{2}}\frac{4a(\cos^{2}(t)-2)dt}{\left({4(1-a^{2})+2a^{2}\cos^{2}(t)}\right)^{\frac{3}{2}}}>0,\quad\forall
a\in(0,1),$

which ends the proof. ∎

###### Proof of Proposition 2.2.

In view of assertion (i) of Proposition 2.1, if $a>1$ then $w$ cannot be a
solution of (2.1). In view of assertion (ii), for $a\in(0,1)$, the function
$w_{a}$ is a solution of (2.1) (i.e., solves the boundary condition
$w^{\prime}(L)=0$) if and only if its antiperiod is a submultiple of $L$, that
is, $\Pi_{a}=L/n$ for some integer $n\geq 1$. This implies that:

$L\geq\Pi_{a}\geq\frac{\pi}{2}.$

We exclude the case $L=\pi/2$ since then no non-trivial solution $w_{a}$
exists. Next, the problem: find $(a_{n})_{n\in\\{1,\dots,m\\}}\in(0,1)^{m}$
s.t.

$\frac{L}{n}=\Pi_{a_{n}}$

admits a unique solution iff $L/n\in[\pi/2,\infty)$ for all
$n\in\\{1,\dots,m\\}$, which is precisely the way $m$ was chosen. This
procedure guarantees the construction of $2m$ distinct solutions of (2.1).

This along with the constant solutions describes the set of solutions of (2.1)
and completes the proof of assertions (i) and (ii). Assertion (iii) follows
immediately by integration. ∎

##  3 Resolvent operator

In order to first establish the local in time existence-uniqueness (by a
suitable fixed point argument), problem (1.5)-(1.6) will be rewritten under
the form

$\left\\{\hskip 5.69054pt\begin{aligned}
(\mu(s)+\varepsilon\partial^{4}_{s})z&=\varepsilon\partial_{s}(F^{\prime}(\partial_{s}z))+\int_{0}^{\infty}z(s,t-\varepsilon
a)\rho(s,a)da,&\;&0<t<T,\ s\in I,\\\
\partial^{2}_{s}z(s,t)&=\partial^{3}_{s}z(s,t)-F^{\prime}(\partial_{s}z(s,t))=0&\;&0<t<T,\
s\in\partial I,\\\ z(t)&=z_{p}(t)&\;&t<0.\end{aligned}\right.$ (3.1)

To study (3.1), for given $\varepsilon>0$, we will need resolvent estimates
for the linear auxiliary problem:

$\left\\{\begin{aligned} &ku+\varepsilon
u^{\prime\prime\prime\prime}=\varepsilon f^{\prime}+g,&s\in I,\\\
&u^{\prime\prime}=0,\,u^{\prime\prime\prime}=f,&s\in\partial
I.\end{aligned}\right.$ (3.2)

Here $k_{2}\geq k_{1}>0$ and $k=k(s)\in L^{\infty}(I)$ is a fixed function
such that

$0<k_{1}\leq k(s)\leq k_{2},\quad s\in I.$ (3.3)

In this section, $K$ denotes a generic positive constant depending only on
$k_{1},k_{2},L$. For given data $(f,g)$ in $L^{2}(I)\times L^{2}(I)$, by a
weak solution of (3.2) we understand a function $u\in H^{2}(I)$ which
satisfies the variational identity

$\int_{I}k(s)u(s)v(s)ds+\varepsilon\int_{I}u^{\prime\prime}(s)v^{\prime\prime}(s)ds=-\varepsilon\int_{I}f(s)v^{\prime}(s)ds+\int_{I}g(s)v(s)ds,\quad\forall
v\in H^{2}(I).$ (3.4)

We start with the basic existence-uniqueness result for (3.2).

######  Proposition 3.1.

Assume (3.3) and let $\varepsilon>0$, $(f,g)\in L^{2}(I)\times L^{2}(I)$.

  * (i)

Problem (3.2) admits a unique weak solution.

  * (ii)

Assume in addition that $f\in H^{1}(I)$. Then a function $u\in H^{2}(I)$ is a
weak solution if and only if it is a strong solution i.e., $u\in H^{4}(I)$,
$u$ satisfies the equation for a.e. $s\in I$ and the boundary conditions
pointwise.

We shall denote by $A_{\varepsilon}:L^{2}(I)\times L^{2}(I)\to H^{2}(I)$ the
resolvent operator, defined by $(f,g)\mapsto u$. Note that, by linearity, we
may write

$A_{\varepsilon}(f,g)=A_{1,\varepsilon}f+A_{2,\varepsilon}g,$ (3.5)

where

$A_{1,\varepsilon}:f\in L^{2}(I)\mapsto A_{\varepsilon}(f,0),\quad
A_{2,\varepsilon}:g\in L^{2}(I)\mapsto A_{\varepsilon}(0,g).$ (3.6)

The following proposition provides the required resolvent estimates.

######  Proposition 3.2.

Assume (3.3) and let $\varepsilon\in(0,1)$.

(i) Let $(f,g)\in L^{2}(I)\times L^{2}(I)$. Then $u=A_{\varepsilon}(f,g)$
satisfies

${\lVert u^{(j)}\rVert}_{2}\leq
K\varepsilon^{-j/4}\bigl{(}\varepsilon^{3/4}{\lVert f\rVert}_{2}+{\lVert
g\rVert}_{2}\bigr{)},\quad j\in\\{0,1,2\\}.$ (3.7)

(ii) Let $q\in[2,\infty]$ and $(f,g)\in W^{1,q}(I)\times L^{q}(I)$. Then, for
any integer $j\in[0,4]$, we have

$\|A_{1,\varepsilon}f\|_{W^{j,q}}\leq
K\varepsilon^{(3-j)/4}\left(\|f\|_{q}+\varepsilon^{1/4}\|f^{\prime}\|_{q}\right)$
(3.8)

and

$\|A_{2,\varepsilon}g\|_{W^{j,q}}\leq K\varepsilon^{-j/4}\|g\|_{q}.$ (3.9)

(iii) Let $K_{1}>0$ and assume that

$k\in H^{2}(I),\quad\|k\|_{H^{2}(I)}\leq K_{1}.$ (3.10)

Then $A_{2,\varepsilon}\in{\mathcal{L}}(H^{2}(I),H^{4}(I))$ and

$\|A_{2,\varepsilon}\|_{\mathcal{L}(H^{2}(I),H^{2+j}(I))}\leq\tilde{K}\varepsilon^{-j/4},\quad
j\in\\{0,1,2\\},$ (3.11)

where the constant $\tilde{K}$ depends only on $k_{1},k_{2},K_{1},L$.

###### Proof of Proposition 3.1.

(i) This is an immediate consequence of Lax-Milgram’s Theorem.

(ii) The weak solution $u$ satisfies the differential equation in (3.2) in the
sense of distributions and, assuming $f\in H^{1}(I)$, we have
$u^{\prime\prime\prime\prime}=\varepsilon^{-1}(g+\varepsilon f^{\prime}-ku)\in
L^{2}(I)$, so that $u\in H^{4}(I)\subset C^{3}(\overline{J})$. As one has
enough regularity, starting from the weak formulation and integrating by parts
one shows that

$\bigl{[}(u^{\prime\prime\prime}-f)v-u^{\prime\prime}v^{\prime}\bigr{]}_{0}^{1}=\int_{I}\bigl{(}u^{\prime\prime\prime\prime}-f^{\prime}+\varepsilon^{-1}(ku-g)\bigr{)}v=0\quad\hbox{
for all $v\in H^{2}(I)$.}$

It follows easily that $u^{\prime\prime\prime}=f$ and $u^{\prime\prime}=0$ on
$\partial I$.

Conversely, if $f\in H^{1}(I)$ and $u$ a strong solution, by multiplying by
$v\in H^{2}(I)$ and integrating by parts, we easily obtain (3.4). ∎

###### Proof of Proposition 3.2.

(i) Recall the following interpolation inequality (which is a consequence of
[3, Theorem 7.37, p.198]): If $|J|\geq L_{0}>0$, there exists $K=K(L_{0})>0$
such that, for all $\eta>0$

$\|u^{\prime}\|_{L^{2}(J)}\leq\eta\|u^{\prime\prime}\|_{L^{2}(J)}+K(1+\eta^{-1})\|u\|_{L^{2}(J)}.$
(3.12)

For $\delta_{1},\delta_{2},\eta>0$ to be fixed, the choice $v=u$ in (3.4)
yields

$\displaystyle\int_{I}$
$\displaystyle\bigl{(}ku^{2}+\varepsilon{u^{\prime\prime}}^{2}\bigr{)}(s)ds\leq\delta_{1}\varepsilon{\lVert
u^{\prime}\rVert}_{L^{2}}^{2}+\delta_{1}^{-1}\varepsilon{\lVert
f\rVert}_{2}^{2}+\delta_{2}{\lVert u\rVert}_{2}^{2}+\delta_{2}^{-1}{\lVert
g\rVert}_{2}^{2}$ $\displaystyle\leq\delta_{1}\varepsilon\eta{\lVert
u^{\prime\prime}\rVert}_{L^{2}}^{2}+(C_{0}(1+\eta^{-1})\delta_{1}\varepsilon+\delta_{2}){\lVert
u\rVert}_{L^{2}}^{2}+\delta_{1}^{-1}\varepsilon{\lVert
f\rVert}_{2}^{2}+\delta_{2}^{-1}{\lVert g\rVert}_{2}^{2},$

with $C_{0}=C_{0}(k_{1},k_{2})>0$. Choosing
$\delta_{1}=\min(\frac{1}{2},\frac{k_{1}}{6C_{0}})\varepsilon^{-1/2}$,
$\eta=(2\delta_{1})^{-1}$ and $\delta_{2}=\frac{k_{1}}{3}$, we get

$C_{0}(1+\eta^{-1})\delta_{1}\varepsilon\leq
C_{0}(1+2\delta_{1})\frac{k_{1}}{6C_{0}}\varepsilon^{1/2}\leq
2C_{0}\varepsilon^{-1/2}\frac{k_{1}}{6C_{0}}\varepsilon^{1/2}=\frac{k_{1}}{3},$

so that

$k_{1}{\lVert u\rVert}_{L^{2}}^{2}+\varepsilon{\lVert
u^{\prime\prime}\rVert}_{L^{2}}^{2}\leq\frac{\varepsilon}{2}{\lVert
u^{\prime\prime}\rVert}_{L^{2}}^{2}+\frac{2k_{1}}{3}{\lVert
u\rVert}_{L^{2}}^{2}+\frac{3}{k_{1}}{\lVert
g\rVert}_{2}^{2}+2\varepsilon^{3/2}{\lVert f\rVert}_{2}^{2},$

hence (3.7) for $j\in\\{0,2\\}$. The case $j=1$ follows from (3.12) with
$\eta=\varepsilon^{1/4}$.

(ii) First consider the case $q=2$. For $j\in\\{0,1,2\\}$, (3.8) and (3.9)
follow from assertion (i). For $j=4$ it is then a consequence of
$u^{\prime\prime\prime\prime}=\varepsilon^{-1}(g-ku)+f^{\prime}$. The result
for $j=3$ is then obtained by interpolating through (3.12).

We next consider the case $q=\infty$ and establish the $L^{\infty}$ estimate
by means of a contradiction and rescaling argument. Thus assume that (3.8) or
(3.9) with $j=0$ fails. Then there exist sequences $\varepsilon_{i}\in(0,1)$,
$k_{i},f_{i}\in L^{\infty}(I)$, $g_{i}\in W^{1,\infty}(I)$ such that, denoting
by $u_{i}$ the corresponding solutions, we have

$k_{1}\leq k_{i}(s)\leq
k_{2},\quad\|g_{i}\|_{\infty}+\varepsilon_{i}^{3/4}\|f_{i}\|_{\infty}+\varepsilon_{i}\|f_{i}^{\prime}\|_{\infty}=1,\quad
M_{i}:=\|u_{i}\|_{\infty}\to\infty.$

Pick $s_{i}\in\overline{I}$ such that $|u_{i}(s_{i})|=M_{i}$ and define the
rescaled functions

$v_{i}(\sigma)=M_{i}^{-1}u_{i}(s_{i}+\varepsilon_{i}^{1/4}\sigma),\quad\sigma\in
J_{i}=[a_{i},b_{i}]:=[-\varepsilon_{i}^{-1/4}s_{i},\varepsilon_{i}^{-1/4}(L-s_{i})].$

Since $u_{i}$ is a strong solution by Proposition 3.1(ii), we have $v_{i}\in
W^{4,\infty}(J_{i})$ and $v_{i}$ satisfies

$\left\\{\hskip 5.69054pt\begin{aligned}
&\tilde{k}_{i}v_{i}+v_{i}^{\prime\prime\prime\prime}=\varepsilon_{i}^{3/4}\tilde{f}_{i}^{\prime}+\tilde{g}_{i},&\quad&a.e.\
\sigma\in J_{i},\\\
&v_{i}^{\prime\prime}=0,\;v_{i}^{\prime\prime\prime}=\varepsilon_{i}^{3/4}\tilde{f}_{i},&\quad&\sigma\in\partial
J_{i},\end{aligned}\right.$ (3.13)

with $\tilde{k}_{i}(\sigma)=k_{i}(s_{i}+\varepsilon_{i}^{1/4}\sigma)$,
$\tilde{f}_{i}(\sigma)=M_{i}^{-1}f_{i}(s_{i}+\varepsilon_{i}^{1/4}\sigma)$ and
$\tilde{g}_{i}(\sigma):=M^{-1}_{i}g_{i}(s_{i}+\varepsilon_{i}^{1/4}\sigma)$.
Then $\|v_{i}\|_{L^{\infty}(J_{i})}=|v_{i}(0)|=1$ and

$\displaystyle\varepsilon_{i}^{3/4}\|\tilde{f}_{i}^{\prime}\|_{L^{\infty}(J_{i})}$
$\displaystyle+\varepsilon_{i}^{3/4}\|\tilde{f}_{i}\|_{L^{\infty}(J_{i})}+\|\tilde{g}_{i}\|_{L^{\infty}(J_{i})}$
(3.14)
$\displaystyle=M_{i}^{-1}\bigl{(}\varepsilon_{i}\|f_{i}^{\prime}\|_{L^{\infty}(I)}+\varepsilon_{i}^{3/4}\|f_{i}\|_{L^{\infty}(I)}+\|g_{i}\|_{L^{\infty}(I)}\bigr{)}\leq
M_{i}^{-1}.$

In particular, $\|v_{i}^{\prime\prime\prime\prime}\|_{L^{\infty}(J_{i})}\leq
c$. Here and in the rest of the proof, $c$ denotes a generic positive constant
independent of $i$. We deduce

$\|v_{i}\|_{W^{4,\infty}(J_{i})}\leq c$ (3.15)

by interpolation (note that $|J_{i}|\geq L$ due to $\varepsilon_{i}<1$). Let
$\alpha<\beta$ be such that $[\alpha,\beta]\subset J_{i}$. Multipying the
differential equation (3.13) with $v_{i}$ and integrating by parts, we obtain

$\int_{\alpha}^{\beta}\tilde{k}_{i}v_{i}^{2}+(v_{i}^{\prime\prime})^{2}=\bigl{[}v_{i}^{\prime}v_{i}^{\prime\prime}-v_{i}v_{i}^{\prime\prime\prime}\bigr{]}_{\alpha}^{\beta}+\int_{\alpha}^{\beta}(\varepsilon_{i}^{3/4}\tilde{f}_{i}^{\prime}+\tilde{g}_{i})v_{i}d\sigma.$
(3.16)

$\bullet$ First consider the case when $a_{i}$ and $b_{i}$ are bounded.
Applying (3.16) with $\alpha=a_{i}$ and $\beta=b_{i}$ and using (3.14),
(3.15), along with the boundary conditions in (3.13), yields

$\displaystyle 1$ $\displaystyle=\|v_{i}\|_{L^{\infty}(J_{i})}\leq
c\int_{a_{i}}^{b_{i}}v_{i}^{2}+(v_{i}^{\prime\prime})^{2}\leq
c\varepsilon_{i}^{3/4}\bigl{|}\bigl{[}\tilde{f}_{i}v_{i}\bigr{]}_{a_{i}}^{b_{i}}\bigr{|}+c\int_{a_{i}}^{b_{i}}(\varepsilon_{i}^{3/4}\tilde{f}_{i}^{\prime}+\tilde{g}_{i})^{2}$
$\displaystyle\leq
c\varepsilon_{i}^{3/4}\|\tilde{f}_{i}v_{i}\|_{L^{\infty}(J_{i})}+(b_{i}-a_{i})\bigl{(}\varepsilon_{i}^{3/4}\|\tilde{f}_{i}^{\prime}\|_{L^{\infty}(J_{i})}+\|\tilde{g}_{i}\|_{L^{\infty}(J_{i})}\bigr{)}^{2}\to
0,\quad i\to\infty,$

a contradiction.

$\bullet$ Next assume that $a_{i}\to-\infty$ along a subsequence and that
$b_{i}\geq 0$ is bounded. Owing to (3.15), by passing to a further
subsequence, we may assume that there exists $w\in W^{4,\infty}((-\infty,0])$
such that $v_{i}\to w$ in $W^{3,\infty}_{loc}((-\infty,0])$, hence in
particular

$w(0)=1.$ (3.17)

Fix any $\alpha\in(-\infty,0)$. Applying (3.16) with $\beta=b_{i}$ and using
the boundary conditions in (3.13) at $\sigma=b_{i}$ and (3.14), (3.15), we
get, for $i\geq i_{0}(\alpha)$ large enough,

$\displaystyle\int_{\alpha}^{0}v_{i}^{2}+(v_{i}^{\prime\prime})^{2}$
$\displaystyle\leq\int_{\alpha}^{b_{i}}v_{i}^{2}+(v_{i}^{\prime\prime})^{2}\leq
c\bigl{|}(v_{i}^{\prime}v_{i}^{\prime\prime}-v_{i}v_{i}^{\prime\prime\prime})(\alpha)\bigr{|}+c\varepsilon_{i}^{3/4}\bigl{|}(\tilde{f}_{i}v_{i})(b_{i})\bigr{|}+c\int_{\alpha}^{b_{i}}(\varepsilon_{i}^{3/4}\tilde{f}_{i}^{\prime}+\tilde{g}_{i})^{2}$
$\displaystyle\leq
c\bigl{(}|v_{i}(\alpha)|+|v^{\prime\prime}_{i}(\alpha)|\bigr{)}+cM_{i}^{-1}+cM_{i}^{-2}(b_{i}-\alpha).$

By Fatou’s lemma, it follows that

$\int_{\alpha}^{0}w^{2}+(w^{\prime\prime})^{2}\leq
c\bigl{(}|w(\alpha)|+|w^{\prime\prime}(\alpha)|\bigr{)}\leq c\quad\hbox{for
all $\alpha\in(-\infty,0)$.}$ (3.18)

Consequently, $w\in H^{2}((-\infty,0])$, hence there exists a sequence
$\alpha_{j}\to-\infty$ such that
$\eta_{j}:=|w(\alpha_{j})|+|w^{\prime\prime}(\alpha_{j})|\to 0$. Going back to
(3.18) with $\alpha=a_{j}$, we deduce that $\int_{\alpha_{j}}^{0}w^{2}\leq
c\eta_{j}$, hence $w\equiv 0$ on $(-\infty,0]$ uppon letting $j\to\infty$. But
this contradicts (3.17).

$\bullet$ The cases when $b_{i}\to\infty$ along a subsequence, with $a_{i}$
either bounded or unbounded, are treated similarly.

We conclude that (3.8) and (3.9) are true for $j=0$ and $q=\infty$.

Now, using $u^{\prime\prime\prime\prime}=\varepsilon^{-1}(-ku+g+\varepsilon
f^{\prime})$ a.e. in $I$ (cf. Proposition 3.1(ii)), we deduce that

$\|u^{\prime\prime\prime\prime}\|_{\infty}\leq
K\varepsilon^{-1}\left(\|g\|_{\infty}+\varepsilon^{3/4}\|f\|_{\infty}+\varepsilon\|{f^{\prime}}\|_{\infty}\right)$

i.e., (3.8), (3.9) for $j=4$. The case $j\in\\{1,2,3\\}$ and $q=\infty$ then
follows by interpolation.

Finally, the case $q\in(2,\infty)$ follows by interpolating between the cases
$q=2$ and $q=\infty$.

(iii) Let $g\in H^{2}$ and set $u=A_{2,\varepsilon}g$. Since $\varepsilon
u^{\prime\prime\prime\prime}=g-ku\in H^{2}$, we have $u\in H^{6}$.
Differentiating twice, we see that $w=u^{\prime\prime}$ satisfies $\varepsilon
w^{\prime\prime\prime\prime}=g^{\prime\prime}-k^{\prime\prime}u-2k^{\prime}u^{\prime}-ku^{\prime\prime}$,
i.e.

$kw+\varepsilon
w^{\prime\prime\prime\prime}=\hat{g}:=g^{\prime\prime}-k^{\prime\prime}u-2k^{\prime}u^{\prime}.$

Moreover, $w=w^{\prime}=0$ on $\partial I$. Multipying with $w$ and
integrating by parts, we obtain

$\displaystyle\int_{0}^{L}(k_{1}w^{2}+\varepsilon w^{\prime\prime 2})ds$
$\displaystyle\leq\int_{0}^{L}(kw^{2}+\varepsilon w^{\prime\prime
2})ds=\bigl{[}w^{\prime}w^{\prime\prime}-ww^{\prime\prime\prime}\bigr{]}_{0}^{L}+\int_{0}^{L}\hat{g}wds\leq\frac{k_{1}}{2}\int_{0}^{L}w^{2}ds+\frac{1}{2k_{1}}\int_{0}^{L}\hat{g}^{2}ds.$

This combined with $\|u\|_{2}\leq K\|g\|_{2}$ (cf. assertion (i)) yields

$\|u\|_{2}^{2}+\|u^{\prime\prime}\|_{2}^{2}+\varepsilon\|u^{\prime\prime\prime\prime}\|_{2}^{2}\leq
K\|\hat{g}\|_{2}^{2}\leq K\|k\|^{2}_{H^{2}}\|g\|^{2}_{H^{2}}.$

This implies (3.11) for $j\in\\{0,2\\}$ and the case $j=1$ follows by
interpolation. ∎

##  4 Local existence-uniqueness

For given $\tau\geq 0$, we define our working space

$X_{\tau}=L^{\infty}(-\infty,\tau;H^{2}(I))$

with norm

$\|z\|_{X_{\tau}}=\sup_{t\in(-\infty,\tau)}\|z(t)\|_{H^{2}(I)},\quad\tau\geq
0.$

Fix $\varepsilon\in(0,1)$ and $\tau>0$. A (weak) solution of (1.4) on
$(0,\tau)$ is a function $z\in X_{\tau}$ such that

$\left\\{\begin{aligned}
\int_{I}{\mathcal{L}}_{\varepsilon}[z](t)vds+\int_{I}\bigl{(}z^{\prime\prime}(t)v^{\prime\prime}+F^{\prime}(z^{\prime}(t))v^{\prime}\bigr{)}ds&=0,\quad\forall
v\in H^{2}(I),&&a.e.\ t\in(0,\tau),\\\ z(t)&=z_{p}(t),&&a.e.\
t<0\end{aligned}\right.$ (4.1)

(here we omit the variable $s$ and the subscript $\varepsilon$ without risk of
confusion).

Let the operators $A_{1,\varepsilon},A_{2,\varepsilon}$ be defined by
Proposition 3.1 with $k(s)=\mu(s)$. If $z\in X_{\tau}$ is a solution of (4.1)
on $(0,\tau)$, then it satisfies

$z(t)=\begin{cases}A_{1,\varepsilon}F^{\prime}(z^{\prime}(t))+A_{2,\varepsilon}\displaystyle\int_{0}^{\infty}z(t-\varepsilon
a)\rho(\cdot,a)da,&a.e.\ t\in(0,\tau),\\\ \vskip 2.84526pt\cr z_{p}(t),&{a.e.\
}t<0.\end{cases}$ (4.2)

Conversely, for a given function $z\in X_{\tau}$, we have
$F^{\prime}(z^{\prime}(t))\in L^{\infty}(I)$ and
$\int_{0}^{\infty}z(t-\varepsilon a)\rho(\cdot,a)da\in L^{\infty}(I)$ for a.e.
$t\in(0,\tau)$, and (4.2) implies (4.1).

The following theorem is our basic local existence-uniqueness result, which in
turn defines a suitable notion of maximal solution.

######  Theorem 3.

Assume (1.7), (1.9) and $z_{p}\in X_{0}$. There exist constants $c_{i}>0$,
with $c_{2}<1$, depending only on $\rho,L$, such that the following properties
are true for all $\varepsilon\in(0,\varepsilon_{1}]$, where

$\varepsilon_{1}:=c_{2}R_{0}^{-8}\ \hbox{ and }\
R_{0}:=1+c_{1}\|z_{p}\|_{X_{0}}.$ (4.3)

  * (i)

For $T=c_{3}\varepsilon$, problem (4.2) admits a (unique) solution $z\in
X_{T}$ such that

$\|z\|_{X_{T}}\leq R_{0}.$

  * (ii)

For any $\tau>0$, problem (4.2) admits at most one solution $z\in X_{\tau}$
such that

$\|z\|_{X_{\tau}}\leq c_{4}\varepsilon^{-1/8}.$ (4.4)

  * (iii)

Let $z$ be the local solution given by assertion (i) and set

$T^{*}=\sup\bigl{\\{}\tau>0;\ \hbox{$z$ extends to a solution satisfying
\eqref{BUalt2}}\bigr{\\}}.$

If $T^{*}$ is finite, then

$\|z\|_{X_{T^{*}}}\geq c_{5}\varepsilon^{-1/8}.$ (4.5)

######  Remark 4.1.

  1. (i)

Note that, in a standard blow-up alternative, one would normally have $\infty$
in the RHS of (4.5). The current definition reflects the lack of parabolicity
of the problem.

  2. (ii)

The solution $z$ is generally discontinuous at $t=0$, unless a suitable
compatibility condition is imposed on $z_{p}$. An upper estimate of the jump
in terms of $\varepsilon$ is given in Proposition 5.3.

In the rest of the paper, by the solution $z=z_{\varepsilon}$ of (3.1), we
will mean the maximal solution defined in Theorem 3(iii).

Our next result provides, under additional assumptions on the past data,
further time regularity of the solution for $t>0$, which will be required in
order to derive the key energy estimates in Section 5 .

######  Proposition 4.1.

Assume (1.7), (1.9), (1.12) and (4.3). Then

$z_{\varepsilon}\in W^{1,\infty}_{loc}([0,T^{*});H^{2}(I))$ (4.6)

and we moreover have

$\|\partial_{t}z_{\varepsilon}(t)\|_{H^{2}(I)}\leq
C(N_{t}\varepsilon^{-1}+R_{2})\exp\bigl{\\{}C\varepsilon^{-1}t\bigr{\\}},\quad\text{a.e.}\
t\in(0,T^{*}),$ (4.7)

with $N_{t}=\|z_{\varepsilon}\|_{X_{t}}$ and
$R_{2}:=\|z_{p}\|_{W^{1,\infty}(-\infty,0;H^{2}(I))}$.

The proof of Theorem 3 will be carried out by a fixed point argument on
problem (4.2).

###### Proof of Theorem 3.

We first note that, owing to (1.7), (1.9), assumptions (3.3), (3.10) of
Proposition 3.2 with $k(s)=\mu(s)$ are satisfied.

(i) For $T>0$, $R\geq 1+\|z_{p}\|_{X_{0}}$ to be determined below, we set

$B_{T,R}=\left\\{z\in X_{T},\ \|z\|_{X_{T}}\leq R\hbox{ and $z(t)=z_{p}$ for
a.e.~{}$t<0$}\right\\},$

and consider the fixed point operator $\mathcal{T}:B_{T,R}\to X_{T}$ defined
by

$[\mathcal{T}(z)](t)=\begin{cases}A_{1,\varepsilon}F^{\prime}(z^{\prime}(t))+A_{2,\varepsilon}\displaystyle\int_{0}^{\infty}z(t-\varepsilon
a)\rho(\cdot,a)da,&a.e.\ t\in(0,T),\\\ \vskip 2.84526pt\cr
z_{p}(t),&a.e.~{}t<0.\end{cases}$ (4.8)

Note that $B_{T,R}$, being a closed subset of the Banach space $X_{T}$, is a
complete metric space. Recalling (1.7), we set

$M_{1}=1+\|\rho\|_{L_{a}^{\infty}(0,\infty;H^{2}(I))}+\|\rho\|_{L_{a}^{1}(0,\infty;H^{2}(I))}.$
(4.9)

In what follows we shall respectively denote by $\|\cdot\|_{\infty}$ and
$\|\cdot\|_{H^{2}}$ the norms in $L^{\infty}(I)$ and $H^{2}(I)$, and keep the
variable $s\in I$ implicit. Let $z\in B_{T,R}$. Using
$F^{\prime}(z^{\prime})=4z^{\prime}(1-{z^{\prime}}^{2})$, estimates (3.7),
(3.11), the Sobolev inequality $\|v\|_{W^{1,\infty}(I)}\leq C\|v\|_{H^{2}(I)}$
and the fact that $\|fg\|_{H^{2}}\leq C\|f\|_{H^{2}}\|g\|_{H^{2}}$, we obtain,
for a.e. $t\in(0,T)$,

$\displaystyle\|\mathcal{T}(z)(t)\|_{H^{2}}$ $\displaystyle\leq
C\|A_{1,\varepsilon}F^{\prime}(z^{\prime}(t))\|_{H^{2}}+\Big{\|}A_{2,\varepsilon}\int_{0}^{t/\varepsilon}z(t-\varepsilon
a)\rho(a)da\Bigr{\|}_{H^{2}}+\Big{\|}A_{2,\varepsilon}\int_{t/\varepsilon}^{\infty}z_{p}(t-\varepsilon
a)\rho(a)da\Bigr{\|}_{H^{2}}$ $\displaystyle\leq
C\varepsilon^{1/4}\|F^{\prime}(z^{\prime}(t))\|_{L^{2}}+C\Big{\|}\int_{0}^{t/\varepsilon}z(t-\varepsilon
a)\rho(a)da\Bigr{\|}_{H^{2}}+C\Big{\|}\int_{t/\varepsilon}^{\infty}z_{p}(t-\varepsilon
a)\rho(a)da\Bigr{\|}_{H^{2}}$ $\displaystyle\leq
C\varepsilon^{1/4}(1+\|z^{\prime}(t)\|^{2}_{\infty})\|z^{\prime}(t)\|_{\infty}+CM_{1}\varepsilon^{-1}T\sup_{\sigma\in(0,t)}\|z(\sigma)\|_{H^{2}}+CM_{1}\sup_{\sigma<0}\|z_{p}(\sigma)\|_{H^{2}}.$

It follows that

$\|\mathcal{T}(z)\|_{X_{T}}\leq
C_{0}\bigl{(}\varepsilon^{1/4}R^{3}+M_{1}\varepsilon^{-1}TR+M_{1}\|z_{p}\|_{X_{0}}\bigr{)},$
(4.10)

where $C_{0}=C_{0}(\rho,L)\geq 1$ (note that, for a.e. $t<0$, we have
$\|z(t)\|_{H^{2}}=\|z_{p}(t)\|_{H^{2}}\leq C_{0}M_{1}\|z_{p}\|_{X_{0}}$ since
$C_{0},M_{1}\geq 1$). Similarly, for all $z_{1},z_{2}\in B_{T,R}$, using also

$|X^{3}-Y^{3}|\leq 2(X^{2}+Y^{2})|X-Y|,\quad X,Y\in\mathbb{R},$ (4.11)

we get

$\displaystyle\|\mathcal{T}(z_{1})(t)-\mathcal{T}(z_{2})(t)\|_{H^{2}}$
$\displaystyle\leq
C\bigl{\|}A_{1,\varepsilon}\bigl{(}F^{\prime}(z^{\prime}_{1}(t))-F^{\prime}(z^{\prime}_{2}(t))\bigr{)}\bigr{\|}_{H^{2}}+\Big{\|}A_{2,\varepsilon}\int_{0}^{t/\varepsilon}[z_{1}(t-\varepsilon
a)-z_{2}(t-\varepsilon a)]\rho(a)da\Bigr{\|}_{H^{2}}$ $\displaystyle\leq
C\varepsilon^{1/4}\bigl{\|}F^{\prime}(z^{\prime}_{1}(t))-F^{\prime}(z^{\prime}_{2}(t))\bigr{\|}_{L^{2}}+\Big{\|}A_{2,\varepsilon}\int_{0}^{t/\varepsilon}[z_{1}(t-\varepsilon
a)-z_{2}(t-\varepsilon a)]\rho(a)da\Bigr{\|}_{H^{2}}$ $\displaystyle\leq
C\varepsilon^{1/4}\bigl{(}1+\|z^{\prime}_{1}(t)\|_{\infty}+\|z^{\prime}_{2}(t)\|_{\infty}\bigr{)}^{2}\|(z^{\prime}_{1}-z^{\prime}_{2})(t)\|_{\infty}+C\Big{\|}\int_{0}^{t/\varepsilon}[z_{1}(t-\varepsilon
a)-z_{2}(t-\varepsilon a)]\rho(a)da\Bigr{\|}_{H^{2}}$ $\displaystyle\leq
C\varepsilon^{1/4}R^{2}\|(z^{\prime}_{1}-z^{\prime}_{2})(t)\|_{\infty}+CM_{1}\varepsilon^{-1}T\sup_{\sigma\in(0,t)}\|(z_{1}-z_{2})(\sigma)\|_{H^{2}}.$

It follows (taking $C_{0}=C_{0}(\rho,L)$ larger if necessary) that

$\|\mathcal{T}(z_{1})-\mathcal{T}(z_{2})\|_{X_{T}}\leq
C_{0}\bigl{(}\varepsilon^{1/4}R^{2}+M_{1}\varepsilon^{-1}T\bigr{)}\|z_{1}-z_{2}\|_{X_{T}}.$
(4.12)

Now set

$c_{1}:=2C_{0}M_{1}\quad c_{2}:=4^{-8}C_{0}^{-4},\quad
c_{3}=(4C_{0}M_{1})^{-1},\quad c_{4}:=\textstyle(4C_{0})^{-1/2}$ (4.13)

and

$R_{0}:=1+c_{1}\|z_{p}\|_{X_{0}}.$

For $\varepsilon\in(0,1]$, choose $T:=c_{3}\varepsilon$ and any $R$ such that

$R_{0}\leq R\leq c_{4}\varepsilon^{-1/8},$ (4.14)

we have

$C_{0}\bigl{\\{}\varepsilon^{1/4}R^{2}+M_{1}\varepsilon^{-1}T\bigr{\\}}\leq
1/2$ (4.15)

and

$C_{0}\bigl{(}\varepsilon^{1/4}R^{3}+M_{1}\varepsilon^{-1}TR+M_{1}\|z_{p}\|_{X_{0}}\bigr{)}-R\leq\bigl{\\{}C_{0}\bigl{(}\varepsilon^{1/4}R^{2}+M_{1}\varepsilon^{-1}T\bigr{)}-\textstyle\frac{1}{2}\bigr{\\}}R\leq
0.$ (4.16)

It follows from (4.15)-(4.16) that

$\mathcal{T}$ is a contraction mapping on $B_{T,R}$. (4.17)

Consequently, (4.2) admits a unique fixed point in $B_{T,R}$. Assuming
$\varepsilon\in(0,\varepsilon_{1}]$ with $\varepsilon_{1}:=c_{2}R_{0}^{-8}$
and making the particular choice $R=R_{0}$ in (4.14), this proves assertion
(i).

(ii) Let $\tau>0$ and let $z_{1},z_{2}\in X_{\tau}$ be solutions satisfying
(4.4). Let

$\tau_{0}=\sup\,\Bigl{\\{}t\in[0,\tau);\ z_{1}=z_{2}\ \hbox{a.e. on
$(-\infty,t)$}\Bigr{\\}}.$

By (4.17) we know that $\tau_{0}>0$. Assume for contradiction that
$\tau_{0}<\tau$. Since $\|z_{i}\|_{X_{\tau}}\leq R:=c_{4}\varepsilon^{-1/8}$
and $z_{1}=z_{2}$ for $t<\tau_{0}$, the argument leading to (4.13), with
$t/\varepsilon$ replaced by $(t-\tau_{0})/\varepsilon$ implies

$\|z_{1}-z_{2}\|_{X_{t}}\leq
C_{0}\bigl{\\{}R^{2}\varepsilon^{1/4}+M_{1}\varepsilon^{-1}(t-\tau_{0})\bigr{\\}}\|z_{1}-z_{2}\|_{X_{t}},\quad\tau_{0}<t<\tau.$

Since $C_{0}R^{2}\varepsilon^{1/4}\leq\ 1/4$ (cf. (4.13)), we deduce that
$z_{1}(t)=z_{2}(t)$ for $t\geq\tau_{0}$ close to $\tau_{0}$: a contradiction.

(iii) By the definition of $T^{*}$, for each integer $j>{T^{*}}^{-1}$, there
exists a solution $z_{j}\in X_{T^{*}-j^{-1}}$ satisfying (4.4) with
$\tau=T^{*}-j^{-1}$. By the uniqueness statement in assertion (ii), we have
$z_{j}=z_{j+1}$ for $t\leq T^{*}-j^{-1}$. The desired solution is thus
obtained by setting $z:=z_{j}$ for $t<T^{*}-j^{-1}$ and $j>{T^{*}}^{-1}$.

Next assume for contradiction that $T^{*}<\infty$ and $\|z\|_{X_{T^{*}}}\leq
c_{5}\varepsilon^{-1/8}$, where $c_{5}=c_{4}/2c_{1}$. Since
$\varepsilon\leq\varepsilon_{1}\leq c_{2}=(c_{4}/2)^{8}$, we have
$\varepsilon^{1/8}\leq c_{4}/2=c_{1}c_{5}$. It follows that

$R_{1}:=1+c_{1}\|z\|_{X_{T^{*}}}\leq 1+c_{1}c_{5}\varepsilon^{-1/8}\leq
2c_{1}c_{5}\varepsilon^{-1/8}=c_{4}\varepsilon^{-1/8}.$

Set $\hat{z}_{p}(t)=z(t+T^{*})$ for a.e. $t<0$. We may thus apply the above
fixed point argument with $\tilde{z}_{p}\in X_{0}$ instead of $z_{p}$ and
$R=R_{1}$ instead of $R=R_{0}$ in (4.14). This provides a time $T>0$ and a
function $\hat{z}\in X_{T}$ such that $\hat{z}=\hat{z}_{p}$ for a.e. $t<0$ and

$\hat{z}(t)=A_{1,\varepsilon}F^{\prime}(\hat{z}^{\prime}(t))+A_{2,\varepsilon}\displaystyle\int_{0}^{\infty}\hat{z}(t-\varepsilon
a)\rho(s,a)da,\quad a.e.\ t\in(0,T).$

Letting $\tilde{z}(t):=\hat{z}(t-T^{*})$ for a.e. $t<T^{*}+T$, it follows that
$\tilde{z}$ satisfies $\tilde{z}(t)=z(t)$ for a.e. $t<T^{*}$ and

$\tilde{z}(t)=A_{1,\varepsilon}F^{\prime}(\tilde{z}^{\prime}(t))+A_{2,\varepsilon}\displaystyle\int_{0}^{\infty}\tilde{z}(t-\varepsilon
a)\rho(s,a)da,\quad a.e.\ t\in(0,T^{*}+T),$

along with (4.4): a contradiction. ∎

###### Proof of Proposition 4.1.

We must show that, more precisely, there exists a representative $\tilde{z}$
of $z$ (i.e. $\tilde{z}(t)=z(t)$ a.e.) such that, for each $T\in(0,T^{*})$,
$\tilde{z}$ is Lipschitz continuous from $[0,T]$ to $H^{2}(I)$ and satisfies
the estimate in (4.7). Fixing a representative $z$ and $T\in(0,T^{*})$, it
suffices to show that there exists $\Sigma\subset(0,T)$ with
$|(0,T)\setminus\Sigma|=0$, such that

$\|z(t_{2})-z(t_{1})\|_{H^{2}}\leq
C(N_{T}\varepsilon^{-1}+R_{2})|t_{2}-t_{1}|\exp\bigl{\\{}C\varepsilon^{-1}T\bigr{\\}},\quad\text{
for all }t_{1},\ t_{2}\in\Sigma$ (4.18)

(since one then easily sees that $\tilde{z}(t):=\lim_{\Sigma\ni t^{\prime}\to
t}z(t^{\prime})$ exists in $H^{2}$ for all $t\in[0,T]$, coincides with $z$ on
$\Sigma$, and has the desired properties).

To this end, we first note that, by (4.2), there exists $\Sigma\subset(0,T)$
with $|(0,T)\setminus\Sigma|=0$, such that

$z(t)=A_{1,\varepsilon}F^{\prime}(z^{\prime}(t))+A_{2,\varepsilon}\displaystyle\int_{0}^{\infty}z(t-\varepsilon
a)\rho({s,a})da\quad\text{ for all }t\in J.$ (4.19)

Fix $t_{1},\ t_{2}\in\Sigma$ with $t_{1}<t_{2}$ and set $h:=t_{2}-t_{1}$ and
$N:=\|z\|_{X_{T}}$. Using (4.19), (3.7), (3.11), (4.11) with
$X=z^{\prime}(s,t)$, $Y=z^{\prime}(s,t-h)$, the Sobolev inequality
$\|v\|_{W^{1,\infty}(I)}\leq C\|v\|_{H^{2}(I)}$ and the fact that
$\|fg\|_{H^{2}}\leq C\|f\|_{H^{2}}\|g\|_{H^{2}}$, we deduce that, for all
$t\in\Sigma\cap(h+J)$,

$\displaystyle\bigl{\|}z(t)-z(t-h)\bigr{\|}_{H^{2}}$ $\displaystyle\leq
4\Bigl{\|}A_{1,\varepsilon}\Bigl{\\{}[z^{\prime}(1-{z^{\prime}}^{2})](t)-[z^{\prime}(1-{z^{\prime}}^{2})](t-h)\Bigr{\\}}\Bigr{\|}_{H^{2}}+\Big{\|}A_{2,\varepsilon}\int_{0}^{\infty}[z(t-\varepsilon
a)-z(t-h-\varepsilon a)]\rho(a)da\Bigr{\|}_{H^{2}}$ $\displaystyle\leq
C\varepsilon^{1/4}(1+N^{2})\|z^{\prime}(t)-z^{\prime}(t-h)\|_{2}+C\int_{0}^{\infty}\Big{\|}[z(t-\varepsilon
a)-z(t-h-\varepsilon a)]\rho(a)\Bigr{\|}_{H^{2}}da$ $\displaystyle\leq
C_{1}\varepsilon^{1/4}(1+N^{2})\|z(t)-z(t-h)\|_{H^{2}}+C_{1}\int_{0}^{\infty}\Big{\|}[z(t-\varepsilon
a)-z(t-h-\varepsilon a)]\Bigr{\|}_{H^{2}}\|\rho(a)\|_{H^{2}}da,$

with $C_{1}=C_{1}(\rho,L)>0$. On the other hand, setting
$\delta_{h}(t)=\|z(t)-z(t-h)\|_{H^{2}}$, we note that, for each $t\in(0,T)$,
we have

$\delta_{h}(t-\varepsilon a)=\|z_{p}(t-\varepsilon a)-z_{p}(t-\varepsilon
a-h)\|_{H^{2}}\leq R_{2}h,\quad\hbox{ for a.e.~{}}a>t/\varepsilon.$ (4.20)

Moreover, up to replacing $C_{0}$ in (4.10), (4.12) by $\max(C_{0},$ $C_{1})$,
we may assume that

$C_{1}\varepsilon^{1/4}(1+N^{2})\leq
C_{1}\varepsilon_{1}^{1/4}+C_{1}c_{4}^{2}=4^{-2}C_{1}C_{0}^{-1}R_{0}^{-2}+C_{1}(4C_{0})^{-1}\leq\textstyle\frac{1}{2}.$

Using (4.9) and (4.20), we then obtain, for all $t\in\Sigma\cap(h+\Sigma)$,

$\displaystyle(2C_{1})^{-1}\delta_{h}(t)$
$\displaystyle\leq\int_{0}^{\infty}\Big{\|}[z(t-\varepsilon
a)-z(t-h-\varepsilon a)]\Bigr{\|}_{H^{2}}\|\rho(a)\|_{H^{2}}da$
$\displaystyle=\int_{0}^{(t-h)/\varepsilon}\delta_{h}(t-\varepsilon
a)\|\rho(a)\|_{H^{2}}da+\int_{(t-h)/\varepsilon}^{t/\varepsilon}\delta_{h}(t-\varepsilon
a)\|\rho(a)\|_{H^{2}}da+\int_{t/\varepsilon}^{\infty}\delta_{h}(t-\varepsilon
a)\|\rho(a)\|_{H^{2}}da$
$\displaystyle=\varepsilon^{-1}\int_{h}^{t}\delta_{h}(\tau)\|\rho(\varepsilon^{-1}(t-\tau))\|_{H^{2}}d\tau+\int_{(t-h)/\varepsilon}^{t/\varepsilon}\delta_{h}(t-\varepsilon
a)\|\rho(a)\|_{H^{2}}da+\int_{t/\varepsilon}^{\infty}\delta_{h}(t-\varepsilon
a)\|\rho(a)\|_{H^{2}}da$ $\displaystyle\leq
M_{1}\left(\varepsilon^{-1}\int^{t}_{h}\delta_{h}(\tau)d\tau+2N\varepsilon^{-1}h+R_{2}h\right),$

hence

$\delta_{h}(t)\leq
C(N\varepsilon^{-1}+R_{2})h+C\varepsilon^{-1}\int^{t}_{h}\delta_{h}(\tau)d\tau=:G(t),\quad\hbox{for
all $t\in J\cap(h+J)$.}$ (4.21)

Now observing that $|(h,T)\setminus(\Sigma\cap(h+J))|=0$, it follows that the
(absolutely continuous) function $G$ satisfies
$\bigl{(}\exp\bigl{\\{}-C\varepsilon^{-1}t\bigr{\\}}G(t)\bigr{)}^{\prime}\leq
0$ for a.e. $t\in(h,T)$, hence $G(t)\leq
G(h)\exp\bigl{\\{}C\varepsilon^{-1}(t-h)\bigr{\\}}$ for all $t\in(h,T)$. Going
back to (4.21) and noting that $t_{2}\in\Sigma\cap(h+\Sigma)$ owing to
$t_{1},\ t_{2}\in\Sigma$, we obtain

$\|z(t_{2})-z(t_{1})\|_{H^{2}}=\delta_{h}(t_{2})\leq G(t_{2})\leq
G(h)\exp\bigl{\\{}C\varepsilon^{-1}T\bigr{\\}}=C(N\varepsilon^{-1}+R_{2})|t_{2}-t_{1}|\exp\bigl{\\{}C\varepsilon^{-1}T\bigr{\\}},$

hence (4.18). The proposition follows. ∎

##  5 Energy estimates

Let $z=z_{\varepsilon}$ be the solution given by Theorem 3, of existence time
$T^{*}_{\varepsilon}$, and denote

$u_{\varepsilon}(s,t,a):=z_{\varepsilon}(s,t)-z_{\varepsilon}(s,t-\varepsilon
a),\quad 0<t<T^{*}_{\varepsilon},\ a>0.$ (5.1)

We define the energy:

$E_{\varepsilon}(t):=\frac{1}{2\varepsilon}\int_{I}\int_{0}^{\infty}u_{\varepsilon}^{2}(s,a,t)\rho(s,a)dads+\frac{1}{2}\int_{I}(z_{\varepsilon}^{\prime\prime}(t))^{2}ds+\int_{I}F(z_{\varepsilon}^{\prime}(t))ds.$
(5.2)

In the following two propositions, we obtain the monotonicity of the energy
and, as a consequence, derive some key estimates. Special care is given to the
dependence with respect to $\varepsilon$, which will turn out important in the
proof of our main results.

######  Proposition 5.1.

Assume (1.7), (1.8), (1.9), (1.12). Let $\varepsilon\in(0,\varepsilon_{1})$
where $\varepsilon_{1}$ is given by Theorem 3.

(i) We have $E_{\varepsilon}\in W^{1,\infty}_{loc}([0,T^{*}_{\varepsilon}))$
and

$E_{\varepsilon}^{\prime}(t)={\frac{1}{2\varepsilon^{2}}}\int_{I}\int_{0}^{\infty}u_{\varepsilon}^{2}(s,a,t)\partial_{a}\rho(s,a)dads\leq
0,\quad a.e.\ t\in(0,T^{*}_{\varepsilon}).$ (5.3)

(ii) For all $0<t_{0}<t<T^{*}_{\varepsilon}$, we have

$\int_{t_{0}}^{t}\int_{I}\int_{0}^{\infty}u_{\varepsilon}^{2}(s,a,\tau)|\partial_{a}\rho(s,a)|dadsd\tau\leq
2\varepsilon^{2}E_{\varepsilon}(t_{0}),$ (5.4)
$\int_{t_{0}}^{t}\bigl{\\{}\|\partial_{t}z_{\varepsilon}(\tau)\|_{L^{2}(I)}^{2}+\varepsilon\|\partial_{t}z_{\varepsilon}^{\prime\prime}(\tau)\|_{L^{2}(I)}^{2}\bigr{\\}}d\tau\leq
CE_{\varepsilon}(t_{0}),$ (5.5)
$\int_{t_{0}}^{t}\big{\\{}\varepsilon^{1/4}\|\partial_{t}z_{\varepsilon}(\tau)\|_{L^{\infty}(I)}^{2}+\varepsilon^{3/4}\|\partial_{t}z_{\varepsilon}^{\prime}(\tau)\|_{L^{\infty}(I)}^{2}\bigr{\\}}d\tau\leq
CE_{\varepsilon}(t_{0}).$ (5.6)

In the next proposition we make the following assumption (which is a
consequence of the second part of (1.10)):

$\int_{0}^{\infty}a\|\rho(\cdot,a)\|_{2}da<\infty.$ (5.7)

######  Proposition 5.2.

Assume (1.7), (1.8), (1.9), (1.12), (5.7), let

$R:=1+\|z_{p}\|_{X_{0}},\quad\hat{R}:=\|z_{p}\|_{W^{1,\infty}(-\infty,0;L^{\infty}(I))},\quad\bar{R}:=R^{4}+\hat{R}^{2}$

and let $\varepsilon\in(0,\varepsilon_{1})$ where $\varepsilon$ is given by
Theorem 3. Then

$E_{\varepsilon}(t)\leq C\bar{R},\quad 0<t<T_{\varepsilon}^{*},$ (5.8)

and

$\|z_{\varepsilon}^{\prime}(\cdot,t)\|_{L^{\infty}(I)}\leq
C\bar{R}^{1/3},\quad\|z_{\varepsilon}^{\prime\prime}(\cdot,t)\|_{L^{2}(I)}\leq
C\bar{R}^{1/2},\quad 0<t<T_{\varepsilon}^{*}.$ (5.9)

In the process of proving Proposition 5.2, we obtain an estimate for the
discontinuity of $z_{\varepsilon}$ at $t=0$, of independent interest.

######  Proposition 5.3.

Under the assumptions of Proposition 5.2, there exists

$z_{\varepsilon}(0^{+}):=\lim_{t\to 0^{+}}z_{\varepsilon}(t)\quad\hbox{ in the
{strong} $H^{2}(I)$ sense}$ (5.10)

and we have

$\|z_{\varepsilon}(0^{+})-z_{p}(0)\|_{2}\leq C(R+\hat{R})\varepsilon^{1/2}.$
(5.11)

In view of the proofs of these three propositions, we prepare the following
two lemmas.

######  Lemma 5.4.

Let $\ell$ be a positive integer, $v\in H^{2}(I)$ and set

$G(t):=\int_{I}\int_{0}^{\infty}u^{\ell}_{\varepsilon}(s,t,a)\rho(s,a)v(s)dads.$
(5.12)

We have $G\in W^{1,\infty}_{loc}([0,T_{\varepsilon}^{*}))$ and, for a.e.
$t\in(0,T_{\varepsilon}^{*})$,

$G^{\prime}(t)=\ell\displaystyle\int_{I}\int_{0}^{\infty}\partial_{t}z_{\varepsilon}(s,t)u^{\ell-1}_{\varepsilon}(s,t,a)\rho(s,a)v(s)dads+\frac{1}{\varepsilon}\displaystyle\int_{I}\int_{0}^{\infty}u^{\ell}_{\varepsilon}(s,t,a)\partial_{a}\rho(s,a)v(s)dads.$
(5.13)

######  Lemma 5.5.

For a.e. $t\in(0,T_{\varepsilon}^{*})$ and all $v\in H^{2}(I)$, the time
derivative $\partial_{t}z$ satisfies

$\int_{I}\mu(s)\partial_{t}z_{\varepsilon}(s,t)v(s)ds+\varepsilon\int_{I}\bigl{(}\partial_{t}z^{\prime\prime}_{\varepsilon}v^{\prime\prime}+F^{\prime\prime}(z^{\prime}_{\varepsilon})\partial_{t}z^{\prime}_{\varepsilon}v^{\prime}\bigr{)}(s,t)ds+\frac{1}{\varepsilon}\int_{I}\int_{0}^{\infty}u_{\varepsilon}(s,t,a)v(s)\partial_{a}\rho(s,a)dads=0.$
(5.14)

###### Proof of Lemma 5.4.

Setting $\tilde{u}(s,t,b)=z(s,t)-z(s,b)$ and changing variables by
$b=t-\varepsilon a$, we first rewrite
$G(t)=\varepsilon^{-1}\int_{I}\int_{-\infty}^{t}\tilde{u}^{\ell}(s,t,b)\rho(s,\textstyle\frac{t-b}{\varepsilon})v(s)dbds$.
It follows that, for all $t,h$ with $t,t+h\in(0,T^{*}_{\varepsilon})$,

$\displaystyle\varepsilon(G(t+h)-G(t))=\int_{I}\int_{-\infty}^{t}\bigl{(}\tilde{u}^{\ell}(s,t+h,b)-\tilde{u}^{\ell}(s,t,b)\bigr{)}\rho(s,\textstyle\frac{t+h-b}{\varepsilon})vdbds$
$\displaystyle+\int_{I}\int_{-\infty}^{t}\tilde{u}^{\ell}(s,t,b)\bigl{(}\rho(s,\textstyle\frac{t+h-b}{\varepsilon})-\rho(s,\textstyle\frac{t-b}{\varepsilon})\bigr{)}vdbds+\int_{I}\int_{0}^{h}\tilde{u}^{\ell}(s,t+h,t+\tau)\rho(s,\textstyle\frac{h-\tau}{\varepsilon})vd\tau
ds.$

Using assumption (1.8) and the fact that $z\in
L^{\infty}((-\infty,T_{0})\times I)\cap W^{1,\infty}(0,T_{0};L^{\infty}(I))$
for any $T_{0}<T^{*}$ (cf. Proposition 4.1), we deduce that $G\in
W^{1,\infty}_{loc}([0,T^{*}))$ and, taking into account that
$\tilde{u}(\cdot,t,t)=0$, we obtain

$G^{\prime}(t)=\varepsilon^{-1}\displaystyle\int_{I}\int_{-\infty}^{t}\partial_{t}\tilde{u}^{\ell}(s,t,b)\rho(s,\textstyle\frac{t-b}{\varepsilon})vdbds+\varepsilon^{-2}\displaystyle\int_{I}\int_{-\infty}^{t}\tilde{u}^{\ell}(s,t,b)\partial_{a}\rho(s,\textstyle\frac{t-b}{\varepsilon})vdbds,$

for a.e. $t\in(0,T^{*})$. The result follows by going back to the variable
$a$. ∎

###### Proof of Lemma 5.5.

Fix $v\in H^{2}(I)$ and let $G$ be given by (5.12) with $\ell=1$. Since $z\in
W^{1,\infty}_{loc}([0,T^{*});$ $H^{2}(I))$ by Proposition 4.1, we may
differentiate (1.4) in time which, after using (5.13), gives

$\displaystyle-\varepsilon$ $\displaystyle\frac{d}{dt}\int_{I}\
\bigl{(}z^{\prime\prime}v^{\prime\prime}+F^{\prime}(z^{\prime})v^{\prime}\bigr{)}ds=G^{\prime}(t)=\int_{I}\int_{0}^{\infty}\partial_{t}z\rho
vdads+\varepsilon^{-1}\int_{I}\int_{0}^{\infty}u\partial_{a}\rho vdads,$

hence (5.14). ∎

###### Proof of Proposition 5.1.

(i) Set
$E_{1}(t)=\frac{1}{2}\int_{I}|z^{\prime\prime}|^{2}ds+\int_{I}F(z^{\prime})ds$.
Since $z\in W^{1,\infty}_{loc}([0,T^{*});$ $H^{2}(I))$, we have $E_{1}\in
W^{1,\infty}_{loc}([0,T^{*}))$ and, using $z_{t}(\cdot,t)$ as test-function in
the weak formulation (1.4), we get, for a.e. $t\in(0,T^{*})$,

$E^{\prime}_{1}(t)=\int_{I}z^{\prime\prime}\partial_{t}z^{\prime\prime}ds+\int_{i}F^{\prime}(z^{\prime})\partial_{t}z^{\prime}ds=-\frac{1}{\varepsilon}\int_{I}\int_{0}^{\infty}u\partial_{t}z\rho
dads.$ (5.15)

Let now $G$ be given by (5.12) with $\ell=2$ and $v\equiv 1$. Since
$E_{\varepsilon}(t)=E_{1}(t)+\frac{1}{2\varepsilon}G(t)$, we obtain (5.3) by
combining (5.13) and (5.15).

(ii) Property (5.4) readily follows from (5.3). Set

${\mathcal{S}}(s,t):=\int_{0}^{\infty}u^{2}(s,t,a)|\partial_{a}\rho(s,a)|da.$

Note that, by Cauchy-Schwarz and (1.8),

$\Bigl{(}\int_{0}^{\infty}u(s,t,a)|\partial_{a}\rho(s,a)|da\Bigr{)}^{2}\leq{\mathcal{S}}(s,t)\int_{0}^{\infty}|\partial_{a}\rho|da\leq
C{\mathcal{S}}(s,t).$

For any $\lambda>0$, applying (5.14) with $v=\partial_{t}z(\cdot,t)$ and using
the fact that $F^{\prime\prime}\geq-4$, we deduce that, for a.e.
$t\in(0,T^{*})$,

$\displaystyle\int_{I}\bigl{(}\mu|\partial_{t}z|^{2}+\varepsilon|\partial_{t}z^{\prime\prime}|^{2}\bigr{)}ds$
$\displaystyle=-\varepsilon\int_{I}F^{\prime\prime}(z^{\prime})|\partial_{t}z^{\prime}|^{2}-\varepsilon^{-1}\int_{I}\partial_{t}z\left(\int_{0}^{\infty}u\partial_{a}\rho
da\right)ds$ $\displaystyle\leq
4\varepsilon\int_{I}|\partial_{t}z^{\prime}|^{2}ds+\lambda\int_{I}|\partial_{t}z|^{2}ds+C\lambda^{-1}\varepsilon^{-2}\int_{I}{\mathcal{S}}(s,t)ds.$

Next using
$\int_{I}|\partial_{t}z^{\prime}|^{2}\leq\frac{1}{8}\int_{I}|\partial_{t}z^{\prime\prime}|^{2}+C_{2}\int_{I}|\partial_{t}z|^{2}$
with $C_{2}=C_{2}(L)>0$ (cf. (3.12)), it follows that

$\displaystyle(\mu_{min}-\lambda-4C_{2}\varepsilon)\int_{I}|\partial_{t}z|^{2}ds+\frac{\varepsilon}{2}\int_{I}|\partial_{t}z^{\prime\prime}|^{2}ds$
$\displaystyle\leq
C\lambda^{-1}\varepsilon^{-2}\int_{I}\int_{0}^{\infty}u^{2}|\partial_{a}\rho|dads.$

Up to replacing $C_{0}$ in (4.10), (4.12) by
$\max(C_{0},(C_{2}/\mu_{min})^{1/4})$, we may assume that
$\varepsilon\leq\varepsilon_{1}\leq 4^{-8}C_{0}^{-4}\leq\mu_{min}/(8C_{2})$.
Choosing $\lambda=\mu_{min}/4$, we get

$A(t):=\int_{I}|\partial_{t}z|^{2}ds+\varepsilon\int_{I}|\partial_{t}z^{\prime\prime}|^{2}ds\leq
C\varepsilon^{-2}\int_{I}\int_{0}^{\infty}u^{2}|\partial_{a}\rho|dads.$ (5.16)

Integrating this in time and using (5.4) yields (5.5).

Let us finally check (5.6). By (3.12) with $\eta=\varepsilon^{1/2}$, we have

$\|\partial_{t}z^{\prime}\|^{2}_{L^{2}}\leq\varepsilon^{1/2}\|\partial_{t}z^{\prime\prime}\|^{2}_{L^{2}}+C\varepsilon^{-1/2}\|\partial_{t}z\|^{2}_{L^{2}}\leq
C\varepsilon^{-1/2}A(t).$ (5.17)

Using the Sobolev inequality

$\|v\|_{L^{\infty}}\leq\eta\|v^{\prime}\|_{L^{2}}+C(1+\eta^{-1})\|v\|_{L^{2}},\quad
v\in H^{1}(I),$ (5.18)

with $\eta=\varepsilon^{1/4}$, we deduce from (5.17) that

$\|\partial_{t}z^{\prime}\|^{2}_{L^{\infty}}\leq\eta\|\partial_{t}z^{\prime\prime}\|^{2}_{L^{2}}+C\eta^{-1}\varepsilon^{-1/2}A(t)\leq
C\varepsilon^{-3/4}A(t)$ (5.19)

and

$\|\partial_{t}z\|^{2}_{L^{\infty}}\leq\eta\|\partial_{t}z^{\prime}\|^{2}_{L^{2}}+C\eta^{-1}\|\partial_{t}z\|^{2}_{L^{2}}\leq
C\bigl{(}\eta\varepsilon^{-1/2}A(t)+\eta^{-1}\|\partial_{t}z\|^{2}_{L^{2}}\bigr{)}\leq
C\varepsilon^{-1/4}A(t),$

and (5.6) follows from (5.5). ∎

We next prove Proposition 5.3 as a consequence of Proposition 5.1.

###### Proof of Proposition 5.3.

Property (5.10) follows from $z\in W^{1,\infty}_{loc}([0,T^{*});H^{2}(I))$
(cf. Proposition 4.1).

We next prove (5.11). By Theorem 3(i), since
$0<\varepsilon\leq\varepsilon_{1}=c_{2}R^{-8}$, we have $T^{*}\geq
T_{\varepsilon}:=c_{3}\varepsilon$ and

$\|z_{\varepsilon}(t)\|_{H^{2}}\leq CR,\quad-\infty<t\leq T_{\varepsilon}.$
(5.20)

Let now $t\in(0,T_{\varepsilon})$. For all $s\in I$, we write

$\mu(s)(z(s,t)-z_{p}(s,0))=\varepsilon\bigl{(}-z^{\prime\prime\prime\prime}+\partial_{s}(F^{\prime}(z^{\prime}))\bigr{)}+\int_{0}^{\infty}(z(s,t-\varepsilon
a)-z_{p}(s,0))\rho(s,a)da.$ (5.21)

To estimate the first term on the RHS, we use (3.1), (3.8) with $j=4$, (3.11)
with $j=2$, the fact that $\|fg\|_{H^{2}}\leq C\|f\|_{H^{2}}\|g\|_{H^{2}}$,
and (5.20) to get

$\displaystyle\|z(t)\|_{H^{4}}$
$\displaystyle\leq\|A_{1,\varepsilon}F^{\prime}(z^{\prime}(t))\|_{H^{4}}+\Big{\|}A_{2,\varepsilon}\int_{0}^{\infty}z(t-\varepsilon
a)\rho(a)da\Bigr{\|}_{H^{4}}$ $\displaystyle\leq
C\varepsilon^{-1/4}\|F^{\prime}(z^{\prime}(t))\|_{L^{2}}+C\|F^{\prime\prime}(z^{\prime}(t))z^{\prime\prime}(t)\|_{L^{2}}+C\varepsilon^{-1/2}\Big{\|}\int_{0}^{\infty}z(t-\varepsilon
a)\rho(a)da\Bigr{\|}_{H^{2}}$ $\displaystyle\leq
C(1+\|z^{\prime}(t)\|^{2}_{\infty})(\varepsilon^{-1/4}\|z^{\prime}(t)\|_{\infty}+\|z^{\prime\prime}(t)\|_{2})+CM_{1}\varepsilon^{-1/2}\sup_{\sigma<t}\|z(\sigma)\|_{H^{2}}$
$\displaystyle\leq CR^{3}\varepsilon^{-1/4}+CM_{1}R\varepsilon^{-1/2}\leq
CR\varepsilon^{-1/2},$

hence

$\bigl{\|}-z^{\prime\prime\prime\prime}(t)+(F^{\prime}(z^{\prime}(t))^{\prime}\bigr{\|}_{2}\leq
CR\varepsilon^{-1/2}+CR^{3}\leq CR\varepsilon^{-1/2}.$ (5.22)

On the other hand, by dominated convergence, (1.12) and (5.7), we have

$\displaystyle\lim_{t\to 0^{+}}\int_{0}^{\infty}\|z(t-\varepsilon
a)-z_{p}(0)\|_{\infty}\|\rho(\cdot,a)\|_{2}da$
$\displaystyle=\int_{0}^{\infty}\|z_{p}(-\varepsilon
a)-z_{p}(0)\|_{\infty}\|\rho(\cdot,a)\|_{2}da\leq\hat{R}\varepsilon\int_{0}^{\infty}a\|\rho(\cdot,a)\|_{2}da=C\hat{R}\varepsilon.$

Combining this with (5.21) and (5.22), it follows that

$\mu_{min}\limsup_{t\to 0^{+}}\|(z(t)-z_{p}(0)\|_{2}\leq
CR\varepsilon^{1/2}+C\hat{R}\varepsilon,$

hence (5.11). ∎

We finally prove Proposition 5.2 as a consequence of Proposition 5.3.

###### Proof of Proposition 5.2.

By dominated convergence, (1.12) and (5.10), we have

$\lim_{t\to
0^{+}}\int_{I}\int_{0}^{\infty}u_{\varepsilon}^{2}(s,a,t)\rho(s,a)dads=\int_{I}\int_{0}^{\infty}|z(s,0^{+})-z_{p}(s,-\varepsilon
a)|^{2}\rho(s,a)dads.$

Moreover, for each $a>0$, we have

$\displaystyle\int_{I}$ $\displaystyle|z(s,0^{+})-z_{p}(s,-\varepsilon
a)|^{2}ds\leq
2\int_{I}|z(s,0^{+})-z_{p}(s,0)|^{2}ds+2\int_{I}|z_{p}(s,0)-z_{p}(s,-\varepsilon
a)|^{2}ds$ $\displaystyle\leq
C(R+\hat{R})^{2}\varepsilon+C\hat{R}^{2}\min(1,\varepsilon^{2}a^{2})\leq
C(R+\hat{R})^{2}\varepsilon(1+a).$

Therefore

$\displaystyle\lim_{t\to
0^{+}}\int_{I}\int_{0}^{\infty}u_{\varepsilon}^{2}(s,a,t)\rho(s,a)dads$
$\displaystyle\leq
C(R+\hat{R})^{2}\varepsilon\int_{I}\int_{0}^{\infty}(1+a)\rho(s,a)dads\lesssim
C(R+\hat{R})^{2}\varepsilon.$

This, along with (5.20) and the definition of $E_{\varepsilon}$ guarantees
that $\lim_{t\to 0^{+}}E_{\varepsilon}(t)\leq C\bar{R}$ and (5.8) follows from
the monotonicity (5.3). Property (5.9) then follows from (5.8) using the
inequality $\|v\|_{\infty}^{3}\leq
C(\|v^{3}\|_{1}+\|(v^{3})^{\prime}\|_{1})\leq
C(1+\|v\|_{4}^{4}+\|v^{\prime}\|_{2}^{2})$. ∎

##  6 Uniform $L^{\infty}$ bounds and global existence (proof of Theorem 1(i))

To prove global existence we cannot solely rely on the energy estimates from
the previous section, since they give bounds up to $T^{*}$ for
$z^{\prime}_{\varepsilon}$ (and $z^{\prime\prime}_{\varepsilon}$) but not for
$z_{\varepsilon}$ itself. The latter are provided by the following
proposition, which immediately implies Theorem 1(i).

######  Proposition 6.1.

Assume (1.7)–(1.10), (1.12), and let $\varepsilon_{1},\bar{R}$ be respectively
given by Theorem 3 and Proposition 5.2. There exists $c_{0}>0$ depending only
on $\rho,L$ such that, for each $\varepsilon\in(0,\varepsilon_{0}]$ with
$\varepsilon_{0}=c_{0}\bar{R}^{-4}\leq\varepsilon_{1}$, we have

$\|z_{\varepsilon}(t)\|_{L^{\infty}(I)}\leq
C\bar{R}^{1/2},\quad-\infty<t<T^{*}_{\varepsilon}.$ (6.1)

As a consequence, we have

$T^{*}_{\varepsilon}=\infty,$ $\|z_{\varepsilon}(t)\|_{H^{2}(I)}\leq
C\bar{R}^{1/2},\quad t\in\mathbb{R}$ (6.2)
$\int_{0}^{\infty}\bigl{\\{}\|\partial_{t}z_{\varepsilon}(\tau)\|_{L^{2}(I)}^{2}+\varepsilon\|\partial_{t}z_{\varepsilon}^{\prime\prime}(\tau)\|_{L^{2}(I)}^{2}\bigr{\\}}d\tau\leq
C\bar{R}$ (6.3)

and

$\int_{0}^{\infty}\big{\\{}\varepsilon^{1/4}\|\partial_{t}z_{\varepsilon}(\tau)\|_{L^{\infty}(I)}^{2}+\varepsilon^{3/4}\|\partial_{t}z_{\varepsilon}^{\prime}(\tau)\|_{L^{\infty}(I)}^{2}\bigr{\\}}d\tau\leq
C\bar{R}.$ (6.4)

Furthermore, for each $\varepsilon\in(0,\varepsilon_{0}]$, we have

$\sup_{t>0}\|z_{\varepsilon}(t)\|_{W^{4,\infty}(I)}<\infty.$ (6.5)

The key to the uniform estimate (6.1) of $z_{\varepsilon}$ is the existence of
a conserved quantity, given by the following past time average of $z$:

$\Theta_{\varepsilon}(t)=\int_{\tau=0}^{\infty}\int_{I}z_{\varepsilon}(s,t-\varepsilon\tau)\varphi(s,\tau)dsd\tau,\quad\hbox{
where }\varphi(s,\tau)=\int_{\tau}^{\infty}\rho(s,a)da,$ (6.6)

for $t\in(0,T^{*})$. The function $\Theta_{\varepsilon}$ will be used in
connection with the scalar product $(z(t),\mu_{1})$, which turns out to
correspond with $\Theta_{\varepsilon}(t)$ without time shift. The following
lemma provides the crucial properties of $\Theta_{\varepsilon}$. We recall
that the constants $\kappa_{\varepsilon}$ are defined in (1.14).

######  Lemma 6.2.

Let the assumptions of Proposition 6.1 be in force.

(i) The function $\Theta_{\varepsilon}(t)$ is constant, namely

$\Theta_{\varepsilon}(t)=\kappa_{\varepsilon},\quad
t\in(0,T_{\varepsilon}^{*}),$ (6.7)

(ii) We have

$\sup_{t\in(0,T_{\varepsilon}^{*})}\bigl{|}(z_{\varepsilon}(t),\mu_{1})-\kappa_{\varepsilon}\bigr{|}\leq
C(\bar{R}\varepsilon)^{1/2}.$ (6.8)

###### Proof.

(i) Integrating the equation in space (i.e., taking $v=1$ as test-function in
(1.4)), we obtain

$\int_{I}\int_{0}^{\infty}(z(s,t)-z(s,t-\varepsilon a))\rho(s,a)dads=0,\quad
0<t<T_{\varepsilon}^{*}.$ (6.9)

Fix any $0<t<t^{\prime}<T_{\varepsilon}^{*}$ and set
$\ell=\varepsilon^{-1}(t^{\prime}-t)$. We denote
$z(t)\cdot\varphi(\tau)=\int_{I}z(s,t)\varphi(s,\tau)ds$ for conciseness.
Putting $\tau=y+\ell$ in the definition of $\Theta(t^{\prime})$, and denoting
$y_{+}=\max(y,0)$, we obtain

$\displaystyle\Theta(t^{\prime})-$
$\displaystyle\Theta(t)=\int_{-\ell}^{\infty}z(t-\varepsilon
y)\cdot\varphi(y+\ell)dy-\int_{0}^{\infty}z(t-\varepsilon\tau)\cdot\varphi(\tau)d\tau$
$\displaystyle=\int_{-\ell}^{\infty}z(t-\varepsilon
y)\cdot\bigl{(}\varphi(y+\ell)-\varphi(y_{+})\bigr{)}dy+\int_{-\ell}^{0}z(t-\varepsilon
y)\cdot\varphi(0)dy$
$\displaystyle=\int_{-\ell}^{0}\int_{0}^{\infty}z(t-\varepsilon
y)\cdot\rho(a)dady-\int_{-\ell}^{\infty}\int_{y_{+}}^{y+\ell}z(t-\varepsilon
y)\cdot\rho(a)dady\equiv I_{1}-I_{2}.$

Noting that $I_{2}=\int_{0}^{\infty}\int_{a-\ell}^{a}z(t-\varepsilon
y)\cdot\rho(a)dyda=\int_{-\ell}^{0}\int_{0}^{\infty}z(t-\varepsilon
y-\varepsilon a)\cdot\rho(a)dady$ by Fubini, and putting $\sigma=t-\varepsilon
y$, it follows that

$\displaystyle\Theta(t^{\prime})-\Theta(t)$
$\displaystyle=\int_{-\ell}^{0}\int_{0}^{\infty}\bigl{(}z(t-\varepsilon
y)-z(t-\varepsilon y-\varepsilon
a)\bigr{)}\cdot\rho(a)dady=\varepsilon^{-1}\int_{t}^{t^{\prime}}\left(\int_{0}^{\infty}(z(\sigma)-z(\sigma-\varepsilon
a))\cdot\rho(a)da\right)d\sigma,$

hence $\Theta(t^{\prime})=\Theta(t)$ in view of (6.9).

On the other hand, for each $s\in I$, $\tau>0$, we have
$z(s,t^{\prime}-\varepsilon\tau)\to z_{p}(s,-\varepsilon\tau)$ as
$t^{\prime}\to 0^{+}$, owing to (1.12). Moreover, for
$t^{\prime}\in(0,T_{\varepsilon})$, we have
$|z(s,t^{\prime}-\varepsilon\tau)|\rho(s,a)\leq\bigl{(}\sup_{\tau\in(-\infty,T_{\varepsilon})}\|z(\tau)\|_{\infty}\bigr{)}\rho(s,a)\in
L^{1}(I\times(0,\infty))$. Property (6.7) follows by dominated convergence.

(ii) By (6.3) we have

$\displaystyle\bigl{\|}z(t)-z(t_{0})\bigr{\|}_{2}$
$\displaystyle\leq\int_{t_{0}}^{t}\|z_{t}(\sigma)\|_{2}d\sigma\leq(t-t_{0})^{1/2}\Bigl{(}\int_{t_{0}}^{t}\|z_{t}(\sigma)\|^{2}_{2}d\sigma\Bigr{)}^{1/2}\leq
C(t-t_{0})^{1/2}\bar{R}^{1/2},\quad 0<t_{0}<t<T_{\varepsilon}^{*}.$ (6.10)

Next, for $-\infty<t_{0}<0<t<T_{\varepsilon}^{*}$, we may write

$\displaystyle\bigl{\|}z(t)-z(t_{0})\bigr{\|}_{2}$
$\displaystyle\leq\|z(t)-z(0^{+})\|_{2}+\|z(0^{+})-z_{p}(0)\|_{2}+\|z_{p}(0)-z_{p}(t_{0})\|_{2}\leq
C\bar{R}^{1/2}t^{1/2}+C(R+\hat{R})\varepsilon^{1/2}+C\hat{R}\,{\rm
min}(1,|t_{0}|),$

owing to (6.10), (5.11) and (1.12). Using ${\rm
min}(1,|t_{0}|)\leq|t_{0}|^{1/2}$, we obtain

$\displaystyle\bigl{\|}z(t)-z(t_{0})\bigr{\|}_{2}$ $\displaystyle\leq
C\bar{R}^{1/2}((t-t_{0})^{1/2}+\varepsilon^{1/2}),\quad-\infty<t_{0}<0<t<T_{\varepsilon}^{*}.$
(6.11)

Now, by the definition of $\varphi,\mu_{1}$, the first part of assumption
(1.10) and (6.7), we have

$\displaystyle\int_{0}^{\infty}\bigl{(}1+\tau^{1/2}\bigr{)}\|\varphi(\cdot,\tau)\|_{2}d\tau$
$\displaystyle\leq\int_{0}^{\infty}\int_{\tau}^{\infty}\bigl{(}1+\tau^{1/2}\bigr{)}\|\rho(\cdot,a)\|_{2}dad\tau\leq
C\int_{0}^{\infty}\bigl{(}1+a^{3/2}\bigr{)}\|\rho(\cdot,a)\|_{2}da<\infty$
(6.12)

and

$(z(t),\mu_{1})-\kappa_{\varepsilon}=\int_{0}^{\infty}\int_{I}\bigl{(}z(s,t)-z(s,t-\varepsilon\tau)\bigr{)}\varphi(s,\tau)dsd\tau,\quad
0<t<T_{\varepsilon}^{*}.$

Using (6.10)-(6.12), we obtain

$\displaystyle\bigl{|}(z(t),\mu_{1})-\kappa_{\varepsilon}\bigr{|}$
$\displaystyle\leq\int_{0}^{\infty}\|z(t)-z(t-\varepsilon\tau)\|_{2}\|{\varphi}(\cdot,\tau)\|_{2}d\tau\leq
C(\bar{R}\varepsilon)^{1/2}\int_{0}^{\infty}\bigl{(}1+\tau^{1/2}\bigr{)}\|{\varphi}(\cdot,\tau)\|_{2}d\tau,\quad
0<t<T_{\varepsilon}^{*},$ (6.13)

hence (6.8). ∎

###### Proof of Proposition 6.1.

We claim that

$\min_{s\in(0,L)}|z(s,t)|\leq CR+C(\bar{R}\varepsilon)^{1/2},\quad
0<t<T_{\varepsilon}^{*}.$ (6.14)

If $z(s,t)=0$ for some $s\in(0,L)$, then there is nothing to prove. We may
thus assume that $z(s,t)\neq 0$ for all $s\in(0,L)$. By continuity we may
assume $z(\cdot,t)>0$ in $(0,L)$ (the case $z<0$ is similar). Setting

$L_{1}:=\int_{I}\mu_{1}(s)ds=\int_{I}\int_{\tau=0}^{\infty}\varphi(s,\tau)d\tau=\int_{I}\int_{a=0}^{\infty}a\rho(s,a)dads>0,$

and using (6.8), we obtain

$L_{1}\min_{s\in(0,L)}z(s,t)\leq\int_{I}z(s,t)\mu_{1}(s)ds\leq|\kappa_{\varepsilon}|+C(\bar{R}\varepsilon)^{1/2},$

hence (6.14).

It follows from (5.9) and (6.14) that

$\|z(t)\|_{\infty}\leq CR+C\bar{R}^{1/3}+C(\bar{R}\varepsilon)^{1/2}\leq
C_{3}\bar{R}^{1/2},\quad 0<t<T_{\varepsilon}^{*},$

for some $C_{3}=C_{3}(\rho,L)>0$. This proves (6.1), hence (6.2) owing to
estimates (5.9) in Proposition 5.2.

Now, we may choose $c_{0}>0$ depending only on $\rho,L$ such that
$\varepsilon_{0}:=c_{0}\bar{R}^{-4}\leq\varepsilon_{1}$ and
$c_{5}\varepsilon_{0}^{-1/8}>C_{3}\bar{R}^{1/2}$. As a direct consequence of
Theorem 3(iii), we deduce that $T_{\varepsilon}^{*}=\infty$. Properties (6.3),
(6.4) then follow from (5.5)-(5.8).

Finally, using (5.9), (6.1) and the differential equation (1.5), we get

$\sup_{t>0}\|z_{\varepsilon}(t)\|_{H^{4}(I)}<\infty,$

hence in particular
$\sup_{t>0}\|z_{\varepsilon}(t)\|_{W^{2,\infty}(I)}<\infty$. Going back to
(1.5) we obtain (6.5). ∎

##  7 Hölder continuity with respect to $\varepsilon$ (proof of Theorem 1(ii))

It is a direct consequence of:

######  Proposition 7.1.

Assume (1.7)–(1.10), (1.12), and let $\varepsilon_{0}$ be given by Proposition
6.1. We have

$\|(z_{\varepsilon}-z_{\bar{\varepsilon}})(t)\|_{H^{2}}\leq
C\bar{\varepsilon}^{-1/4}\bar{R}({\bar{\varepsilon}}-\varepsilon)^{1/4}\exp\bigl{\\{}\bar{\varepsilon}^{-1}\|\rho\|_{\infty}t\bigr{\\}}.\quad
0<\varepsilon<\bar{\varepsilon}<\varepsilon_{0},$ (7.1)

###### Proof of Proposition 7.1.

Step 1. First estimate of the difference. Let
$0<\varepsilon<\bar{\varepsilon}<\varepsilon_{0}$ and $t>0$. We claim that

$\displaystyle{\bar{\varepsilon}}\int_{I}$
$\displaystyle(z^{\prime\prime}_{\varepsilon}-z^{\prime\prime}_{\bar{\varepsilon}})^{2}(t)ds+\int_{I}(z_{\varepsilon}-z_{\bar{\varepsilon}})^{2}(t)ds\leq
C\bar{R}^{2}({\bar{\varepsilon}}-\varepsilon)+C\int_{I}\int_{0}^{\infty}(z_{\varepsilon}-z_{\bar{\varepsilon}})^{2}(t-\bar{\varepsilon}a)\rho(s,a)dads$
(7.2) $\displaystyle+C\int_{I}\int_{0}^{\infty}(z_{\varepsilon}(t-\varepsilon
a)-z_{\varepsilon}(t-{\bar{\varepsilon}}a))^{2}\rho(s,a)dads.$

To this end, subtracting the equation (1.4) for $z_{\varepsilon}$ and for
$z_{\bar{\varepsilon}}$ and choosing
$v=(z_{\varepsilon}-z_{\bar{\varepsilon}})(t)$, we obtain (omitting the
variable $s$ without risk of confusion)

$\displaystyle{\bar{\varepsilon}}\int_{I}(z^{\prime\prime}_{\varepsilon}-z^{\prime\prime}_{\bar{\varepsilon}})^{2}(t)ds+\int_{I}{\mu}(z_{\varepsilon}-z_{\bar{\varepsilon}})^{2}(t)ds={\bar{\varepsilon}}\int_{I}(F^{\prime}(z^{\prime}_{\varepsilon})-F^{\prime}(z^{\prime}_{\bar{\varepsilon}}))(t)(z_{\varepsilon}-z_{\bar{\varepsilon}})^{\prime}(t)ds$
(7.3)
$\displaystyle\quad+({\bar{\varepsilon}}-\varepsilon)\left\\{\int_{I}z^{\prime\prime}_{\varepsilon}(t)(z_{\varepsilon}-z_{\bar{\varepsilon}})^{\prime\prime}(t)ds-\int_{I}(F^{\prime}(z^{\prime}_{\varepsilon}(t)))(z_{\varepsilon}-z_{\bar{\varepsilon}})^{\prime}(t)\right\\}ds$
$\displaystyle\quad+\int_{I}\int_{0}^{\infty}(z_{\varepsilon}-z_{\bar{\varepsilon}})(t-\bar{\varepsilon}a)\rho(s,a)(z_{\varepsilon}-z_{\bar{\varepsilon}})(t)dads+\int_{I}\int_{0}^{\infty}(z_{\varepsilon}(t-\varepsilon
a)-z_{\varepsilon}(t-{\bar{\varepsilon}}a))\rho(s,a)(z_{\varepsilon}-z_{\bar{\varepsilon}})(t)dads.$

To estimate the first term on the RHS of (7.3), we use (5.9), (6.2) and the
Sobolev embedding to write

$\left|F^{\prime}(z^{\prime}_{\varepsilon})-F^{\prime}(z^{\prime}_{\bar{\varepsilon}})\right|\leq
4\left\\{\left|z^{\prime}_{\varepsilon}-z_{\bar{\varepsilon}}^{\prime}\right|+\left|(z^{\prime}_{\varepsilon})^{3}-(z_{\bar{\varepsilon}}^{\prime})^{3}\right|\right\\}\leq
C\bar{R}\left|z^{\prime}_{\varepsilon}-z_{\bar{\varepsilon}}^{\prime}\right|.$

Next, using (3.12), one writes as above:

$\left|\int_{I}(F^{\prime}(z^{\prime}_{\varepsilon})-F^{\prime}(z^{\prime}_{\bar{\varepsilon}}))(z^{\prime}_{\varepsilon}-z^{\prime}_{\bar{\varepsilon}})ds\right|\leq
C\bar{R}\int_{I}\left|z^{\prime}_{\varepsilon}(s,t)-z_{\bar{\varepsilon}}^{\prime}(s,t)\right|^{2}ds\leq
C_{4}\bar{R}\left\\{\delta\|z_{\varepsilon}^{\prime\prime}(t)-z_{\bar{\varepsilon}}^{\prime\prime}(t)\|_{2}^{2}+(1+\delta^{-1})\|z_{\varepsilon}(t)-z_{\bar{\varepsilon}}(t)\|_{2}^{2}\right\\}$

with $C_{4}=C_{4}(\rho,L)>0$. Choosing $\delta=(2C_{4}\bar{R})^{-1}$, using
$\bar{R}^{2}\bar{\varepsilon}\leq c_{0}\bar{R}^{-2}\leq c_{0}$ and taking
$c_{0}=c_{0}(\rho,L)>0$ smaller in Proposition 6.1 if necessary, we obtain

$\displaystyle{\bar{\varepsilon}}$
$\displaystyle\left|\int_{I}(F^{\prime}(z^{\prime}_{\varepsilon}(t))-F^{\prime}(z^{\prime}_{\bar{\varepsilon}}(t)))(z_{\varepsilon}-z_{\bar{\varepsilon}})^{\prime}(t)ds\right|\leq
C\bar{R}^{2}\bar{\varepsilon}\int_{I}(z_{\varepsilon}-z_{\bar{\varepsilon}})^{2}(t)ds+\frac{{\bar{\varepsilon}}}{2}\int_{I}(z^{\prime\prime}_{\varepsilon}-z^{\prime\prime}_{\bar{\varepsilon}})^{2}(t)ds$
$\displaystyle\qquad\leq\frac{\mu_{min}}{2}\int_{I}(z_{\varepsilon}-z_{\bar{\varepsilon}})^{2}(t)ds+\frac{{\bar{\varepsilon}}}{2}\int_{I}(z^{\prime\prime}_{\varepsilon}-z^{\prime\prime}_{\bar{\varepsilon}})^{2}(t)ds.$

Next using (6.2), we can estimate the third term on the RHS of (7.3) by

$({\bar{\varepsilon}}-\varepsilon)\int_{I}\bigl{|}z^{\prime\prime}_{\varepsilon}(t)(z_{\varepsilon}-z_{\bar{\varepsilon}})^{\prime\prime}(t)\bigr{|}ds+\int_{I}\bigl{|}F^{\prime}(z^{\prime}_{\varepsilon}(t))(z_{\varepsilon}^{\prime}-z_{\bar{\varepsilon}}^{\prime})(t)\bigr{|}ds\leq
C\bar{R}^{2}({\bar{\varepsilon}}-\varepsilon).$

Estimating the remaining two terms by Young’s inequality, we get (7.2).

Step 2. Control of the time shift term in the RHS of (7.2). We claim that

$\int_{I}\int_{0}^{\infty}\bigl{|}z_{\varepsilon}(t-\varepsilon
a)-z_{\varepsilon}(t-{\bar{\varepsilon}}a)\bigr{|}^{2}\rho(s,a)dads\leq
C\bar{R}\bigl{(}\bar{\varepsilon}({\bar{\varepsilon}}-\varepsilon)\bigr{)}^{1/2}.$
(7.4)

To this end, recalling that
$z_{\varepsilon}(\tau)=z_{\bar{\varepsilon}}(\tau)=z_{p}$ for $\tau<0$, we
split the LHS of (7.4) as

$\displaystyle\int_{I}$
$\displaystyle\int_{0}^{\infty}\bigl{|}z_{\varepsilon}(t-\varepsilon
a)-z_{\varepsilon}(t-{\bar{\varepsilon}}a)\bigr{|}^{2}\rho(s,a)dads=\int_{I}\int_{0}^{t/{\bar{\varepsilon}}}\bigl{|}z_{\varepsilon}(t-\varepsilon
a)-z_{\varepsilon}(t-{\bar{\varepsilon}}a)\bigr{|}^{2}\rho(s,a)dads$
$\displaystyle\qquad+\int_{I}\int_{t/{\bar{\varepsilon}}}^{t/\varepsilon}\bigl{|}z_{\varepsilon}(t-\varepsilon
a)-z_{\varepsilon}(t-{\bar{\varepsilon}}a)\bigr{|}^{2}\rho(s,a)dads.$

By (6.10), we have

$\int_{I}\int_{0}^{t/{\bar{\varepsilon}}}\bigl{|}z_{\varepsilon}(t-\varepsilon
a)-z_{\varepsilon}(t-{\bar{\varepsilon}}a)\bigr{|}^{2}\rho(s,a)dads\leq
C({\bar{\varepsilon}}-\varepsilon)\bar{R}\int_{I}\int_{0}^{\infty}a\rho(s,a)dads.$
(7.5)

On the other hand, by (6.11), we have

$\int_{I}\bigl{|}z_{\varepsilon}(t-\varepsilon
a)-z_{\varepsilon}(t-{\bar{\varepsilon}}a)\bigr{|}^{2}ds\leq
C\bar{R}\bigl{(}(\bar{\varepsilon}-\varepsilon)a+\varepsilon\bigr{)},\quad
t/\bar{\varepsilon}<a<t/\varepsilon.$

Since, using the second part of assumption (1.10),

$\displaystyle\int_{t/{\bar{\varepsilon}}}^{t/\varepsilon}\|\rho(\cdot,a)\|_{\infty}da$
$\displaystyle\leq
C\min\left\\{\frac{\bar{\varepsilon}}{t}\int_{0}^{\infty}a\|\rho(\cdot,a)\|_{\infty}da,\frac{t({\bar{\varepsilon}}-\varepsilon)}{\varepsilon{\bar{\varepsilon}}}\|\rho\|_{\infty}\right\\}\leq
C\min\left\\{\frac{\bar{\varepsilon}}{t},\frac{t({\bar{\varepsilon}}-\varepsilon)}{\varepsilon{\bar{\varepsilon}}}\right\\}\leq
C\Bigl{(}\frac{{\bar{\varepsilon}}-\varepsilon}{\varepsilon}\Bigr{)}^{1/2},$

it follows that

$\displaystyle\int_{I}$
$\displaystyle\int_{t/{\bar{\varepsilon}}}^{t/\varepsilon}\bigl{|}z_{\varepsilon}(t-\varepsilon
a)-z_{\varepsilon}(t-{\bar{\varepsilon}}a)\bigr{|}^{2}\rho(s,a)dads\leq\int_{t/{\bar{\varepsilon}}}^{t/\varepsilon}C\bar{R}\bigl{(}(\bar{\varepsilon}-\varepsilon)a+\varepsilon\bigr{)}\|\rho(\cdot,a)\|_{\infty}da$
$\displaystyle\leq
C\bar{R}(\bar{\varepsilon}-\varepsilon)\int_{0}^{\infty}a\|\rho(\cdot,a)\|_{\infty}da+C\bar{R}\varepsilon\int_{t/{\bar{\varepsilon}}}^{t/\varepsilon}\|\rho(\cdot,a)\|_{\infty}da\leq
C\bar{R}(\bar{\varepsilon}-\varepsilon)+C\bar{R}\varepsilon\Bigl{(}\frac{{\bar{\varepsilon}}-\varepsilon}{\varepsilon}\Bigr{)}^{1/2}\leq
C\bar{R}\bigl{(}\bar{\varepsilon}({\bar{\varepsilon}}-\varepsilon)\bigr{)}^{1/2}.$

Claim (7.4) follows by adding this with (7.5).

Step 3. Conclusion. Letting
$\delta=C\bar{R}^{2}\bigl{(}\bar{\varepsilon}({\bar{\varepsilon}}-\varepsilon)\bigr{)}^{1/2}$,

$\Lambda_{\bar{\varepsilon},\varepsilon}(t)={\bar{\varepsilon}}\int_{I}(z^{\prime\prime}_{\varepsilon}-z^{\prime\prime}_{\bar{\varepsilon}})^{2}(t)ds+\int_{I}(z_{\varepsilon}-z_{\bar{\varepsilon}})^{2}(t)ds,$

combining (7.2), (7.4), and using that
$z_{\varepsilon}(\tau)=z_{\bar{\varepsilon}}(\tau)=z_{p}$ for $\tau<0$, we
obtain

$\displaystyle\Lambda_{\bar{\varepsilon},\varepsilon}(t)$
$\displaystyle\leq\delta+C\int_{0}^{t/\bar{\varepsilon}}\Lambda_{\bar{\varepsilon},\varepsilon}(t-\bar{\varepsilon}a)\|\rho\|_{\infty}da\leq\delta+C\bar{\varepsilon}^{-1}\int_{0}^{t}\Lambda_{\bar{\varepsilon},\varepsilon}(\tau)\|\rho\|_{\infty}d\tau.$
(7.6)

It follows from Gronwall’s lemma that
$\Lambda_{\bar{\varepsilon},\varepsilon}(t)\leq\delta\exp\bigl{\\{}\bar{\varepsilon}^{-1}\|\rho\|_{\infty}t\bigr{\\}}$,
hence (7.1). ∎

##  8 Proof of convergence as $t\to\infty$ (Theorem 1(iii))

We denote by $\mathcal{S}$ the set of steady states, i.e. solutions of (1.16).
The $\omega$-limit set of $z_{\varepsilon}$ is defined by

$\omega(z_{\varepsilon}):=\left\\{Z\in H^{2}(I);\,\exists t_{n}\to\infty,\
\lim_{n}\|z_{\varepsilon}(t_{n})-Z\|_{H^{2}(I)}=0\right\\}.$ (8.1)

Also, recalling (1.13), we define

$\mathcal{S}_{K}:=\bigl{\\{}Z\in\mathcal{S};\ (Z,\mu_{1})=K\bigr{\\}},\quad
K\in\mathbb{R}.$

For fixed $\varepsilon\in(0,\varepsilon_{0}]$, the convergence as $t\to\infty$
will be a direct consequence of the following quasiconvergence property, along
with the structure of steady states.

######  Lemma 8.1.

Let the assumptions of Theorem 1 be in force and let
$\varepsilon\in(0,\varepsilon_{0}]$.

(i) The set $\omega(z_{\varepsilon})$ is a nonempty compact connected subset
of $H^{2}(I)$.

(ii) We have
$\omega(z_{\varepsilon})\subset\mathcal{S}_{\kappa_{\varepsilon}}$.

Whereas the proof of Lemma 8.1(i) is standard, the proof of assertion (ii)
relies on two ingredients: the decay of the delay term
${\mathcal{L}}_{\varepsilon}[z_{\varepsilon}]$ as $t\to\infty$ and the good
properties of the inner product of $z_{\varepsilon}(t)$ with $\mu_{1}$.

######  Lemma 8.2.

Let the assumptions of Theorem 1 be in force.

  * (i)

Let $p\in[1,\infty)$. The function
${\mathcal{L}}_{\varepsilon}[z_{\varepsilon}]$ satisfies the decay property

$\lim_{t\to\infty}\|{\mathcal{L}}_{\varepsilon}[z_{\varepsilon}](\cdot,t)\|_{L^{p}(I)}=0,\quad\hbox{for
each $\varepsilon\in(0,\varepsilon_{0}]$.}$ (8.2)

  * (ii)

We have the convergence

$\lim_{t\to\infty}(z_{\varepsilon}(t),\mu_{1})=\kappa_{\varepsilon},\quad\hbox{for
each $\varepsilon\in(0,\varepsilon_{0}]$.}$ (8.3)

###### Proof.

(i) We have

$\int_{I}|{\mathcal{L}}_{\varepsilon}z(s,t)|ds\leq\int_{a=0}^{\infty}\int_{I}H_{\varepsilon}(t,s,a)dsda,\quad\hbox{
where }H_{\varepsilon}(t,s,a):=\varepsilon^{-1}|z(s,t)-z(s,t-\varepsilon
a)|\rho(s,a).$

For each fixed $(s,a)\in I\times(0,\infty)$, inequality (6.4) guarantees that,
for all $\varepsilon\in(0,\varepsilon_{0}]$ and $t>\varepsilon a$,

$H_{\varepsilon}(t,s,a)\leq\sqrt{\frac{a}{\varepsilon}}\,\rho(s,a)\Bigl{(}\int_{t-\varepsilon
a}^{t}\|z_{t}(\tau)\|^{2}_{\infty}d\tau\Bigr{)}^{1/2}\leq
C\bar{R}^{1/2}\varepsilon^{-5/8}\sqrt{a}\rho(s,a).$ (8.4)

Moreover, the first inequality in (8.4) implies
$\lim_{t\to\infty}H_{\varepsilon}(t,s,a)=0$ for each fixed $\varepsilon,s,a$.
Since $\sqrt{a}\rho(s,a)\in L^{1}(I\times(0,\infty))$, by dominated
convergence, we obtain (8.2) for $p=1$. Since, on the other hand,
$\|{\mathcal{L}}_{\varepsilon}z(\cdot,t)\|_{\infty}\leq
C\bar{R}^{1/2}\varepsilon^{-1}\|\rho\|_{L^{1}(I\times(0,\infty))}$ by (6.1),
we deduce (8.2) for all finite $p$ by interpolation.

(ii) Fix $\varepsilon\in(0,\varepsilon_{0}]$. Recalling (6.13) we have

$\bigl{|}(z(t),\mu_{1})-\kappa_{\varepsilon}\bigr{|}\leq\int_{0}^{\infty}\|z(t)-z(t-\varepsilon\tau)\|_{2}\|{\varphi}(\cdot,\tau)\|_{2}d\tau$

and, for each fixed $\tau>0$, (6.4) guarantees that

$\|z(t)-z(t-\varepsilon\tau)\|_{2}\leq\sqrt{\varepsilon\tau}\Bigl{(}\int_{t-\varepsilon\tau}^{t}\|z_{t}(\sigma)\|^{2}_{2}d\sigma\Bigr{)}^{1/2}\to
0,\,\text{as}\,\to\infty.$

On the other hand, we have
$\|z(t)-z(t-\varepsilon\tau)\|_{2}\|\varphi(\cdot,\tau)\|_{2}\leq
C\bar{R}\|\varphi(\cdot,\tau)\|_{2}\in L^{1}_{\tau}(0,\infty)$, owing to (6.1)
and (6.12). Property (8.3) then follows by dominated convergence. ∎

###### Proof of Lemma 8.1.

(i) It is easy to verify that

$\omega(z_{\varepsilon})={\mathop{\scalebox{1.5}{$\cap$}}_{n\in\mathbb{N}^{*}}}\overline{K_{n}},\quad\hbox{where
}K_{n}=\\{z_{\varepsilon}(s);\ s\geq n\\}.$

On the other hand, it follows from (6.5) that each $\overline{K_{n}}$ is
compact (for the $H^{2}(I)$ topology). Moreover, since $z\in
C((0,\infty);H^{2}(I))$ as a consequence of (4.6), $K_{n}$ hence
$\overline{K_{n}}$ is connected. We conclude that $\omega(z_{\varepsilon})$ is
a nonempty compact connected set, as the intersection of a nonincreasing
sequence of such sets.

(ii) Let $\varepsilon\in(0,\varepsilon_{0}]$. For any $v\in H^{2}(I)$ and
$t>0$, we have

$\displaystyle\Bigl{|}\int_{I}\bigl{(}z_{\varepsilon}^{\prime\prime}(s,t)v^{\prime\prime}(s)+F^{\prime}(z_{\varepsilon}^{\prime}(s,t))v^{\prime}(s)\bigr{)}ds\Bigr{|}$
$\displaystyle=\Bigl{|}\int_{I}{\mathcal{L}}_{\varepsilon}z_{\varepsilon}(s,t)v(s)ds\Bigr{|}\leq\|{\mathcal{L}}_{\varepsilon}z_{\varepsilon}(\cdot,t)\|_{L^{2}(I)}\|v\|_{L^{2}(I)}.$
(8.5)

Let $Z\in\omega(z_{\varepsilon})$. There exists a sequence $t_{n}\to\infty$
such that $\lim_{n}\|z_{\varepsilon}(t_{n})-Z\|_{H^{2}(I)}=0$. Passing to the
limit in (8.5) for $t=t_{n}$ with help of (8.2) with $p=2$, it follows that

$\int_{I}\bigl{(}Z^{\prime\prime}(s)v^{\prime\prime}(s)+F^{\prime}(Z^{\prime}(s))v^{\prime}(s)\bigr{)}ds=0.$

Consequently, $Z\in\mathcal{S}$. Taking $\kappa_{\varepsilon}$ given by Lemma
8.2, property (8.3) then guarantees that $(Z,\mu_{1})=\kappa_{\varepsilon}$,
hence $Z\in\mathcal{S}_{\kappa_{\varepsilon}}$. ∎

###### Proof of Theorem 1(iii).

The set $\omega(z_{\varepsilon})$ is nonempty and connected by Lemma 8.1 (i).
On the other hand, it is finite since
$\omega(z_{\varepsilon})\subset\mathcal{S}_{\kappa_{\varepsilon}}$ by Lemma
8.1(ii) and $\mathcal{S}_{\kappa_{\varepsilon}}$ is finite by Proposition 2.2.
The set $\omega(z_{\varepsilon})$ is thus a singleton, which proves the
theorem. ∎

##  9 Convergence when $\varepsilon$ goes to $0$ (proof of first part of
Theorem 2(i))

The result will be a consequence of the following two lemmas.

######  Lemma 9.1.

Assume (1.7)–(1.10), (1.12) and pick any sequence $\varepsilon_{j}\to 0^{+}$.

(i) There exists a subsequence $\varepsilon^{\prime}_{j}$ and

$\tilde{z}_{0}\in
L^{\infty}([0,\infty);H^{2}(I))\cap\,C^{\nu}_{loc}([0,\infty);C^{1}(\overline{I}))\cap\,H^{1}_{loc}([0,\infty);L^{2}(I))$
(9.1)

for all $\nu\in(0,1/8)$, such that,
$\lim_{j}z_{\varepsilon^{\prime}_{j}}=\tilde{z}_{0}$ where, for each $T>0$,
the convergence is strong in $C([0,T];$ $C^{1}(\overline{I})$, weak in
$H^{1}((0,T);L^{2}(I))$, and weak-* in $L^{\infty}(0,T;H^{2}(I))$.

(ii) The function $\tilde{z}_{0}$ is a global solution of (1.15) (cf.
Definition A.1) with $b=\mu_{1}$ and $\tilde{z}(0)={z_{p}}(0)$.

###### Proof.

(i) As a consequence of estimates (6.2), (6.3), similar to [11, Lemma 5,
p.976], we see that $\\{z_{\varepsilon}\\}$ is bounded in
$L^{\infty}([0,\infty);H^{2}(I))\cap H^{1}_{loc}([0,\infty);L^{2}(I))$ hence,
by interpolation, it is bounded in
$W^{\theta,2/\theta}_{loc}([0,\infty);H^{2(1-\theta)}(I))\hookrightarrow$
$C^{(\theta-\eta)/2}_{loc}$ $([0,\infty);$ $C^{1}(\overline{I}))$ for all
$\theta\in(0,1/4)$ and $\eta>0$. The conclusion then follows from standard
compactness properties.

(ii) Let $\varepsilon:=\varepsilon^{\prime}_{j}$ be as in assertion (i) and
fix $\psi\in C^{1}([0,T];H^{2}(I))$. Taking $v=\psi(\cdot,t)$ in (1.4) and
integrating in time, we obtain

${\mathcal{D}}_{\varepsilon}(\rho,{z}_{\varepsilon},\psi)+\int_{0}^{T}\int_{I}\bigl{(}z_{\varepsilon}^{\prime\prime}\psi^{\prime\prime}+F^{\prime}(z_{\varepsilon}^{\prime})\psi^{\prime}\bigr{)}dsdt=0,$
(9.2)

where

${\mathcal{D}}_{\varepsilon}(\rho,{z}_{\varepsilon},\psi):={\frac{1}{\varepsilon}}\int_{I}\int_{0}^{T}\int_{0}^{\infty}\rho(s,a)\left({z}_{\varepsilon}(s,t)-{z}_{\varepsilon}(s,t-\varepsilon
a)\right)da\psi(s,t)dtds.$

With the decomposition

$\displaystyle{\mathcal{D}}_{\varepsilon}$
$\displaystyle(\rho,{z}_{\varepsilon},\psi)={\frac{1}{\varepsilon}}\int_{I}\int_{0}^{T}\int_{\frac{T-t}{\varepsilon}}^{\infty}\rho(s,a){z}_{\varepsilon}(s,t)\psi(s,t)dadtds-\;\frac{1}{\varepsilon}\int_{I}\int_{0}^{T}\int_{0}^{\frac{T-t}{\varepsilon}}{z}_{\varepsilon}(s,t)(\psi(s,t+\varepsilon
a)-\psi(s,t))\,\rho(s,a)dadtds$
$\displaystyle-\;\frac{1}{\varepsilon}\int_{I}\int_{0}^{T}\int_{\frac{t}{\varepsilon}}^{\infty}\rho(s,a)z_{p}(s,t-\varepsilon
a)\psi(s,t)dadtds=:{\mathcal{D}}^{1}_{\varepsilon}-{\mathcal{D}}^{2}_{\varepsilon}-{\mathcal{D}}^{3}_{\varepsilon},$

we use an argument similar to [6, Propositions A.1 and A.2. pp.41-42]. Namely,
using the change of variables $(t,a)=(T-\varepsilon h,a)$ (resp.,
$(t,a)=(\varepsilon(a-\tau),a)$), the convergence
$z_{\varepsilon}\to\tilde{z}_{0}$ in $L^{\infty}(I\times(0,T))$, the
regularity of $\psi$ and assumptions (1.10), (1.12), it follows by dominated
convergence that

$\displaystyle{\mathcal{D}}^{1}_{\varepsilon}$
$\displaystyle=\int_{I}\int_{0}^{T/\varepsilon}\int_{h}^{\infty}\rho(s,a)(z_{\varepsilon}\psi)(s,T-\varepsilon
h)dadhds$
$\displaystyle\xrightarrow[j\to\infty]{}\int_{I}\int\int_{\\{0<h<a<\infty\\}}\rho(s,a)(\tilde{z_{0}}\psi)(s,T)dadhds=\int_{I}(\tilde{z_{0}}\psi)(s,T)\mu_{1}(s)ds,$
$\displaystyle{\mathcal{D}}^{3}_{\varepsilon}$
$\displaystyle=\int_{I}\int_{0}^{\infty}\int_{0}^{a}\rho(s,a)z_{p}(s,-\varepsilon\tau)\psi(s,\varepsilon(a-\tau))d\tau
dads$
$\displaystyle\xrightarrow[j\to\infty]{}\int_{I}\int\int_{\\{0<\tau<a<\infty\\}}\rho(s,a)(z_{p}\psi)(s,0)d\tau
dads=\int_{I}(z_{p}\psi)(s,0)\mu_{1}(s)ds,$
$\displaystyle{\mathcal{D}}^{2}_{\varepsilon}$
$\displaystyle\xrightarrow[j\to\infty]{}\int_{I}\int_{0}^{T}\int_{0}^{\infty}z_{0}(s,t)\partial_{t}\psi(s,t)a\rho(s,a)dadtds=\int_{0}^{T}\int_{I}\tilde{z}_{0}\partial_{t}\psi\mu_{1}dsdt.$

Therefore,

$\displaystyle{\mathcal{D}}_{\varepsilon}(\rho,{z}_{\varepsilon},\psi)\xrightarrow[j\to\infty]{}$
$\displaystyle\int_{I}\tilde{z_{0}}(T)\psi(T)\mu_{1}ds-\int_{0}^{T}\int_{I}\tilde{z}_{0}\partial_{t}\psi\mu_{1}dsdt-\int_{I}z_{p}(0)\psi(0)\mu_{1}ds$
$\displaystyle=\int_{0}^{T}\int_{I}\partial_{t}\tilde{z}_{0}\psi\mu_{1}dsdt+\int_{I}(\tilde{z}_{0}-z_{p})(0)\psi(0)\mu_{1}ds,$

where the integration by parts in time is allowed since $\tilde{z}_{0}\in
H^{1}(0,T;L^{2}(I))$. On the other hand, thanks to the weak and strong
convergence properties in assertion (i), we may pass to the limit in the
integral term in (9.2), and we get

$\int_{0}^{T}\int_{I}\partial_{t}\tilde{z}_{0}\psi\mu_{1}dsdt+\int_{I}(\tilde{z}_{0}-z_{p})(0)\psi(0)\mu_{1}ds+\int_{0}^{T}\int_{I}\bigl{(}\tilde{z_{0}}^{\prime\prime}\psi^{\prime\prime}+F^{\prime}(\tilde{z_{0}}^{\prime})\psi^{\prime}\bigr{)}dsdt=0.$

By density, this remains true for all $\psi\in L^{2}(0,T);H^{2}(I))$. Now, for
any $\theta\in H^{2}(I)$, by choosing $\psi_{n}(s,t)=(1-nt)_{+}\theta(s)$ and
letting $n\to\infty$, it follows that
$\int_{I}(\tilde{z}_{0}-z_{p})(0)\theta\mu_{1}ds=0$, hence
$\tilde{z}_{0}(0)=z_{p}(0)$ and $\tilde{z}_{0}$ is a global solution of
(1.15). ∎

######  Lemma 9.2.

Under the assumptions of Proposition 1.2, problem (1.15) has at most one
solution in the class $X:=L^{\infty}_{loc}([0,\infty);H^{2}(I))\cap
H^{1}_{loc}([0,\infty);L^{2}(I))$.

###### Proof.

Let $\tilde{z}_{0},\bar{z}_{0}\in X$ be two solutions (cf. Definition A.1).
Then, for each $T>0$, subtracting the equations, using
$|F^{\prime}(\tilde{z}_{0}^{\prime})-F^{\prime}(\bar{z}_{0}^{\prime})|\leq
C(T)|w^{\prime}|$ and taking $\psi=w:=\tilde{z}_{0}-\bar{z}_{0}$, we see that
$w$ satisfies

$\int_{0}^{t}\int_{I}b(s)w\partial_{t}wdsd\tau\leq
C(T)\int_{0}^{t}\int_{I}(w^{\prime})^{2}dsd\tau-\int_{0}^{t}\int_{I}(w^{\prime\prime})^{2}dsd\tau,\quad
0<t<T.$

Using the interpolation inequality (3.12) and $w(0)=0$, we obtain, for all
$t\in(0,T)$,

$\displaystyle\int_{I}b(s)$ $\displaystyle
w^{2}(t)ds=2\int_{0}^{t}\int_{I}b(s)w\partial_{t}wdsd\tau\leq
C_{1}(T)\int_{0}^{t}\int_{I}w^{2}dsd\tau\leq
C_{2}(T)\int_{0}^{t}\int_{I}b(s)w^{2}dsd\tau.$

By Gronwall’s Lemma, we deduce that $w\equiv 0$. ∎

###### Proof of first part of Theorem 2(i).

Let $\tilde{z}_{0}$, in the class (9.1), be any cluster point of the family
$\\{z_{\varepsilon},\ \varepsilon\in(0,\varepsilon_{0}]\\}$ for the notion of
convergence in Lemma 9.1. By that lemma, there exists at least one, and
$\tilde{z}_{0}$ is a global solution of (1.15) with $b=\mu_{1}$.

On the other hand, by (1.9), setting
$a_{0}=(2\|\rho\|_{\infty})^{-1}\mu_{\min}$, we get

$\mu_{1}(s)=\int_{0}^{\infty}a\rho(s,a)da\geq
a_{0}\int_{a_{0}}^{\infty}\rho(s,a)da\geq
a_{0}\bigl{(}\mu_{\min}-a_{0}\|\rho\|_{\infty}\bigr{)}\geq\frac{\mu_{\min}^{2}}{4\|\rho\|_{\infty}}>0,\
\ s\in I.$

It follows from Lemma 9.2 that (1.15) with $b=\mu_{1}$ has at most one global
weak solution, hence the cluster point is unique, which implies the desired
convergence as $\varepsilon\to 0$. ∎

##  10 Proof of Theorem 1(iv) and end of proof of Theorem 2: strong $H^{2}$
convergence and stablity of affine steady states with respect to $\varepsilon$

The energy associated with the solution $z_{0}$ of (1.15) is given by

$E_{0}(t):=\frac{1}{2}\int_{I}(z_{0}^{\prime\prime}(t))^{2}ds+\int_{I}F(z_{0}^{\prime}(t))ds.$

We shall use the following properties (see A for the proof).

######  Proposition 10.1.

Let the assumptions of Proposition 1.2 be in force.

(i) For all $t_{2}>t_{1}\geq 0$, we have

$E_{0}(t_{2})-E_{0}(t_{1})=-\int_{t_{1}}^{t_{2}}\int_{I}b(\partial_{t}z_{0})^{2}dsdt\leq
0.$

In particular, $E_{0}\in W^{1,p}_{loc}([0,\infty))$ for all $p\in(1,\infty)$
and

$E^{\prime}_{0}(t)=-\int_{I}b(\partial_{t}z_{0})^{2}ds,\quad a.e.~{}t>0.$

(ii) For all $t>0$, we have $(z_{0}(t),b)=K_{0}:=(\phi,b)$.

Our next result, which requires the additional coercivity assumption (1.11) on
the kernel, yields the strong $L^{2}_{t}(H^{2}_{s})$ convergence statement in
Theorem 2(i). It also provides the energy convergence property which will be
the key to the proof of Theorem 2(ii). We recall that $E_{\varepsilon}$ and
$u_{\varepsilon}$ are defined in (5.1)-(5.2).

######  Proposition 10.2.

Assume (1.7)–(1.12). Then, for all $T>0$, we have

$\lim_{\varepsilon\to 0}z_{\varepsilon}=z_{0}\ \hbox{ strongly in
$L^{2}(0,T;H^{2}(I))$}$ (10.1)

and

$\lim_{\varepsilon\to 0}E_{\varepsilon}(t)=E_{0}(t),\quad\hbox{uniformly for
$t>0$ bounded.}$ (10.2)

Moreover, one has

$\lim_{\varepsilon\to 0}E^{\prime}_{\varepsilon}(t)\equiv\lim_{\varepsilon\to
0}\frac{1}{2\varepsilon^{2}}\int_{I}\int_{0}^{\infty}u^{2}_{\varepsilon}(s,a,t)\partial_{a}\varrho(s,a)dads=-\int_{I}\mu_{1}(s)\left|\partial_{t}{z_{0}}(s,t)\right|^{2}ds\equiv
E^{\prime}_{0}(t),$ (10.3)

the convergence taking place weakly in ${\mathcal{M}}(0,T)$, i.e. in the space
of finite Radon measures on $(0,T)$.

######  Remark 10.1.

(i) The equality in (10.3) can be explained heuristically as follows: when
$\varepsilon\to 0$, one has formally that

$\frac{1}{2\varepsilon^{2}}\int_{0}^{\infty}\partial_{a}\rho(s,a)u_{\varepsilon}^{2}(s,a,t)da\to\frac{1}{2}\int_{0}^{\infty}\partial_{a}\rho(s,a)u_{0}^{2}(s,a,t)da,\quad\hbox{where
$u_{0}(s,a,t):=a\partial_{t}{z_{0}}(s,t)$.}$

Then the identity $\partial_{a}(a^{2}\rho)=2a\rho+a^{2}\partial_{a}\rho$,
because the integral of the LHS vanishes, implies that
$\mu_{1}(s)\equiv\int_{0}^{\infty}a\rho(s,a)da=-\frac{1}{2}\int_{0}^{\infty}\partial_{a}\rho(s,a)a^{2}da$,
giving the desired limit.

(ii) With some additional work, it can be shown that the same result holds in
the finite dimensional case [8], as well as for the linear delayed heat
equation [10] and also in the case of delayed harmonic maps [6].

The proof is based on the following two lemmas. The first one is a higher
order a priori estimate, uniform for $\varepsilon>0$:

######  Lemma 10.3.

Assume (1.7)–(1.12). Then, for each $T>0$, we have

$\sup_{\varepsilon\in(0,\varepsilon_{0}]}\|z_{\varepsilon}\|_{L^{2}(0,T;H^{3}(I))}<\infty$
(10.4)

and

$\sup_{\varepsilon\in(0,\varepsilon_{0}]}\|z_{\varepsilon}\|_{H^{1/6}(0,T;H^{5/2}(I))}<\infty.$
(10.5)

In particular, the family $\\{z_{\varepsilon},\
\varepsilon\in(0,\varepsilon_{0}]\\}$ is precompact in $L^{2}(0,T;H^{2}(I))$.

Our second lemma shows that the first term in the energy $E_{\varepsilon}$
vanishes as $\varepsilon\to 0$.

######  Lemma 10.4.

Assume (1.7)–(1.10), (1.12) and let

$G_{\varepsilon}(t):=\varepsilon^{-1}\int_{I}\int_{0}^{\infty}u_{\varepsilon}^{2}(t,s,a)\rho(s,a)dads.$

Then we have

$\int_{T}^{T+1}G_{\varepsilon}(t)dt\leq
C\bar{R}\bigl{(}\eta(\varepsilon)+\sqrt{\varepsilon}\bigl{(}1+T^{-1/2}\bigr{)}\bigr{)},\quad\hbox{for
all }T>0,$ (10.6)

where
$\eta(\varepsilon):=\int_{1/\sqrt{\varepsilon}}^{\infty}a\|\rho(\cdot,a)\|_{\infty}da\to
0$, as $\varepsilon\to 0$.

###### Proof of Lemma 10.3.

Recalling (1.22), multiplying (1.5) with $-z^{\prime\prime}$, integrating by
parts, using (5.9) and the Sobolev inequality
$\|z^{\prime\prime}\|_{\infty}\leq
C(\|z^{\prime\prime}\|_{2}+\|z^{\prime\prime\prime}\|_{2})$, we get, for all
$t>0$,

$\displaystyle\|z^{\prime\prime\prime}\|_{2}^{2}$
$\displaystyle=-\int_{I}(F^{\prime}(z^{\prime}))^{\prime}z^{\prime\prime}+\int_{I}(\mathcal{L}_{\varepsilon}z)z^{\prime\prime}\leq
C\int_{I}(1+(z^{\prime})^{2})(z^{\prime\prime})^{2}+\|\mathcal{L}_{\varepsilon}z\|_{1}\|z^{\prime\prime}\|_{\infty}$
$\displaystyle\leq
C\bar{R}^{5/3}+C\|\mathcal{L}_{\varepsilon}z\|_{1}(\|z^{\prime\prime\prime}\|_{2}+\bar{R}^{1/2})\leq
C\bar{R}^{5/3}+\frac{1}{2}\|z^{\prime\prime\prime}\|_{2}^{2}+C\bar{R}+C\|\mathcal{L}_{\varepsilon}z\|_{1}^{2},$

hence

$\|z^{\prime\prime\prime}\|_{2}^{2}\leq
C\bar{R}^{5/3}+C\|\mathcal{L}_{\varepsilon}z\|_{1}^{2}.$ (10.7)

On the other hand, by Cauchy-Schwarz, we have

$\displaystyle\|\mathcal{L}_{\varepsilon}z\|_{1}^{2}$
$\displaystyle\leq\varepsilon^{-2}\left(\int_{I}\int_{0}^{\infty}u(s,t,a)\rho(s,a)dads\right)^{2}=\varepsilon^{-2}\left(\int_{I}\int_{0}^{\infty}\bigl{(}u(s,t,a)|\rho_{a}(s,a)|^{1/2})\rho(s,a)|\rho_{a}(s,a)|^{-1/2}dads\right)^{2}$
$\displaystyle\leq\varepsilon^{-2}\int_{I}\int_{0}^{\infty}u^{2}(t,s,a)|\rho_{a}(s,a)|dads\int_{I}\int_{0}^{\infty}\rho^{2}(s,a)|\rho_{a}(s,a)|^{-1}dads$

(where we defined $\rho(s,a)|\rho_{a}(s,a)|^{-1/2}:=0$ if
$\rho(s,a)=\rho_{a}(s,a)=0$). Consequently, by (1.11), (5.4) and (5.8),

$\int_{0}^{T}\|\mathcal{L}_{\varepsilon}z\|_{1}^{2}dt\leq\varepsilon^{-2}\int_{0}^{T}\int_{I}\int_{0}^{\infty}u^{2}(t,s,a)|\rho_{a}(s,a)|dads\leq
C\bar{R}.$

From this, (5.9) and (10.7), we deduce (10.4). Since, on the other hand, for
each $T>0$, (6.2) implies

$\sup_{\varepsilon\in(0,\varepsilon_{0}]}\|z_{\varepsilon}\|_{H^{1}(0,T;L^{2}(I))}<\infty,$

it follows by interpolation with (10.4) that
$\sup_{\varepsilon\in(0,\varepsilon_{0}]}\|z_{\varepsilon}\|_{H^{1-\nu}(0,T;H^{3\nu}(I))}<\infty$
for each $\nu\in(0,1)$, hence (10.5). ∎

###### Proof of Lemma 10.4.

We have

$\displaystyle G_{\varepsilon}(t)\leq$
$\displaystyle\varepsilon^{-1}\int_{I}\int_{0}^{t/\varepsilon}\left(\int_{t-\varepsilon
a}^{t}|z_{t}(s,\sigma)|d\sigma\right)^{2}\rho(s,a)dads+\varepsilon^{-1}\int_{I}\int_{t/\varepsilon}^{\infty}u^{2}(t,s,a)\rho(s,a)dads\equiv
G^{(1)}_{\varepsilon}(t)+G^{(2)}_{\varepsilon}(t).$

To estimate $G^{(1)}_{\varepsilon}(t)$, we write

$\displaystyle G^{(1)}_{\varepsilon}(t)$
$\displaystyle\leq\int_{I}\int_{0}^{t/\varepsilon}\left(\int_{t-\varepsilon
a}^{t}|z_{t}(s,\sigma)|^{2}d\sigma\right)a\rho(s,a)dads\leq\int_{0}^{t/\varepsilon}\int_{t-\varepsilon
a}^{t}\|z_{t}(\sigma)\|_{2}^{2}a\|\rho(\cdot,a)\|_{\infty}d\sigma da$ (10.8)
$\displaystyle\leq\int_{0}^{t}\|z_{t}(\sigma)\|_{2}^{2}\left(\int_{(t-\sigma)/\varepsilon}^{t/\varepsilon}a\|\rho(\cdot,a)\|_{\infty}da\right)d\sigma$
$\displaystyle\leq\int_{0}^{(t-\sqrt{\varepsilon})_{+}}\|z_{t}(\sigma)\|_{2}^{2}\left(\int_{1/\sqrt{\varepsilon}}^{\infty}a\|\rho(\cdot,a)\|_{\infty}da\right)d\sigma+C\int_{(t-\sqrt{\varepsilon})_{+}}^{t}\|z_{t}(\sigma)\|_{2}^{2}d\sigma,$

where we used the second part of (1.10) in the last inequality. To handle the
last integral, we compute

$\int_{T}^{T+1}\int_{(t-\sqrt{\varepsilon})_{+}}^{t}\|z_{t}(\sigma)\|_{2}^{2}d\sigma
dt\leq\int_{0}^{T+1}\|z_{t}(\sigma)\|_{2}^{2}\biggl{(}\int_{\sigma}^{\sigma+\sqrt{\varepsilon}}dt\biggr{)}d\sigma\\\
\leq\sqrt{\varepsilon}\int_{0}^{T+1}\|z_{t}(\sigma)\|_{2}^{2}d\sigma.$

Going back to (10.8) and using (6.3), we get

$\int_{T}^{T+1}G^{(1)}_{\varepsilon}(t)dt\leq
C\bigl{(}\eta(\varepsilon)+\sqrt{\varepsilon}\bigr{)}\int_{0}^{T+1}\|z_{t}(\sigma)\|_{2}^{2}d\sigma\leq
C\bigl{(}\eta(\varepsilon)+\sqrt{\varepsilon}\bigr{)}\bar{R}.$ (10.9)

To estimate $G^{(2)}_{\varepsilon}(t)$, we use (6.1) and the first part of
(1.10) to write

$\displaystyle G^{(2)}_{\varepsilon}(t)$ $\displaystyle\leq
C\bar{R}\varepsilon^{-1}\int_{I}\int_{t/\varepsilon}^{\infty}\rho(s,a)dads\leq
C\bar{R}\varepsilon^{1/2}t^{-3/2}\int_{I}\int_{t/\varepsilon}^{\infty}a^{3/2}\rho(s,a)dads\leq
C\bar{R}\varepsilon^{1/2}t^{-3/2}.$

Integrating the latter for $t\in(T,T+1)$ and combining with (10.9), we get
(10.6). ∎

###### Proof of Proposition 10.2.

By Lemma 10.3, for any sequence $\varepsilon_{j}\to 0^{+}$, there exists a
subsequence converging in $L^{2}_{loc}([0,\infty);H^{2}(I))$ to some
$\tilde{z}$, and we moreover have $\tilde{z}\in H^{1}_{loc}([0,\infty);$
$L^{2}(I))$ owing to Lemma 9.1(i). By the proof of Lemma 9.1(ii), $\tilde{z}$
is a weak solution of (1.15), hence $\tilde{z}=z_{0}$ by Lemma 9.2. This shows
(10.1).

Next write

$E_{\varepsilon}(t)=\hat{E}_{\varepsilon}(t)+\frac{1}{2}G_{\varepsilon}(t),\quad\hbox{
with
}\hat{E}_{\varepsilon}(t):=\frac{1}{2}\int_{I}(z_{\varepsilon}^{\prime\prime}(t))^{2}ds+\int_{I}F(z_{\varepsilon}^{\prime}(t))ds.$

By (10.1) and Lemma 10.4, for any $t_{2}>t_{1}>0$ we have

$\lim_{\varepsilon\to
0}\int_{t_{1}}^{t_{2}}E_{\varepsilon}(\sigma)d\sigma=\lim_{\varepsilon\to
0}\int_{t_{1}}^{t_{2}}\hat{E}_{\varepsilon}(\sigma)d\sigma+\frac{1}{2}\lim_{\varepsilon\to
0}\int_{t_{1}}^{t_{2}}G_{\varepsilon}(\sigma)d\sigma=\int_{t_{1}}^{t_{2}}E_{0}(\sigma)d\sigma.$

By the time monotonicity of $E_{0}$ and $E_{\varepsilon}$, dividing by
$t_{2}-t_{1}$, it follows that

$\liminf_{\varepsilon\to 0}E_{\varepsilon}(t_{1})\geq
E_{0}(t_{2}),\quad\limsup_{\varepsilon\to 0}E_{\varepsilon}(t_{2})\leq
E_{0}(t_{1}).$ (10.10)

On the other hand, by (1.17), we have $E_{0}\in C([0,\infty))$. For fixed
$t>0$, taking $t_{1}=t$ (resp., $t_{2}=t$) and letting $t_{2}\to t^{+}$
(resp., $t_{1}\to t^{-}$) in the first (resp., second) inequality of (10.10),
we obtain

$\liminf_{\varepsilon\to 0}E_{\varepsilon}(t)\geq
E_{0}(t)\geq\limsup_{\varepsilon\to 0}E_{\varepsilon}(t),$

hence (10.2) (the convergence being uniform for bounded $t$, owing to the
monotonicity of $t\mapsto E_{\varepsilon}(t)$ and Dini’s theorem).

Finally, by (5.3) and (5.4), we have $E^{\prime}_{\varepsilon}\in L^{1}(0,T)$
and
$\sup_{\varepsilon\in(0,\varepsilon_{0})}\|E^{\prime}_{\varepsilon}\|_{L^{1}(0,T)}<\infty$.
For any sequence $\varepsilon_{i}\to 0$, some subsequence of
$E^{\prime}_{\varepsilon_{i}}$ thus converges weakly in the sense of measures
to some limit $\mu$. On the other hand, we know from (10.2) that
$E_{\varepsilon}$ converges to $E_{0}$ in ${\mathcal{D}}^{\prime}((0,T))$ as
$\varepsilon\to 0$. By uniqueness of limits $\mu=E_{0}^{\prime}$ and (10.3)
follows. ∎

###### Proof of Theorem 2(ii).

First note that, since $\mathcal{S}$ is finite up to additive constants, we
have

$\eta_{1}=\min\Bigl{\\{}\tilde{E}(W);\ W\in\mathcal{S},\
|W^{\prime}|\not\equiv 1\Bigr{\\}}>0,\quad\hbox{where
}\tilde{E}(W)=\frac{1}{2}\int_{I}(W^{\prime\prime})^{2}ds+\int_{I}F(W^{\prime})ds.$
(10.11)

Also, since the imbedding $H^{2}(I)\subset C^{1}(\overline{I})$ is compact, we
have

$\eta_{2}=\inf\Bigl{\\{}\tilde{E}(W);\ W\in H^{2}(I),\
W^{\prime}(0)=0\Bigr{\\}}>0.$ (10.12)

Set $\eta_{0}=\min(\eta_{1},\eta_{2})$.

Assume that $Z^{\prime}_{0}\equiv 1$ (the case $Z^{\prime}_{0}\equiv-1$ is
similar). Since $\lim_{t\to\infty}E_{0}(t)=\tilde{E}(Z_{0})=0$ and
$\lim_{t\to\infty}\|z^{\prime}_{0}(t)-1\|_{\infty}=0$, there exists $t_{0}>0$
such that $E_{0}(t_{0})<\eta_{0}/2$ and $z^{\prime}_{0}(0,t_{0})>1/2$. By the
convergence property (10.2) of the energy and the fact that
$z_{\varepsilon}\to z_{0}$ in $C([0,t_{0}];C^{1}([0,L]))$ (cf. Lemma 9.1),
there exists $\bar{\varepsilon}_{0}\in(0,\varepsilon_{0})$ such that, for all
$\varepsilon\in(0,\bar{\varepsilon}_{0})$,
$E_{\varepsilon}(t_{0})\leq\eta_{0}/2$ and
$z^{\prime}_{\varepsilon}(0,t_{0})\geq 1/2$. In particular,

$\tilde{E}(Z_{\varepsilon})\leq E_{\varepsilon}(Z_{\varepsilon})\leq
E_{\varepsilon}(t_{0})\leq\eta_{0}/2<\eta_{1},$

hence $Z^{\prime}_{\varepsilon}\equiv\pm 1$. Moreover, for any
$\varepsilon\in(0,\bar{\varepsilon}_{0}]$, the case
$Z^{\prime}_{\varepsilon}\equiv-1$ cannot occur, since otherwise, by the
continuity of $t\mapsto z_{\varepsilon}(t)$ in $C^{1}(\bar{I})$ (cf.
Proposition 4.1), there would exist $t_{1}>t_{0}$ such that
$z^{\prime}_{\varepsilon}(0,t_{1})=0$, hence

$\eta_{2}\leq\tilde{E}(z_{\varepsilon}(t_{1}))\leq E_{\varepsilon}(t_{1})\leq
E_{\varepsilon}(t_{0})\leq\eta_{0}/2:$

a contradiction. We have thus shown that, for all
$\varepsilon\in(0,\bar{\varepsilon}_{0}]$, $Z^{\prime}_{\varepsilon}\equiv 1$.
The proof is complete. ∎

###### Proof of Theorem 1(iv).

The argument is completely similar to that in the proof of Theorem 2(ii), but
easier (and without requiring assumption (1.11)), since we can just rely on
the continuity property in Theorem 1(ii) instead of Proposition 10.2). We
therefore skip the details. ∎

##  Appendix A Proof of Propositions 1.2 and 10.1

Consider the linear inhomogeneous problem

$b\partial_{t}u+u^{\prime\prime\prime\prime}=f^{\prime}\ \hbox{in $Q_{T}$,\
with $u^{\prime\prime}=u^{\prime\prime\prime}-f=0$ on $\partial I$, \
$u(0)=u_{0}$,}$ (A.1)

where $T>0$, $Q_{T}=(0,T)\times I$, $f\in L^{2}(0,T;H^{1})$ and $u_{0}\in
L^{2}$.

######  Definition A.1.

(i) A solution of (A.1) on $(0,T)$ is a function $u$ such that

$\left\\{\begin{aligned} &u\in H^{1}(0,T;L^{2})\cap L^{2}(0,T;H^{2}),\quad
u(0)=u_{0},\\\
&\int_{0}^{T}\int_{I}\bigl{(}b(s)\partial_{t}u\psi+u^{\prime\prime}\psi^{\prime\prime}+f\psi^{\prime}\bigr{)}dsdt=0,\quad\forall\psi\in
L^{2}(0,T;H^{2}(I)).\end{aligned}\right.$ (A.2)

Observe that the initial condition in (A.2) makes sense owing to
$H^{1}(0,T;L^{2})\subset C([0,T];L^{2})$.

(ii) A solution $z_{0}$ of (1.15) on $(0,T)$ is a solution of (A.1) with
$f=F^{\prime}(z_{0}^{\prime})$ and $u_{0}=\phi$ (note that $z_{0}\in
L^{2}(0,T;H^{2})$ implies $F^{\prime}(z_{0}^{\prime})\in L^{2}(0,T;H^{1})$).

######  Remark A.1.

(i) The definition (A.1) of (weak) solution is actually equivalent to that of
$u$ being a strong solution, i.e.

$\left\\{\begin{aligned} &u\in H^{1}(0,T;L^{2})\cap L^{2}(0,T;H^{4}),\quad
u(0)=u_{0},\\\ &b\partial_{t}u+u^{\prime\prime\prime\prime}-f^{\prime}=0\hbox{
in $L^{2}(I)$ for a.e.~{}$t\in(0,T)$,}\\\
&u^{\prime\prime}=u^{\prime\prime\prime}-f=0\hbox{ on $\partial I$ for
a.e.~{}$t\in(0,T)$.}\end{aligned}\right.$ (A.3)

Indeed, (A.2) implies $u^{\prime\prime\prime\prime}=f^{\prime}-b\partial_{t}u$
in the distributional sense in $(0,T)\times I$, hence $u\in L^{2}(0,T;H^{4})$,
and the other two conditions in (A.3) are satisfied owing to the identity

$\int_{0}^{T}\int_{I}\bigl{(}u^{\prime\prime\prime\prime}-f^{\prime}\bigr{)}\psi=\int_{0}^{T}\int_{I}\bigl{(}u^{\prime\prime}\psi^{\prime\prime}+f\psi^{\prime}\bigr{)}+\int_{0}^{T}[(u^{\prime\prime\prime}-f)\psi-u^{\prime\prime}\psi^{\prime}]_{0}^{L},\
\ \psi\in L^{2}(0,T;H^{2}).$

Also, the converse implication readily follows from this identity.

(ii) Equation (A.1) has a most one solution (just subtract the equations for
$u_{1}$ and $u_{2}$ and take $\psi=u_{1}-u_{2}$).

We shall use the following linear result for (A.1). In what follows, for
$1<p<\infty$, we denote

$X_{p,T}:=W^{1,p}(0,T;L^{2})\cap L^{p}(0,T;H^{4}),\quad
X_{p,T,loc}:=W^{1,p}_{loc}((0,T];L^{2})\cap L^{p}_{loc}((0,T];H^{4})$

and $B:L^{2}(0,T;H^{1})\to L^{2}(0,T;\mathbb{R}^{2})$ is the trace operator
(namely, $(Bf)(t)=\\{f(t,0),f(t,L)\\}$ for a.e. $t\in(0,T)$).

######  Lemma A.1.

Let $T>0$, $p>2$, $u_{0}\in H^{2}$, $f\in L^{p}(0,T;H^{1})$ and assume that
$Bf\in W^{\theta,p}(0,T;\mathbb{R}^{2})$ with $\theta>1/8$. Then there exists
a strong solution $u\in C([0,T];H^{2})\cap X_{2,T}\cap X_{p,T,loc}$ of (A.1)
such that $u(0)=u_{0}$. Moreover, for each $\eta\in(0,T)$, we have the
estimate

$\displaystyle\|\partial_{t}u\|_{L^{p}(\eta,T;L^{2})}$
$\displaystyle+\|u^{\prime\prime\prime\prime}\|_{L^{p}(\eta,T;L^{2})}\leq
C(\eta,T)\bigl{(}\|f^{\prime}\|_{L^{p}(0,T;L^{2})}+\|Bf\|_{W^{\theta,p}(0,T;\mathbb{R}^{2})}+\|u\|_{L^{p}(0,T;L^{2})}\bigr{)}.$
(A.4)

###### Proof.

Set $a=1/b\in H^{2}(I)$. We approximate the initial data by a sequence of
smooth functions $u_{0,n}\in C^{\infty}(\bar{I})$ such that
$\lim_{n\to\infty}\|u_{0,n}-u_{0}\|_{H^{2}}=0$ and consider the problem

$\partial_{t}u_{n}+au_{n}^{\prime\prime\prime\prime}=af^{\prime},\quad\hbox{with
$u_{n}^{\prime\prime}=u_{n}^{\prime\prime\prime}-f=0$ on $\partial I$ and
$u_{n}(0)=u_{0,n}$.}$ (A.5)

We claim that, by [2, Theorem 2.3], (A.5) admits a (unique) solution $u_{n}\in
X_{p,T}$. Indeed we check the applicability of that theorem with $m=q=2$ and
$\kappa_{j}=\frac{7-2j}{8}$ for $j=2,3$. Since $f^{\prime}\in
L^{p}(0,T;L^{2})$ and since no compatibility conditions for the initial and
boundary data are required owing to $\kappa_{j}\leq 1/q$, the result applies
provided the boundary trace $Bf$ satisfies $Bf\in
F^{\kappa}_{p,2}(0,T;\mathbb{R}^{2})$ with $\kappa=\kappa_{3}=1/8$, where
$F^{\kappa}_{p,q}$ denotes the Triebel-Lizorkin space. To this end it suffices
to use the fact that

$W^{\theta,p}(0,T;\mathbb{R}^{2})=F^{\theta}_{p,p}(0,T;\mathbb{R}^{2})\subset
F^{\kappa}_{p,2}(0,T;\mathbb{R}^{2}),\quad\theta>\kappa>0,\ 1<p<\infty$

(see [18, Chapter 2] and, e.g., [17, section 1] for the first part and [5,
Theorem 1.2] for the second part. These are stated there in the whole
Euclidean space, but the case of a smooth domain – here just the interval
$(0,T)$ – follows by a standard extension property; see [19, Chapter 2]).

Now, by interpolation, we have $u_{n}\in W^{1/2,p}(0,T;H^{2})\subset
C([0,T];H^{2})$. We claim that

$\max_{t\in[0,T]}\|(u_{n}-u_{k})(t)\|_{H^{2}}+\int_{0}^{T}\int_{I}|\partial_{t}(u_{n}-u_{k})|^{2}dsdt\leq
C\|u_{0,n}-u_{0,k}\|_{H^{2}}.$ (A.6)

Indeed, set $w=u_{n}-u_{k}$, which is the solution of
$\partial_{t}w+aw^{\prime\prime\prime\prime}=0$ with
$w^{\prime\prime}=w^{\prime\prime\prime}=0$ on $\partial I$ and
$w(0)=u_{0,n}-u_{0,k}$. For $0<t<T$ and $0<h<\min(t,T-t)$, applying (A.2) with
$u(t)=w(t+h)+w(t)$, $f=0$ and $\psi=h^{-1}(w(\tau+h)-w(\tau))$, we obtain

$\displaystyle h^{-1}\int_{t}^{t+h}$
$\displaystyle\int_{I}(w^{\prime\prime})^{2}-h^{-1}\int_{0}^{h}\int_{I}(w^{\prime\prime})^{2}=h^{-1}\int_{0}^{t}\int_{I}(w^{\prime\prime}(\tau+h))^{2}-(w^{\prime\prime}(\tau))^{2}$
$\displaystyle=\int_{0}^{t}\int_{I}(w(\tau+h)+w(\tau))^{\prime\prime}\frac{(w(\tau+h)-w(\tau))^{\prime\prime}}{h}=-\int_{0}^{t}\int_{I}b(w(\tau+h)+w(\tau))_{t}\frac{w(\tau+h)-w(\tau)}{h}.$

Letting $h\to 0$ and using $u_{n}\in C([0,T];H^{2})\cap H^{1}(0,T;L^{2})$, we
obtain

$\int_{I}(w^{\prime\prime})^{2}(t)+\int_{0}^{t}\int_{I}b|\partial_{t}w(t)|^{2}=\int_{I}(w^{\prime\prime})^{2}(0).$

Since also $\int_{I}w^{2}(t)\leq
2\int_{I}w^{2}(0)+2T\int_{0}^{T}\int_{I}|\partial_{t}w|^{2}$, we deduce (A.6).

It follows from (A.6) that $(u_{n})$ is a Cauchy sequence, hence converges, in
$C([0,T];$ $H^{2})$ $\cap H^{1}(0,T;L^{2})$. Passing to the limit in the
integral identity (A.2) for $u_{n}$, we obtain a solution $u$ of (A.1) in that
class. Now let $\varphi\in C^{1}([0,T])$, with $\varphi=0$ on $[0,\eta/2]$ and
$\varphi=1$ on $[\eta,T]$. The function $v:=u\varphi$ satisfies
$\partial_{t}v+av^{\prime\prime\prime\prime}=g=af^{\prime}\varphi+u\partial_{t}\varphi$
with $v^{\prime\prime}=0$, $v^{\prime\prime\prime}=\tilde{f}\equiv f\varphi$
on $\partial I$ and $v(0)=0$. Since $u\in C([0,T];H^{2})\subset
L^{\infty}(Q_{T})$, hence $g\in L^{p}(0,T;L^{2})$, and $B\tilde{f}\in
W^{\theta,p}(0,T;\mathbb{R}^{2})$, it follows from [2, Theorem 2.3] and
uniqueness of solutions (cf. Remark A.1(ii)) that $v\in X_{p,T}$, hence $u\in
X_{p,T,loc}$, with

$\|\partial_{t}v\|_{L^{p}(0,T;L^{2})}+\|v^{\prime\prime\prime\prime}\|_{L^{p}(0,T;L^{2})}\leq
C(T)\bigl{(}\|g\|_{L^{p}(0,T;L^{2})}+\|B\tilde{f}\|_{W^{\theta,p}(0,T;\mathbb{R}^{2})}\bigr{)},$

which yields (A.4). ∎

We now turn to the proof of Proposition 1.2(i). Although one might directly
apply a fixed point argument on problem (1.15), it will be convenient to prove
existence by taking advantage of the convergence result in Lemma 9.1 for
problem (1.4), applied to a suitably defined kernel $\rho$ and past data
$z_{p}$. We stress that we did not use Proposition 1.2 in the proof of Lemma
9.1, so there is no circular reasoning.

###### Proof of Proposition 1.2(i).

We define

$\rho(s,a)=e^{-a}b(s)\quad\hbox{ and }\quad z_{p}(s,t)=\phi(s)\ \hbox{ for all
$t\in(-\infty,0]$.}$ (A.7)

Then $\rho,z_{p}$ satisfy all the assumptions of Lemma 9.1 and
$\mu_{1}(s)=\int_{0}^{\infty}ae^{-a}b(s)da=b(s)$. It follows that there exists
a global solution

$z_{0}\in C([0,T];C^{1}(\overline{I}))\cap L^{\infty}(0,\infty;H^{2}(I)),$
(A.8)

with

$z_{0}\in H^{1}(0,T;L^{2}(I))\cap L^{2}(0,T;H^{4}(I)),\quad 0<T<\infty,$ (A.9)

of

$\left\\{\begin{aligned}
b(s)\partial_{t}z_{0}+z_{0}^{\prime\prime\prime\prime}&=\left(F^{\prime}(z_{0}^{\prime})\right)^{\prime},&\hbox{
in $Q_{\infty}$},\\\ z_{0}^{\prime\prime\prime}=0,\
z_{0}^{\prime\prime\prime}&=F^{\prime}(z_{0}^{\prime}),&s\in\partial I,\
t>0,\\\ z_{0}(s,0)&=\phi(s),&s\in I,\end{aligned}\right.$

obtained as limit as $\varepsilon\to 0$ of solutions $z_{\varepsilon}$ of
problems (1.4).

The uniqueness part of Proposition 1.2(i) is a consequence of Lemma 9.2.

We next prove the additional regularity properties of the solution. Let $T>0$,
fix any $p\in(2,\infty)$ and set $f:=F^{\prime}(z_{0}^{\prime})$. By (A.8),
interpolation and Sobolev embedding, we have $z_{0}\in
H^{1-\nu}(0,T;H^{4\nu})$ for all $\nu\in(0,1)$, hence $f\in
L^{\infty}(0,\infty;H^{1})$, as well as $Bf\in H^{k}(0,T;\mathbb{R}^{2})$ for
all $k\in(0,5/8)$, with $\sup_{t\geq
1}\|Bf\|_{H^{k}(t,t+1;\mathbb{R}^{2})}<\infty$. Therefore, by Sobolev
embedding, there exists $\theta_{p}>1/8$ such that $Bf\in
W^{\theta_{p},p}(0,T;\mathbb{R}^{2})$, with moreover

$\sup_{t\geq 1}\ \|Bf\|_{W^{\theta_{p},p}(t,t+1;\mathbb{R}^{2})}<\infty.$
(A.10)

By Lemma A.1, there exists a solution $u\in C([0,T];H^{2})\cap X_{2,T}\cap
X_{p,T,loc}$ of (A.1). By uniqueness of solutions (cf. Remark A.1(ii)), we
deduce that $u=z_{0}$. Moreover, owing to estimate (A.4) in Lemma A.1 and the
fact that $z_{0}\in L^{\infty}(0,\infty;H^{2}(I))$, we deduce (1.18). ∎

We next give the proof of the energy identity for problem (1.15). Although, at
a formal level, the identity would readily follow by multiplying by
$\partial_{t}z_{0}$ and integrating by parts, more care is needed in view of
the available regularity of $z_{0}$.

###### Proof of Proposition 10.1.

(i) Let $t_{2}>t_{1}\geq 0$ and $0<h<t_{2}-t_{1}$. We write

$\displaystyle
2h^{-1}\int_{t_{2}}^{t_{2}+h}E_{0}(t)dt-2h^{-1}\int_{t_{1}}^{t_{1}+h}E_{0}(t)dt=h^{-1}\int_{t_{2}}^{t_{2}+h}\int_{I}((z_{0}^{\prime\prime})^{2}+2F(z_{0}^{\prime}))-h^{-1}\int_{t_{1}}^{t_{1}+h}\int_{I}((z_{0}^{\prime\prime})^{2}+2F(z_{0}^{\prime}))$
$\displaystyle=h^{-1}\int_{t_{1}}^{t_{2}}\int_{I}\bigl{[}(z_{0}^{\prime\prime}(\tau+h))^{2}+2F(z_{0}^{\prime}(\tau+h))\bigr{]}-\bigl{[}(z_{0}^{\prime\prime}(\tau))^{2}+2F(z_{0}^{\prime}(\tau))\bigr{]}$
$\displaystyle=\int_{t_{1}}^{t_{2}}\int_{I}(z_{0}(\tau+h)+z_{0}(\tau))^{\prime\prime}\frac{(z_{0}(\tau+h)-z_{0}(\tau))^{\prime\prime}}{h}+2\frac{F(z_{0}^{\prime}(\tau+h))-F(z_{0}^{\prime}(\tau))}{h}.$

On the other hand, by the weak formulation of (1.15) (cf. Definition A.1), for
all $\psi\in L^{\infty}(t_{1},t_{2};H^{2}(I))$, we have

$\displaystyle\int_{t_{1}}^{t_{2}}\int_{I}$
$\displaystyle\left(z_{0}(\tau+h)+z_{0}(\tau)\right)^{\prime\prime}\psi^{\prime\prime}+\bigl{(}F^{\prime}(z_{0}^{\prime}(\tau+h))+F^{\prime}(z_{0}^{\prime}(\tau))\bigr{)}\psi^{\prime}+b\left(\partial_{t}z_{0}(\tau+h)+\partial_{t}z_{0}(\tau)\right)\psi=0.$

Taking $\psi=h^{-1}(z_{0}(\tau+h)-z_{0}(\tau))$, we get

$\displaystyle
2h^{-1}\int_{t_{2}}^{t_{2}+h}E_{0}(t)dt-2h^{-1}\int_{t_{1}}^{t_{1}+h}E_{0}(t)dt=-\int_{t_{1}}^{t_{2}}\int_{I}b\bigl{(}\partial_{t}z_{0}(\tau+h)+\partial_{t}z_{0}(\tau)\bigr{)}\frac{z_{0}(\tau+h)-z_{0}(\tau)}{h}$
$\displaystyle+\int_{t_{1}}^{t_{2}}\int\Bigl{[}2\frac{F(z_{0}^{\prime}(\tau+h))-F(z_{0}^{\prime}(\tau))}{z_{0}^{\prime}(\tau+h)-z_{0}^{\prime}(\tau)}-F^{\prime}(z_{0}^{\prime}(\tau+h))-F^{\prime}(z_{0}^{\prime}(\tau))\Bigr{]}\frac{z_{0}^{\prime}(\tau+h)-z_{0}^{\prime}(\tau)}{h}\equiv
J_{1}+J_{2}.$

Since $z_{0}\in H^{1}(0,T;L^{2})$, we have $\lim_{h\to
0}J_{1}=-2\int_{t_{1}}^{t_{2}}\int_{I}b(\partial_{t}z_{0})^{2}dsdt$. On the
other hand, letting $M:=\sup_{Q_{T}}|z^{\prime}_{0}|<\infty$ and using that

$\Bigl{|}\frac{F(X)-F(Y)}{X-Y}-F^{\prime}(X)\Bigr{|}\leq
c|X-Y|,\quad|X|,|Y|\leq M,$

for some constant $c=c(M,F)>0$, we get

$|J_{2}|\leq
2c\int_{t_{1}}^{t_{2}}\int_{I}\frac{|z^{\prime}_{0}(\tau+h)-z^{\prime\prime}_{0}(\tau)|^{2}}{h}\leq
2cT\,\biggl{(}\,\sup_{\tau\in(t_{1},T)}h^{-1/2}\|z^{\prime}_{0}(\tau+h)-z^{\prime}_{0}(\tau)\|_{L^{2}}\biggr{)}^{2}.$

Now, by (A.9) and interpolation, we have $z\in W^{3/4,p}_{loc}(0,T;H^{1})$ for
all $p\in(2,\infty)$, hence $z^{\prime}\in C^{\alpha}_{loc}(0,T;L^{2})$ for
all $\alpha\in(0,3/4)$, so that $\lim_{h\to 0}J_{2}=0$. Using that $z_{0}\in
C([0,T];H^{2})$, hence $E_{0}\in C([0,T])$, the assertion follows.

(ii) This follows immediately by integrating the equation in space. ∎

We shall finally prove Proposition 1.2(ii). The $\omega$-limit set
$\omega(z_{0})$ is defined by (8.1) with $\varepsilon=0$ and we set

$\tilde{\mathcal{S}}_{K}:=\\{Z\in\mathcal{S};\ (Z,b)=K\\},\quad
K\in\mathbb{R},$

where, as before, $\mathcal{S}$ is the set of steady states, i.e. solutions of
(1.16). We use the following properties of $\omega(z_{0})$.

######  Proposition A.2.

Let the assumptions of Proposition 1.2 be in force.

(i) The set $\omega(z_{0})$ is a nonempty compact connected subset of
$H^{2}(I)$.

(ii) We have $\omega(z_{0})\subset\tilde{\mathcal{S}}_{K_{0}}$, where
$K_{0}:=(\phi,b)$.

###### Proof.

(i) This follows from the same argument as Lemma 8.1(i).

(ii) By Proposition 10.1(i), we have

$\int_{0}^{\infty}\int_{I}b(s)|\partial_{t}z_{0}|^{2}dsdt\leq
E_{0}(0)<\infty.$ (A.11)

Let $Z\in\omega(z_{0})$. There exists a sequence $t_{n}\to\infty$ such that