# Traveling fronts in a reaction-diffusion equation
with a memory term††thanks: Research partially supported by Deutsche
Forschungsgemeinschaft via subproject A5 within SFB 910 _Control of self-
organizing nonlinear systems_ (Project no. 163436311).

Alexander Mielke∗,† and Sina Reichelt∗
∗ Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin †
Institut für Mathematik, Humboldt-Universität zu Berlin

(26. April 2021)

###### Abstract

Based on a recent work on traveling waves in spatially nonlocal reaction-
diffusion equations, we investigate the existence of traveling fronts in
reaction-diffusion equations with a memory term. We will explain how such
memory terms can arise from reduction of reaction-diffusion systems if the
diffusion constants of the other species can be neglected. In particular, we
show that two-scale homogenization of spatially periodic systems can induce
spatially homogeneous systems with temporal memory.

The existence of fronts is proved using comparison principles as well as a
reformulation trick involving an auxiliary speed that allows us to transform
memory terms into spatially nonlocal terms. Deriving explicit bounds and
monotonicity properties of the wave speed of the arising traveling front, we
are able to establish the existence of true traveling fronts for the original
problem with memory. Our results are supplemented by numerical simulations.

In memory of Pavel Brunovskỳ

##  1 Introduction

The study of traveling fronts in scalar reaction-diffusion equations with a
bistable nonlinearity is a classical topic and there is a rich literature
concerning the existence, uniqueness, and asymptotic stability of such fronts,
see e.g. [FiM77] and [Che97, AcK15] and the references therein. The last two
references are actually devoted to scalar, spatially nonlocal, parabolic
equations of the form

$\dot{u}=Lu+F(u)+G\big{(}u,{\mathbb{J}}S(u)\big{)},\quad t>0,\
x\in{\mathbb{R}},$ (1.1)

where traveling fronts of the form
$u(t,x)={\mathcal{U}}(x{-}ct)\in{\mathbb{R}}$ are investigated. In particular,
[Che97] studies the case $Lu=Du_{xx}$, assumes that
$G(u,\cdot):{\mathbb{R}}\to{\mathbb{R}}$ and $S:{\mathbb{R}}\to{\mathbb{R}}$
are monotonously increasing and that ${\mathbb{J}}$ is a convolution with a
smooth, nonnegative kernel $J\geq 0$, i.e.
$({\mathbb{J}}f)(x)=(J{*}f)(x)=\int_{\mathbb{R}}J(x{-}y)f(y)\;\\!\mathrm{d}y$.
The paper [AcK15] considers the case $G\equiv 0$ and
$Lu={\mathrm{D}}^{\alpha}_{\theta}u$, which is a Riesz-Feller operator with
$\alpha\in{]1,2]}$ and $|\theta|\leq\min\\{\alpha,2{-}\alpha\\}$. In both
cases, the nonlinearity $F$ is of bistable type such that (1.1) has exactly
three homogeneous steady states $u\equiv u_{\alpha}$ with
$\alpha\in\\{-,{\mathrm{m}},+\\}$, where $u_{\pm}$ are stable while the middle
state $u_{\mathrm{m}}$ is unstable.

In [Che97, AcK15] it is shown that, under suitable technical assumptions,
equation (1.1) admits traveling fronts $u(t,x)={\mathcal{U}}(x{-}ct)$ with
${\mathcal{U}}(\xi)\to u_{\pm}$ for $\xi\to\pm\infty$, and that these
solutions are even stable up to translation. The crucial tool for the analysis
in these two papers are comparison principles that are flexible enough to
survive nonlocal operators with positive kernels and monotone operators. The
only non-monotonicity occurs in the local function $u\mapsto F(u)$.

Here we treat a similar type of equation but allow for memory terms
(nonlocality in time), namely the equation

$\dot{u}(t,x)=Du_{xx}(t,x)+F(u(t,x))+\int_{0}^{\infty}\widehat{\Gamma}(\tau)\,u(t{-}\tau,x)\;\\!\mathrm{d}\tau,\quad
t>0,\ x\in{\mathbb{R}},$ (1.2)

with an integrable, nonnegative memory kernel $\widehat{\Gamma}$. A more
general, nonlinear memory kernel of the type
$G\big{(}u,\widehat{\Gamma}{*}S(u(\tau,x))\big{)}$ could be treated as well
(cf. Section 3.5), but we avoid this complexity to keep the main arguments
simple. Of course, memory terms of this type destroy any kind of comparison
principles, so the ideas in [Che97, AcK15] cannot be applied directly.
Traveling fronts in parabolic equations with discrete time delay are treated
in [WuZ01], but not including the bistable case considered here.

However, introducing the auxiliary wave speed $\mathtt{v}\in{\mathbb{R}}$, we
look at a corresponding equation with spatial nonlocality, namely

$\dot{u}(t,x)=Du_{xx}(t,x)+F(u(t,x))+\int_{0}^{\infty}\widehat{\Gamma}(\tau)\,u(t,x{+}\mathtt{v}\tau)\;\\!\mathrm{d}\tau,\quad
t>0,\ x\in{\mathbb{R}}.$ (1.3)

This nonlocal equation has now traveling fronts if the function
$\widehat{F}:u\mapsto F(u)+\gamma u$ with
$\gamma=\int_{0}^{\infty}\widehat{\Gamma}(\tau)\;\\!\mathrm{d}\tau$ is a
bistable nonlinearity and the associated fronts will have a wave speed
$c=C(\mathtt{v})$. The main observation is now that in the special case
$\mathtt{v}=C(\mathtt{v})$ the traveling wave
$u(t,x)={\mathcal{U}}(x{-}\mathtt{v}t)$ of (1.3) also solves the original
memory equation (1.2), see Proposition 2.1.

Hence, existence of traveling fronts for the memory equation (1.2) will be
established by providing conditions that guarantee that the function
$\mathtt{v}\mapsto C(\mathtt{v})$ has a fixed point. Indeed, we again use the
comparison principle for the spatially nonlocal equation (1.3) to derive
continuity of $C$ and certain monotonicities. In Theorem 2.8 we can conclude
that the memory equation (1.2) has always a traveling front if
$\widehat{F}:u\mapsto F(u)+\gamma u$ is a bistable nonlinearity. In Corollary
2.9 we provide bounds on the associated wave speed in terms of the wave speed
of the local equation with nonlinearity $\widehat{F}$.

In Section 3 we discuss several derivations of the memory equation from
classical reaction-diffusion systems. In all cases, the memory appears by a
coupling to ODEs or PDEs acting locally in $x\in{\mathbb{R}}$ but induce
memory through its internal dynamics. The simplest case occurs in the coupled
PDE-ODE system considered extensively in [GuR18]:

$\dot{u}=Du_{xx}+F(u)+\sum_{i=1}^{m_{\mathrm{m}}}a_{i}w_{i},\qquad\dot{w}_{i}=-\lambda_{i}w_{i}+b_{i}u\
\text{ for }i=1,\ldots,m_{\mathrm{m}},$ (1.4)

where $a_{i},\ b_{i}$, and $\lambda_{i}$ are fixed real parameters with
$\lambda_{i}>0$. Clearly, the linear ODEs can be solved by
$w_{i}(t)=\int_{0}^{\infty}\mathrm{e}^{-\lambda_{i}\tau}b_{i}u(t{-}\tau)\;\\!\mathrm{d}\tau$,
and we obtain (1.2) with

$\widehat{\Gamma}(\tau)=\sum_{i=1}^{m_{\mathrm{m}}}\mathrm{e}^{-\lambda_{i}\tau}a_{i}b_{i},\quad\text{
giving
}\gamma=\int_{0}^{\infty}\\!\widehat{\Gamma}(\tau)\;\\!\mathrm{d}\tau=\sum_{i=1}^{m_{\mathrm{m}}}\frac{a_{i}b_{i}}{\lambda_{i}}.$

We emphasize that our theory is applicable also in cases where some of the
products $a_{i}b_{i}$ are negative, as long as $\widehat{\Gamma}$ is
nonnegative. We also discuss possible nonlinear couplings that still allow for
the application of [Che97].

A second motivation for deriving memory equations is the study of traveling
pulses and fronts in situations where the coefficients in the system are
rapidly oscillating, thus modeling a periodic heterostructure of the medium on
a microscopic scale. In the following, the homogenization parameter
$\varepsilon>0$ denotes the ratio between the characteristic length scales of
the microscopic and macroscopic structure. A typical system considered in
Section 3.2 is the heterogeneous FitzHugh-Nagumo system

$\displaystyle\dot{u}$
$\displaystyle=\big{(}D_{\mathrm{u}}(\tfrac{1}{\varepsilon}x)u_{x}\big{)}_{x}+F(\tfrac{1}{\varepsilon}x,u)-\alpha\big{(}\tfrac{1}{\varepsilon}x\big{)}w,$
(1.5) $\displaystyle\dot{w}$
$\displaystyle=\big{(}\varepsilon^{2}D_{\mathrm{w}}(\tfrac{1}{\varepsilon}x)w_{x}\big{)}_{x}-\lambda\big{(}\tfrac{1}{\varepsilon}x\big{)}w+\beta\big{(}\tfrac{1}{\varepsilon}x\big{)}u,$

where all dependence on $\tfrac{1}{\varepsilon}x$ is assumed to be 1-periodic.
Of course, exact traveling waves cannot exist, but one still expects
periodically modulated pulses and fronts traveling through the heterogeneous
medium. Indeed, there exists a vast literature on the study of traveling
fronts for reaction-diffusion equations in continuous periodic media, e.g.
[HuZ95, BeH02], fronts in discrete periodic media [GuH06, CGW08], and fronts
in perforated domains [Hei01]. We refer to [Xin00] for a review including
references to earlier works. Most of the latter results share a common
approach based on comparison principles, and so do we. However, our approach
also allows for systems of reaction-diffusion equations.

In [MSU07] reaction-diffusion systems are studied and exponential averaging is
used to show that traveling wave solutions can be described by a spatially
homogeneous equation and exponentially small remainders. The approach based on
center-manifold reduction in [BoM14] applies to traveling waves in parabolic
equations and, moreover, the authors prove the existence of a generalized
oscillating wave that converges to a limiting wave. We point out that all
previously mentioned articles study limit problems of “one-scale” nature, in
contrast to [GuR18] and the present work where traveling pulses in “two-scale
FitzHugh–Nagumo systems” are investigated, see Section 3.3. According to
Theorem 3.1, the solutions $(u_{\varepsilon},w_{\varepsilon})$ of (1.5)
converge to the solution $(U,W)$ of the two-scale system

$\displaystyle\dot{U}(t,x)$
$\displaystyle=D_{\mathrm{u}}^{\text{eff}}U_{xx}(t,x)+\int_{\mathbb{T}}F(y,U(t,x))\;\\!\mathrm{d}y-\int_{\mathbb{T}}\alpha(y)W(t,x,y)\;\\!\mathrm{d}y,$
(1.6) $\displaystyle\dot{W}(t,x,y)$
$\displaystyle=\big{(}D_{\mathrm{w}}(y)W_{y}(t,x,y)\big{)}_{y}-\lambda(y)W(t,x,y)+\beta(y)U(t,x),$

where the microscopic variable $y$ lies in the circle
${\mathbb{T}}={\mathbb{R}}_{\\!/{\mathbb{Z}}}$. The importance of the two-
scale system (1.6) is that the microscopic variable $y$ decouples from the
macroscopic space variable $x$ such that this system admits exact traveling
waves of the form $(U(x{-}ct),W(x{-}ct,y))$. Moreover, the coupling from $U$
to $W$ and its feedback are local in $x\in{\mathbb{R}}$ and hence can be
captured in spatially local memory terms as in (1.3).

The paper is structured as follows. In Section 2 we develop our existence
result for traveling fronts for the memory equation (1.2), see Theorem 2.8
which relies on the comparison principle in the spirit of [Che97, AcK15] and
on new a priori bounds for the front speed $c$ in Theorem 2.5. In Section 3 we
first present the derivation of memory equations from coupled, but homogeneous
systems, and secondly we show that two-scale homogenization can be used to
derive effective two-scale limits, which again lead to the same memory
equation. Finally, Section 4 compares the abstract theory with numerical
results in the special case that $F$ is the classical bistable cubic
polynomial $-u(u{-}a)(u{-}1)$ and the memory kernel is simply given by
$\widehat{\Gamma}(\tau)=-\beta\,\mathrm{e}^{-\tau}$.

##  2 Existence of traveling fronts

We first describe the setup and the assumptions for our theory in Section 2.1.
In Section 2.2 we introduce the spatially nonlocal equation with the auxiliary
speed $\mathtt{v}$ and show that for traveling fronts these equations are
related. The main technical part are Sections 2.3 and 2.4 where we exploit the
comparison principles developed in [Che97]. The main results are presented in
Section 2.5, where we also discuss potential generalizations.

###  2.1 Setup of the memory equation and assumptions

In all of Section 2 we study the Nagumo-type reaction-diffusion equation (cf.
[NAY62, McK70]) with linear memory term in the form

$\dot{u}(t,x)=Du_{xx}(t,x)+F(u(t,x))+\gamma\int_{0}^{\infty}\Gamma(\tau)u(t{-}\tau,x)\;\\!\mathrm{d}\tau.$
(2.1)

Here, the nonnegative memory kernel $\Gamma$ is normalized, and hence
satisfies

$\Gamma\in{\mathrm{C}}^{0}({[0,\infty[}),\quad\Gamma(\tau)\geq 0\text{ for all
}\tau\geq 0,\quad\int_{0}^{\infty}\Gamma(\tau)\;\\!\mathrm{d}\tau=1.$ (2.2)

To formulate the precise conditions on the bistable nonlinearity $F$ we
introduce the tilted function

$F_{\gamma}(u)=F(u)+\gamma u\quad\text{for }u\in{\mathbb{R}}.$

Hypotheses (H):

  * (H1)

We have $F\in{\mathrm{C}}^{2}({\mathbb{R}})$ and there exists $\gamma_{*}>0$
such that for all $\gamma\in[0,\gamma_{*}]$ the function $F_{\gamma}$ has
exactly three zeros, which we assume to be
$u_{-}^{\gamma}<u_{\mathrm{m}}^{\gamma}<u_{+}^{\gamma}$ and which satisfy
$F_{\gamma}^{\prime}(u_{-}^{\gamma})<0$,
$F^{\prime}(u^{\gamma}_{\mathrm{m}})>0$, and
$F_{\gamma}^{\prime}(u^{\gamma}_{+})<0$.

  * (H2)

The memory kernel $\Gamma$ satisfies (2.2) and
$\widehat{g}_{1}:=\int_{0}^{\infty}\Gamma(\tau)\tau\;\\!\mathrm{d}\tau<\infty$.

The assumptions imply that the three constant functions
$u(t,x)=u^{\gamma}_{\alpha}$ with $\alpha\in\\{-,{\mathrm{m}},+\\}$ are indeed
trivial solutions for (2.1). The solutions $u_{\pm}^{\gamma}$ will be stable,
while the middle solution $u^{\gamma}_{\mathrm{m}}$ is unstable. Our aim is to
show the existence of nontrivial traveling fronts
$u(t,x)={\mathcal{U}}(x{-}ct)$ connecting the two stable levels
$u^{\gamma}_{\pm}$, namely ${\mathcal{U}}(\xi)\to u^{\gamma}_{\pm}$ for
$\xi\to\pm\infty$. Of course, then also a reflected traveling front
$u(t,x)={\mathcal{U}}^{\text{refl}}(x{-}c^{\text{refl}}t)$ exists satisfying
${\mathcal{U}}^{\text{refl}}(\xi)\to u^{\gamma}_{\mp}$ for $\xi\to\pm\infty$.
Indeed, using the reflection symmetry $x\to-x$ of (2.1) gives
${\mathcal{U}}^{\text{refl}}(\xi)={\mathcal{U}}({-}\xi)$ and
$c^{\text{refl}}=-c$.

###  2.2 Traveling waves for the auxiliary equation

The works in [Che97, AcK15] allow for nonlocal terms in the reaction-diffusion
equation, but only for spatial nonlocality and not for temporal one. However,
for traveling waves space and time coincide up to a scaling, so we look at an
auxiliary problem, where we choose a corresponding spatial nonlocality. For
this we have to choose an _auxiliary wave speed_ $\mathtt{v}$ and arrive at
the _auxiliary problem with spatial nonlocality_ :

$\dot{\widetilde{u}}(t,x)=D\widetilde{u}_{xx}(t,x)+F(\widetilde{u}(t,x))+\gamma\int_{0}^{\infty}\\!\Gamma(\tau)\,\widetilde{u}(t,x{+}\mathtt{v}\tau)\;\\!\mathrm{d}\tau.$
(2.3)

The basis of our theory is the following simple proposition that connects the
existence of traveling waves for the original problem with that of the
auxiliary one.

###### Proposition 2.1

If (2.3) has a bounded traveling-wave solution
$\widetilde{u}(t,x)={\mathcal{U}}(x{-}\mathtt{c}t)$ and the wave speed
$\mathtt{c}$ matches the auxiliary speed $\mathtt{v}$ occurring as parameter
in (2.3), then $u(t,x)={\mathcal{U}}(x{-}\mathtt{v}t)$ is also a solution of
(2.1).

Proof. Obviously, the partial derivatives and the local function $F(u)$ in
(2.3) and (2.1) coincide, so it remains to match the integral terms. The
straightforward calculation

$\displaystyle\int_{0}^{\infty}\\!\Gamma(\tau)\,\widetilde{u}(t,x{+}\mathtt{v}\tau)\;\\!\mathrm{d}\tau=\int_{0}^{\infty}\\!\Gamma(\tau)\,{\mathcal{U}}(x{+}\mathtt{v}\tau-\mathtt{c}t)\;\\!\mathrm{d}\tau$
$\displaystyle\overset{\mathtt{v}=\mathtt{c}}{=}\int_{0}^{\infty}\\!\Gamma(\tau)\,{\mathcal{U}}(x+\mathtt{v}(t{-}\tau))\;\\!\mathrm{d}\tau=\int_{0}^{\infty}\\!\Gamma(\tau)\,u(t{-}\tau,x)\;\\!\mathrm{d}\tau,$

relying strongly on $\mathtt{c}=\mathtt{v}$, turns the spatial nonlinearity
into a temporal memory. This gives the desired result.

The above result does not specify the form of the traveling wave, hence it is
applicable to traveling pulse, traveling fronts, or to (quasi-)periodic wave
trains.

###  2.3 Comparison principles for the auxiliary equation

We are now in a position to apply the general theory for nonlocal parabolic
equations as developed in [Che97, AcK15] to our auxiliary problem (2.3) for
$\gamma\in[0,\gamma_{*}]$. For this, we define the nonlinear operator
$u\mapsto{\mathcal{A}}_{\gamma,\mathtt{v}}[u]$ via

$\displaystyle{\mathcal{A}}_{\gamma,\mathtt{v}}[u](x)$
$\displaystyle:=Du_{xx}(x)+F(u(x))+\gamma\int_{0}^{\infty}\\!\Gamma(\tau)\,u(x{+}\mathtt{v}\tau)\;\\!\mathrm{d}\tau$
(2.4)

in the notation of [Che97]. We first observe the relation
${\mathcal{A}}_{\gamma,\mathtt{v}}[\lambda\bm{1}]=F_{\gamma}(\lambda)\bm{1}$
for all $\lambda\in{\mathbb{R}}$, where $\bm{1}$ denotes the constant function
$u(x)=1$. Moreover, the Fréchet derivative reads

$\big{(}{\mathcal{A}}^{\prime}_{\gamma,\mathtt{v}}[u](\varphi)\big{)}(x)=D\varphi_{xx}(x)+F^{\prime}(u(x))\varphi(x)+\gamma\int_{0}^{\infty}\\!\Gamma(\tau)\,\varphi(x{+}\mathtt{v}\tau)\;\\!\mathrm{d}\tau.$

Hence, we obtain
$\big{(}{\mathcal{A}}^{\prime}_{\gamma,\mathtt{v}}[u{+}v](\bm{1})\big{)}(x)-\big{(}{\mathcal{A}}^{\prime}_{\gamma,\mathtt{v}}[u](\bm{1})\big{)}(x)=F^{\prime}(u(x){+}v(x))-F^{\prime}(u(x))$.
With this and the assumption $F\in{\mathrm{C}}^{2}({\mathbb{R}})$ from (H1),
the two assumptions (A1) and (A3) in [Che97, Sect. 2] are satisfied.

The crucial and nontrivial assumption (A2) in [Che97] concerns the strong
comparison principle and is the content of the following proposition. To
formulate it we introduce the notion of super- and subsolutions for (2.3): We
call $u^{*}$ a supersolution and $u_{*}$ a subsolution if the relations
$\dot{u}^{*}-{\mathcal{A}}_{\gamma,\mathtt{v}}[u^{*}]\geq 0$ and
$\dot{u}_{*}-{\mathcal{A}}_{\gamma,\mathtt{v}}[u_{*}]\leq 0$ hold,
respectively.

######  Proposition 2.2 (Strong comparison principle)

Let the Hypotheses (H1)–(H2) and $\gamma\in[0,\gamma_{*}]$ hold. If $u^{*}$ is
a supersolution and $u_{*}$ is a subsolution of (2.3) such that
$u^{*}(0,\cdot)\geq u_{*}(0,\cdot)$ and
$u^{*}(0,\cdot){-}u_{*}(0,\cdot)\not\equiv 0$, then we have
$u^{*}(t,\cdot)\gneqq u_{*}(t,\cdot)$ for all $t>0$.

Proof. We follow some ideas in the proof of [Che97, Thm. 5.1] for establishing
condition (C2). By assumption the difference $w(t,x):=u^{*}(t,x)-u_{*}(t,x)$
satisfies $w(0,\cdot)\geq 0$ and

$\displaystyle\dot{w}\geq
Dw_{xx}+F(u^{*})-F(u_{*})+{\mathbb{J}}w\quad\text{with}\quad({\mathbb{J}}w)(t,x):=\gamma\int_{0}^{\infty}\\!\Gamma(\tau)w(t,x{+}\mathtt{v}\tau)\;\\!\mathrm{d}\tau.$

_Step 1._ We show by contradiction that the inequality $\dot{w}\geq
Dw_{xx}+{\mathbb{J}}w$ implies $w\geq 0$. We set $K:=4\gamma_{*}{+}4D$ with
$\gamma_{*}$ from (H1). By assuming $w\ngeq 0$, there exist $\delta>0$ and
$T>0$ such that $w(t,x)>-\delta\mathrm{e}^{Kt}$ for $t\in\left[0,T\right[$ and
$\inf_{x\in{\mathbb{R}}}w(T,x)=-\delta\mathrm{e}^{KT}$. Without loss of
generality, we may assume $w(T,0)<-\frac{3}{4}\delta\mathrm{e}^{KT}$.

Next we define the comparison function
$\varphi_{\sigma}(t,x):=-(\frac{1}{2}+\sigma z(x))\,\delta\mathrm{e}^{Kt}$
with $z(x)=\frac{1+3x^{2}}{1+x^{2}}$, where the parameter $\sigma>0$ will be
fixed later. On the one hand, using $z\geq 1$ we have, for
$\sigma\geq\frac{1}{2}$, the estimate

$\displaystyle\textstyle\varphi_{\sigma}(t,x)\leq-\big{(}\frac{1}{2}+\frac{1}{2}\,1\big{)}\,\delta\,\mathrm{e}^{Kt}=-\delta\mathrm{e}^{Kt}\leq
w(t,x)$

for all $t\in[0,T]$ and $x\in{\mathbb{R}}$. On the other hand, using $z(0)=1$
for $\sigma=1/4$ we find

$\displaystyle\textstyle\varphi_{1/4}(T,0)=-\big{(}\frac{1}{2}+\frac{1}{4}1\big{)}\,\delta\,\mathrm{e}^{KT}=-\frac{3}{4}\delta\mathrm{e}^{KT}>w(T,0).$

We define $\sigma_{*}$ as follows:

$\sigma_{*}:=\inf\Sigma\quad\text{with
}\Sigma:=\big{\\{}\,\sigma>0\>\big{|}\>\forall\,(t,x)\in[0,T]{\times}{\mathbb{R}}:\
\varphi_{\sigma}(t,x)\leq w(t,x)\,\big{\\}}$

and observe that the above estimates imply
$\sigma_{*}\in[\frac{1}{4},\frac{1}{2}]$. By continuity, we see that $\Sigma$
is closed, hence $\sigma_{*}\in\Sigma$, which implies
$\sigma_{*}>\frac{1}{4}$. Using $z(x)>2$ for $|x|>2$ for all
$\sigma\geq\frac{1}{4}$, we have
$\varphi_{\sigma}(t,x)<-(\frac{1}{2}+\frac{1}{4}2)\delta\mathrm{e}^{Kt}\leq
w(t,x)$. Hence, for
$\sigma\in{[\frac{1}{4},\sigma_{*}[}\subset{\mathbb{R}}\setminus\Sigma$ there
exists $(t_{\sigma},x_{\sigma})\in[0,T]{\times}[-1,1]$ such that
$\varphi_{\sigma}(t_{\sigma},x_{\sigma})\geq w(t_{\sigma},x_{\sigma})$.

By the continuity of $w$ and $(\sigma,t,x)\mapsto\varphi_{\sigma}(t,x)$ and by
compactness of $[0,T]{\times}[-1,1]$, we hence find
$(t_{*},x_{*})\in[0,T]{\times}[-1,1]$ such that
$\varphi_{\sigma_{*}}(t_{*},x_{*})\geq w(t_{*},x_{*})$. Obviously, we have
$t_{*}>0$, but $t=T$ may be possible. However $\sigma_{*}\in\Sigma$ means
$\varphi_{\sigma_{*}}(t,x)\leq w(t,x)$ for all
$(t,x)\in[0,T]{\times}{\mathbb{R}}$. Thus, we conclude
$w(t_{*},x_{*})=\varphi_{\sigma_{*}}(t_{*},x_{*})$,
$w_{x}(t_{*},x_{*})=\partial_{x}\varphi_{\sigma_{*}}(t_{*},x_{*})$,

$w_{xx}(t_{*},x_{*})\geq\partial^{2}_{x}\varphi_{\sigma_{*}}(t_{*},x_{*}),\quad\text{and
}\left\\{\begin{array}[]{cl}\dot{w}(t_{*},x_{*})=\dot{\varphi}_{\sigma_{*}}(t_{*},x_{*})&\text{for
}t_{*}\in{]0,T[},\\\
\dot{w}(t_{*},x_{*})\leq\dot{\varphi}_{\sigma_{*}}(t_{*},x_{*})&\text{for
}t_{*}=T.\end{array}\right.$

Using (H2) (nonnegativity of $\Gamma$) and $\gamma\in[0,\gamma_{*}]$ together
with the inequalities (i) $\frac{3}{4}\leq\frac{1}{2}+\sigma_{*}z(x_{*})$ and
(ii) $\frac{1}{2}+\sigma_{*}z(x)\leq 2$ for all $x\in{\mathbb{R}}$, the
following chain of inequalities for the particular point $(t_{*},x_{*})$
holds:

$\displaystyle-\tfrac{3}{4}\delta K\mathrm{e}^{Kt_{*}}$
$\displaystyle\overset{\text{(i)}}{\geq}-\big{(}\tfrac{1}{2}+\sigma_{*}z(x_{*})\big{)}\delta
K\mathrm{e}^{Kt_{*}}=\dot{\varphi}_{\sigma_{*}}(t_{*},x_{*})$
$\displaystyle\geq\dot{w}(t_{*},x_{*})\geq
Dw_{xx}(t_{*},x_{*})+{\mathbb{J}}w(t_{*},x_{*})$
$\displaystyle\\!\\!\\!\overset{w\geq\varphi_{\sigma_{*}}}{\geq}D\partial_{x}^{2}\varphi_{\sigma_{*}}(t_{*},x_{*})+\textstyle\gamma\int_{0}^{\infty}\\!\Gamma(\tau)\,\varphi_{\sigma_{*}}(t_{*},x_{*}{+}\mathtt{v}\tau))\;\\!\mathrm{d}\tau$
$\displaystyle\overset{\text{(ii)}}{\geq}-D\sigma_{*}z^{\prime\prime}(x_{*})\delta\mathrm{e}^{Kt_{0}}-2\gamma\delta\mathrm{e}^{Kt_{*}}\geq-\big{(}2D+2\gamma_{*})\delta\mathrm{e}^{Kt_{*}}=-\tfrac{1}{2}\delta
K\mathrm{e}^{Kt_{*}},$

where we used $z^{\prime\prime}(x)\leq 4$ and $\sigma_{*}\leq\frac{1}{2}$ and
the definition of $K=4\gamma_{*}{+}4D$. Thus, we have reached a contradiction
and the assertion $w\geq 0$ is proven.

_Step 2._ Without loss of generality, we can assume that $u_{*}(t,x)$ and
$u^{*}(t,x)$ only attain values in the bounded interval
$[-\mathtt{C}_{\mathrm{max}},\mathtt{C}_{\mathrm{max}}]$ for some constant
$\mathtt{C}_{\mathrm{max}}>0$ depending on the roots of $F$ according to (H2).
Therefore, we have $F(u^{*})-F(u_{*})\geq K_{0}w$ with
$K_{0}:=-\min_{|u|\leq\mathtt{C}_{\mathrm{max}}}F^{\prime}(u)>0$ and

$\displaystyle\dot{w}\geq Dw_{xx}+F(v^{*})-F(v_{*})+{\mathbb{J}}w\geq
Dw_{xx}-K_{0}w+{\mathbb{J}}w\geq Dw_{xx}-K_{0}w,$

where the last estimate follows because of $w\geq 0$ from Step 1 and (H2),
giving ${\mathbb{J}}w\geq 0$.

Finally, we use that $w(0,\cdot)$ is nonnegative and not identical to $0$.
Hence, the solution $\psi$ of the linear equation
$\dot{\psi}=D\psi_{xx}-K_{0}\psi$ is strictly positive, as it is given by
$\psi(t,\cdot)=\mathrm{e}^{-K_{0}t}H_{D}(t){*}w(0,\cdot)$, where $H_{D}(t)$ is
the strictly positive heat kernel. We now set
$W=\mathrm{e}^{K_{0}t}(w{-}\psi)$ and obtain $W(0,\cdot)\equiv 0$ and
$\dot{W}\geq DW_{xx}$. As in Step 1 (with ${\mathbb{J}}=0$) we obtain $W\geq
0$ and conclude

$u^{*}(t,x)-u_{*}(t,x)=w(t,x)\geq\psi(t,x)>0\quad\text{for all }t>0\text{ and
}x\in{\mathbb{R}}.$

This is the desired strong comparison principle.

As an important technical tool we obtain the following simple result
concerning the speed of traveling fronts.

######  Proposition 2.3 (Comparison of speeds)

Assume that the auxiliary equation (2.3) has a traveling front
$u(t,x)={\mathcal{U}}(x{-}ct)$. If there is a traveling-front subsolution
$w_{*}(t,x)=W_{*}(x{-}c_{*}t)$ satisfying

$\displaystyle\lim_{\xi\to-\infty}W_{*}(\xi)<\lim_{\xi\to-\infty}{\mathcal{U}}(\xi)<\lim_{\xi\to+\infty}W_{*}(\xi)<\lim_{\xi\to+\infty}{\mathcal{U}}(\xi),$
(2.5a) then we have $c\leq c_{*}$. If there is a traveling-front supersolution
$w^{*}(t,x)=W^{*}(x{-}c^{*}t)$ satisfying
$\displaystyle\lim_{\xi\to-\infty}{\mathcal{U}}(\xi)<\lim_{\xi\to-\infty}W^{*}(\xi)<\lim_{\xi\to+\infty}{\mathcal{U}}(\xi)<\lim_{\xi\to+\infty}W^{*}(\xi),$
(2.5b)

then we have $c\geq c^{*}$.

Proof. It suffices to show the result for the subsolution, since the proof for
the supersolutions is analogous.

As the limits of $W_{*}$ and ${\mathcal{U}}$ at $\xi=\pm\infty$ are strictly
ordered, we can shift $W_{*}$ to the right to make it smaller for $t=0$. More
precisely, we set $u^{*}(t,x)={\mathcal{U}}(x{-}ct)$ and
$u_{*}(t,x)=W_{*}(x{-}\xi{-}c_{*}t)$ and find $\xi$ big enough such that
$u^{*}(0,x)={\mathcal{U}}(x)>W_{*}(x{-}\xi)=u_{*}(0,x)$. The comparison
principle in Proposition 2.2 now implies

$u^{*}(t,x)={\mathcal{U}}(x{-}ct)>W_{*}(x{-}\xi{-}c_{*}t)=u_{*}(t,x)\quad\text{
for all }t\geq 0\text{ and }x\in{\mathbb{R}}.$ (2.6)

Now we assume $c>c_{*}$, insert $x=\frac{1}{2}(c{+}c_{*})t$ into (2.6), and
take the limit $t\to\infty$. This leads to
$\lim_{\xi\to-\infty}{\mathcal{U}}(\xi)\geq\lim_{\xi\to+\infty}W_{*}(\xi)$,
which contradicts the middle assumption in (2.5a). Hence the assumption
$c>c_{*}$ is false and $c\leq c_{*}$ is established.

###  2.4 Existence of traveling fronts for the auxiliary equation

We are now in the position to formulate our result concerning the existence of
traveling fronts in the auxiliary equation (2.3). The proof will be a direct
application of the corresponding result [Che97, Thm. 5.1] for spatially
nonlocal equations. We obtain a two-parameter family
${\mathcal{U}}_{\gamma,\mathtt{v}}$ of traveling fronts depending on the
auxiliary speed $\mathtt{v}\in{\mathbb{R}}$ and the strength
$\gamma\in[0,\gamma_{*}]$ of the nonlocal term.

###### Proposition 2.4

Let the Hypotheses (H1) and (H2) hold and let $\gamma\in[0,\gamma_{*}]$. Then,
for all $\mathtt{v}\in{\mathbb{R}}$ there exists a unique (up to translation)
traveling-front solution

$u(t,x)={\mathcal{U}}_{\gamma,\mathtt{v}}\big{(}x{-}\mathtt{c}t\big{)}\text{
solving \eqref{eq:Aux1} with wave speed
}\mathtt{c}=\mathtt{C}(\gamma,\mathtt{v}),$

which is characterized by

$\displaystyle{-}\mathtt{c}\,{\mathcal{U}}^{\prime}_{\gamma,\mathtt{v}}(\xi)=D\,{\mathcal{U}}^{\prime\prime}_{\gamma,\mathtt{v}}(\xi)+F({\mathcal{U}}_{\gamma,\mathtt{v}}(\xi))+\gamma\int_{0}^{\infty}\Gamma(\tau)\,{\mathcal{U}}_{\gamma,\mathtt{v}}(\xi{+}\mathtt{v}\tau)\;\\!\mathrm{d}\tau,$
(2.7a) $\displaystyle{\mathcal{U}}_{\gamma,\mathtt{v}}(\xi)\to u_{-}^{\gamma}\
\text{ for }\xi\to-\infty\quad\text{ and
}\quad{\mathcal{U}}_{\gamma,\mathtt{v}}(\xi)\to u^{\gamma}_{+}\ \text{ for
}\xi\to+\infty.$ (2.7b)

These traveling fronts satisfy the properties

$\displaystyle{\mathcal{U}}_{\gamma,\mathtt{v}}\in{\mathrm{C}}^{2}({\mathbb{R}}),\quad{\mathcal{U}}_{\gamma,\mathtt{v}}^{\prime}(\xi)>0\text{
on }{\mathbb{R}},\quad{\mathcal{U}}^{\prime}_{\gamma,\mathtt{v}}(\xi)\to
0\text{ as }|\xi|\to\infty.$ (2.8)

Moreover, they are globally asymptotically stable in the following sense:
There exists $\kappa>0$ such that for all solutions $u$ of (2.3) satisfying

$\displaystyle\forall\,x\in{\mathbb{R}}:u(0,x)\in[u_{-}^{\gamma}{-}\kappa,u^{\gamma}_{+}{+}\kappa],\qquad\liminf_{x\to+\infty}u(0,x)>u^{\gamma}_{\mathrm{m}},\quad\limsup_{x\to-\infty}u(0,x)<u^{\gamma}_{\mathrm{m}},$
(2.9)

there exist constants $\xi$ and $K$ (depending on $u(0,\cdot)$) such that

$\displaystyle\|u(t,\cdot)-{\mathcal{U}}_{\gamma,\mathtt{v}}(\,\cdot\,{-}\mathtt{c}t{+}\xi)\|_{{\mathrm{L}}^{\infty}({\mathbb{R}})}\leq
Ke^{-\kappa t}\quad\text{for }t\geq 0.$ (2.10)

Proof. For the evolution equation $\dot{u}=Du_{xx}+G(u,J{*}S(u))$ in [Che97,
Eqn. (5.1)], we distinguish the cases $\mathtt{v}=0$ and $\mathtt{v}\neq 0$.
In the former case, there is no nonlocal term, and we set $G(u,p)=F(u)+\gamma
u$ and $J\equiv 0$. In the latter case we identify the quantities

$\displaystyle G(u,p)=F(u)+\gamma p,\quad S(u)=u,\quad
J=J_{v}:s\mapsto\frac{1}{|\mathtt{v}|}\,\Gamma\big{(}{-}\frac{s}{\mathtt{v}}\big{)},$

where $\Gamma(\tau)=0$ for $\tau<0$ is assumed.

It remains to verify the Assumptions (D1)–(D4) of [Che97, Thm. 5.1]:

  * (D1)

Clearly, the operator ${\mathcal{A}}_{\gamma,\mathtt{v}}$ defined in (2.4) is
translation invariant,
${\mathcal{A}}_{\gamma,\mathtt{v}}[u(\,\cdot\,{+}h)](x)={\mathcal{A}}_{\gamma,\mathtt{v}}[u(\cdot)](\,\cdot\,{+}h)$
for all $h\in{\mathbb{R}}$. The desired properties of the function
$F_{\gamma}:u\mapsto F(u)+\gamma u$ characterized by
${\mathcal{A}}_{\gamma,\mathtt{v}}(\lambda\bm{1})=F_{\gamma}(\lambda)\bm{1}$
follow directly from (H1).

  * (D2)

The condition (H2) is stronger than our condition because it additionally asks
$\Gamma\in{\mathrm{C}}^{1}({\mathbb{R}})$ and
$\int_{\mathbb{R}}|\Gamma^{\prime}(\tau)|\;\\!\mathrm{d}\tau<\infty$. However,
a close inspection of the proof of [Che97, Thm. 5.1] reveals that these
additional conditions are not needed in the case $D>0$. Indeed, (D2) is used
to derive (C2) and (C4) there. However, (C2) is the strong comparison
principle, which holds according to Proposition 2.2, while (C4) follows from
classical parabolic regularity theory because of $D>0$.

  * (D3)

The function $G(u,p)=F(u)+\gamma p$ and $S(u)=u$ are smooth with
$\partial_{p}G(u,p)=\gamma\geq 0$ and $S^{\prime}(u)=1>0$.

  * (D4)

This holds because of $D>0$.

Therefore, [Che97, Thm. 5.1] yields the desired existence of traveling fronts.

The comparison principle is not only useful for establishing existence and
uniqueness of traveling fronts. It will also be essential to derive
qualitative properties of the function
$(\gamma,\mathtt{v})\mapsto\mathtt{C}(\gamma,\mathtt{v})$. We first derive
upper and lower bounds for $\mathtt{C}$ and then its continuity, which will be
crucial to construct traveling fronts for the memory equation (2.1). The ideas
of the proof follow [Che97, Lem. 3.2 & Thm. 3.5], but they are much more
explicit, thus providing realistic bounds by assuming reasonable bounds for
$F_{\gamma}$. In Figure 2.1 we display the way in which $F_{\gamma}$ needs to
be estimated.

######  Theorem 2.5 (Bounds on the front speed)

Let the Hypotheses (H1) and (H2) hold and fix $\gamma\in[0,\gamma_{*}]$ and
$\mathtt{v}\in{\mathbb{R}}$. Assume further that $F_{\gamma}$ satisfies the
estimates

$\displaystyle F_{\gamma}(u)\geq-\Phi_{*}\text{ for
}u\in[u^{\gamma}_{-},u^{\gamma}_{\mathrm{m}}]\quad\text{and}\quad
F_{\gamma}(u)\leq\Phi^{*}\text{ for
}u\in[u^{\gamma}_{\mathrm{m}},u^{\gamma}_{+}],$ (2.11)
$\displaystyle\exists\,a_{*},b_{*}\in[u^{\gamma}_{\mathrm{m}},u^{\gamma}_{+}]\
\exists\,\alpha_{*}>0:\quad a_{*}<b_{*}\ \text{ and }\
F_{\gamma}(u)\geq\alpha_{*}(u{-}a_{*})\text{ for }u\in[a_{*},b_{*}],$
$\displaystyle\exists\,a^{*},b^{*}\in[u^{\gamma}_{-},u^{\gamma}_{\mathrm{m}}]\
\exists\,\alpha^{*}>0:\quad b^{*}<a^{*}\ \text{ and }\
F_{\gamma}(u)\leq\alpha^{*}(u{-}a^{*})\text{ for }u\in[b^{*},a^{*}].$

Then, the speed $\mathtt{c}=\mathtt{C}(\gamma,\mathtt{v})$ of the traveling
front ${\mathcal{U}}_{\gamma,\mathtt{v}}$ satisfies

$\displaystyle\mathtt{v}\geq 0:\quad$
$\displaystyle-\max\Big{\\{}\Big{(}\frac{\Phi^{*}D}{b^{*}{-}a^{*}}\Big{)}^{1/2},\,\frac{\Phi^{*}\gamma\widehat{g}_{1}\mathtt{v}}{\alpha^{*}(b^{*}{-}a^{*})}\Big{\\}}-\gamma\widehat{g}_{1}\mathtt{v}\leq\mathtt{C}(\gamma,\mathtt{v})\leq\Big{(}\frac{\Phi_{*}D}{u^{\gamma}_{+}{-}u^{\gamma}_{\mathrm{m}}}\Big{)}^{1/2},$
(2.12a) $\displaystyle\mathtt{v}\leq 0:\quad$
$\displaystyle-\Big{(}\frac{\Phi^{*}D}{u^{\gamma}_{\mathrm{m}}{-}u^{\gamma}_{-}}\Big{)}^{1/2}\leq\mathtt{C}(\gamma,\mathtt{v})\leq\max\Big{\\{}\Big{(}\frac{\Phi_{*}D}{b_{*}{-}a_{*}}\Big{)}^{1/2},\,\frac{\Phi_{*}\gamma\widehat{g}_{1}|\mathtt{v}|}{\alpha_{*}(b_{*}{-}a_{*})}\Big{\\}}+\gamma\widehat{g}_{1}|\mathtt{v}|$
(2.12b)

with $\widehat{g}_{1}=\int_{0}^{\infty}\tau\Gamma(\tau)\;\\!\mathrm{d}\tau>0$
from (H2).

$u$$F_{\gamma}(u)$$u^{\gamma}_{-}$$u^{\gamma}_{\mathrm{m}}\;\,$$u^{\gamma}_{+}$$a_{*}$$b_{*}$$-\Phi_{*}$$a^{*}$$b^{*}$$\Phi^{*}$

Figure 2.1: We show how $F_{\gamma}$ needs to be estimated from above and
below to obtain the upper and lower bounds in Theorem 2.5 for the speed of
traveling fronts ${\mathcal{U}}_{\gamma,\mathtt{v}}$ for the nonlocal equation
(2.3).

Proof. We construct suitable traveling fronts as subsolutions $W_{*}$ and
supersolutions $W^{*}$ with speeds $c_{*}$ and $c^{*}$, respectively. The
comparison principle for the travel speeds in Proposition 2.3 gives the
desired result $c^{*}\leq\mathtt{c}\leq c_{*}$.

Choosing a small $\delta>0$ and a positive slope $m>0$, we set
$\xi_{*}:=(a_{*}{-}u^{\gamma}_{-})/m>0$ and $\lambda_{*}:=m/(b_{*}{-}a_{*})$
and define the function $W_{*}$ in the specific form

$W_{*}(\xi)=\left\\{\begin{array}[]{cl}u^{\gamma}_{-}-\delta m&\text{for
}\xi\leq-\delta,\\\ u^{\gamma}_{-}+m\xi&\text{for }\xi\in[-\delta,\xi_{*}],\\\
b_{*}-(b_{*}{-}a_{*})\mathrm{e}^{-\lambda_{*}(\xi-a_{*}/m)}&\text{for
}\xi>\xi_{*}.\end{array}\right.$

Actually $W_{*}$ could be smoothed out in
$I_{\delta}:=[-\frac{2}{3}\delta,-\frac{1}{3}\delta]$ such that it lies in
${\mathrm{C}}^{2}({\mathbb{R}})$ with $W^{\prime\prime}_{*}(\xi)\geq 0$ for
$\xi\in I_{\delta}$. This smoothening would not destroy the property of a
subsolution.

A main observation is the monotonicity $W^{\prime}_{*}(\xi)\geq 0$ and that
$W_{*}$ and ${\mathcal{U}}_{\gamma,\mathtt{v}}$ satisfy the ordering
conditions (2.5a) for the limits at $\xi=\pm\infty$. It remains to be shown
that there is a speed $c_{*}$ such that $u_{*}(t,x)=W_{*}(x{-}c_{*}t)$ is a
subsolution. To obtain this, we proceed as follows (using $\xi=x{-}c_{*}t$):

$\displaystyle\dot{u}_{*}(\xi)-{\mathcal{A}}_{\gamma,\mathtt{v}}[u_{*}](\xi)$
$\displaystyle=-c_{*}W^{\prime}_{*}(\xi)-DW^{\prime\prime}_{*}(\xi)-F_{\gamma}\big{(}W_{*}(\xi)\big{)}-{\mathcal{L}}_{\gamma,\mathtt{v}}[W_{*}](\xi)$
$\displaystyle\text{ with }{\mathcal{L}}_{\gamma,\mathtt{v}}[w](\xi)$
$\displaystyle:=\gamma\int_{0}^{\infty}\Gamma(\tau)\big{(}w(\xi{+}\mathtt{v}\tau)-w(\xi)\big{)}\;\\!\mathrm{d}\tau.$

For $\mathtt{v}\geq 0$ we have ${\mathcal{L}}_{\gamma,\mathtt{v}}[W_{*}]\geq
0$ because of the monotonicity of $W_{*}$ and the nonnegativity of $\Gamma$.
Hence, we can drop $-{\mathcal{L}}_{\gamma,\mathtt{v}}[W_{*}]$ when showing
that $u_{*}$ is a subsolution.

For $\mathtt{v}<0$ we estimate
$0\leq-{\mathcal{L}}_{\gamma,\mathtt{v}}[W_{*}](\xi)$ from above by
considering two regimes for $\xi$ separately: For $\xi\leq-\delta$ we have
${\mathcal{L}}_{\gamma,\mathtt{v}}[W_{*}](\xi)=0$ since $W_{*}$ is constant on
${]{-}\infty,-\delta]}$. For $\xi>-\delta$ we use
$W^{\prime}_{*}(\xi)\in[0,m]$ and obtain

$0\leq-{\mathcal{L}}_{\gamma,\mathtt{v}}[W_{*}](\xi)\leq-\gamma\int_{0}^{\infty}\Gamma(\tau)\,\mathtt{v}\tau
m\,\;\\!\mathrm{d}\tau=g_{\mathtt{v}}m\quad\text{with
}g_{\mathtt{v}}:=\gamma\widehat{g}_{1}\max\\{-\mathtt{v},0\\}.$

We derive the conditions that $m$ and $c_{*}$ have to satisfy in order to
guarantee that $W_{*}$ is indeed a subsolutions. For this we estimate
$\dot{u}_{*}(\xi)-{\mathcal{A}}_{\gamma,\mathtt{v}}[u_{*}](\xi)$ from above on
the separate domains and collect the corresponding conditions:

$\displaystyle\xi<-\delta:\quad$ $\displaystyle-c_{*}\cdot 0-D\cdot
0-F_{\gamma}\big{(}u^{\gamma}_{-}{-}\delta m\big{)}-0\ \leq 0,$
$\displaystyle\xi\in{]{-}\delta,\xi_{*}[}:\quad$ $\displaystyle-c_{*}\cdot
m-D\cdot 0+\qquad\Phi_{*}\quad+m\,g_{\mathtt{v}}\leq 0,$
$\displaystyle\xi>\xi_{*}:\quad$
$\displaystyle(b_{*}{-}a_{*})\big{(}({-}c_{*}\lambda_{*}+\lambda_{*}^{2}D+\alpha_{*})\mathrm{e}^{-\lambda_{*}(\xi-\xi_{*})}-\alpha_{*}\big{)}+m\,g_{\mathtt{v}}\leq
0.$

The first condition is always true because $F_{\gamma}(u)>0$ for
$u<u^{\gamma}_{-}$. For the second term we simply choose
$c_{*}\geq\Phi_{*}/m+g_{\mathtt{v}}$, where $m$ is still to be determined.
Hence, it remains to analyze the third condition. For $\mathtt{v}\geq 0$, the
last term vanishes and the terms multiplying $\alpha_{*}$ are nonpositive.
Hence, it suffices to take $c_{*}\geq\lambda_{*}D=mD/(b_{*}{-}a_{*})$.
Together with $c_{*}\geq\Phi_{*}/m$ we can choose $m$ optimally and find that
$c_{*}=\big{(}\Phi_{*}D/(b_{*}{-}a_{*})\big{)}^{1/2}$ guarantees that $W_{*}$
is a subsolution. Surprisingly, the result does not depend on the slope
$\alpha_{*}>0$, hence we may optimize $a_{*}$ and $b_{*}$ by pushing them to
their limits $u^{\gamma}_{\mathrm{m}}$ and $u^{\gamma}_{+}$, respectively.
Thus, we obtain the upper estimate for $c$ in (2.12a).

For $\mathtt{v}<0$ the term involving $\mathtt{v}_{\\!\bm{-}}>0$ can only be
compensated by $\alpha_{*}(b_{*}{-}a_{*})$. Setting
$m=\theta\alpha_{*}(b_{*}{-}a_{*})/g_{\mathtt{v}}$ we can rewrite the third
condition in the form

$\textstyle\alpha_{*}(b_{*}{-}a_{*})\Big{(}\big{(}-c_{*}\frac{\theta}{g_{\mathtt{v}}}+\frac{\alpha_{*}\theta^{2}D}{g_{\mathtt{v}}^{2}}+1\big{)}\,\mathrm{e}^{-\lambda_{*}(\xi-\xi_{*})}-1+\theta\Big{)}\leq
0.$

Together with the second condition it remains to satisfy

$c_{*}\geq\max\big{\\{}\tfrac{1}{\theta}\,\tfrac{\Phi_{*}g_{\mathtt{v}}}{\alpha_{*}(b_{*}{-}a_{*})}+g_{\mathtt{v}}\,,\,\theta\,\tfrac{\alpha_{*}D}{g_{\mathtt{v}}}+g_{\mathtt{v}}\big{\\}}.$

For $g_{\mathtt{v}}^{2}\Phi_{*}<\alpha_{*}^{2}(b_{*}{-}a_{*})D$ we find an
optimal $\theta\in{]0,1[}$, whereas otherwise the first term in the maximum
dominates and $\theta=1$ gives the smallest bound for $c_{*}$. This
establishes the upper estimate in (2.12b).

To obtain the lower estimates, we construct a supersolution in a completely
analogous fashion. We emphasize that now we need to estimate
$-{\mathcal{L}}_{\gamma,\mathtt{v}}[W^{*}]$ from below. Since we still have
$(W^{*})^{\prime}(\xi)>0$, we find
$-{\mathcal{L}}_{\gamma,\mathtt{v}}[W^{*}](\xi)\geq 0$ for $\mathtt{v}\leq 0$,
which is now the easy case leading to the simple lower bound in (2.12b). For
$\mathtt{v}>0$ we then use
$-{\mathcal{L}}_{\gamma,\mathtt{v}}[W^{*}](\xi)\geq-\mathtt{v}\gamma\widehat{g}_{1}\,m$
and obtain the lower bound in (2.12a).

###### Lemma 2.6

Let the Hypotheses (H1) and (H2) hold. Then, the function
$\mathtt{C}:[0,\gamma_{*}]{\times}{\mathbb{R}}\to{\mathbb{R}}$ is continuous.

Proof. Consider a sequence $(\gamma_{n},\mathtt{v}_{n})_{n\in{\mathbb{N}}}$
with $(\gamma_{n},\mathtt{v}_{n})\to(\gamma,\mathtt{v})$ as $n\to\infty$.
According to Proposition 2.4, for all $n\in{\mathbb{N}}$ there exists a unique
traveling front $u_{n}(t,x)={\mathcal{U}}_{n}(x{-}\mathtt{c}_{n}t)$ for (2.3)
with $\mathtt{c}_{n}=\mathtt{C}(\gamma_{n},\mathtt{v}_{n})$.

_Step 1: Uniform bounds for the sequence
$(\mathtt{c}_{n},{\mathcal{U}}_{n})$._ Since
$({\mathcal{U}}_{n})_{n\in{\mathbb{N}}}$ is a traveling front, we have
$u^{\gamma_{n}}_{-}<{\mathcal{U}}_{n}(\xi)<u^{\gamma_{n}}_{+}$. By (H1) and
the form $F_{\gamma}(u)=F(u)+\gamma u$, the mappings $\gamma\mapsto
u^{\gamma}_{\pm}$ are uniformly continuous and hence bounded, i.e. there
exists $U_{*}>0$ such that $-U_{*}\leq u^{\gamma}_{-}<u^{\gamma}_{+}\leq
U_{*}$ for all $\gamma\in[0,\gamma_{*}]$. Hence,
$\|{\mathcal{U}}_{n}\|_{{\mathrm{L}}^{\infty}}\leq U_{*}$ and there exists
$R_{*}>0$ such that $|F(u)|\leq R_{*}$ for all $u\in[-U_{*},U_{*}]$.

Moreover, we can apply the speed bounds for
$\mathtt{c}_{n}=\mathtt{C}(\gamma_{n},\mathtt{v}_{n})$ from Theorem 2.5 to
show that $|\mathtt{c}_{n}|\leq C_{0}$. Indeed, in a neighborhood of the limit
$(\gamma,\mathtt{v})$ we can choose the estimating quantities $\Phi_{*}$,
$a_{*}$, $b_{*}$, $\Phi^{*}$, $a^{*}$, and $b^{*}$ uniform for
$(\gamma_{n},\mathtt{v}_{n})$ for all $n\geq n_{0}$.

Next we show that ${\mathcal{U}}_{n}$ is also bounded in
${\mathrm{H}}^{1}_{\text{loc}}({\mathbb{R}})$. For this we use equation (2.7a)
in the form

$-\mathtt{c}_{n}{\mathcal{U}}^{\prime}_{n}=D{\mathcal{U}}^{\prime\prime}_{n}+h_{n}\quad\text{
with
}h_{n}(\xi)=F({\mathcal{U}}_{n}(\xi))+\gamma_{n}\int_{0}^{\infty}\Gamma(\tau){\mathcal{U}}_{n}(\xi{+}\mathtt{v}_{n}\tau)\;\\!\mathrm{d}\tau,$
(2.13)

where now $\|h_{n}\|_{{\mathrm{L}}^{\infty}}\leq H_{*}$ for a suitable
constant $H_{*}$. Moreover, we define the function
$\psi_{0}\in{\mathrm{C}}^{1}({\mathbb{R}})$ with $\psi_{0}(x)=(x^{2}{-}1)^{2}$
for $|x|\leq 1$ and $\psi_{0}(x)=0$ otherwise, which satisfies the estimate
$\psi^{\prime}_{0}(x)^{2}\leq 16\psi_{0}(x)$. For arbitrary
$\zeta_{*}\in{\mathbb{R}}$ we define the test function
$\psi:\xi\mapsto\psi_{0}(\xi{-}\zeta_{*})$ and test equation (2.13) with
$\psi{\mathcal{U}}_{n}$. Setting $\Xi=[\zeta_{*}{-}1,\zeta_{*}{+}1]$ this
leads to the estimate

$\displaystyle\Delta_{n,\zeta_{*}}:=\int_{\Xi}D|{\mathcal{U}}^{\prime}_{n}|^{2}\psi\;\\!\mathrm{d}\xi=-\int_{\Xi}D({\mathcal{U}}^{\prime\prime}_{n}{\mathcal{U}}_{n}\psi+{\mathcal{U}}_{n}{\mathcal{U}}^{\prime}_{n}\psi^{\prime})\;\\!\mathrm{d}\xi$
$\displaystyle\overset{\text{\eqref{eq:TF.hn}}}{=}\int_{\Xi}\big{(}(h_{n}{+}\mathtt{c}_{n}{\mathcal{U}}^{\prime}_{n}){\mathcal{U}}_{n}\psi-D{\mathcal{U}}_{n}{\mathcal{U}}^{\prime}_{n}\psi^{\prime})\;\\!\mathrm{d}\xi$
$\displaystyle\leq
H_{*}U_{*}\|\psi\|_{{\mathrm{L}}^{1}(\Xi)}\\!+\big{\|}(D\psi)^{1/2}{\mathcal{U}}^{\prime}_{n}\big{\|}_{{\mathrm{L}}^{2}1(\Xi)}\big{\|}\mathtt{c}_{n}(\tfrac{\psi}{D})^{1/2}{\mathcal{U}}_{n}-(\tfrac{D}{\psi})^{1/2}\psi^{\prime}{\mathcal{U}}_{n}\big{\|}_{{\mathrm{L}}^{2}1(\Xi)}$
$\displaystyle\leq\tfrac{16}{15}H_{*}U_{*}+\Delta_{n,\zeta_{*}}^{1/2}\Big{(}\big{(}\tfrac{16}{15D}\big{)}^{1/2}C_{0}U_{*}+8D^{1/2}U_{*}\Big{)}.$

Since $\zeta_{*}$ was arbitrary and $D\psi(x)\geq D/2$ for $|x|\leq 1/2$, we
find the uniform estimate

$\sup_{\zeta_{*}\in{\mathbb{R}}}\int_{\zeta_{*}{-}1/2}^{\zeta_{*}+1/2}|{\mathcal{U}}^{\prime}_{n}(\xi)|^{2}\;\\!\mathrm{d}\xi\leq\tfrac{64}{15D}H_{*}U_{*}+\tfrac{8}{D}\big{(}\tfrac{16}{15D}C_{0}^{2}U_{*}^{2}+64DU_{*}^{2}\big{)},$

which together with $\|{\mathcal{U}}_{n}\|_{{\mathrm{L}}^{\infty}}\leq U_{*}$
is the desired uniform bound in ${\mathrm{H}}^{1}_{\text{loc}}({\mathbb{R}})$.

Inserting the last result into (2.13), we first obtain a uniform bound for
${\mathcal{U}}_{n}$ in ${\mathrm{H}}^{2}_{\text{loc}}({\mathbb{R}})$, and
inserting again we find
$\|{\mathcal{U}}^{\prime\prime}_{n}\|_{{\mathrm{L}}^{\infty}}\leq
C_{2}<\infty$ for all $n\in{\mathbb{N}}$.

_Step 2: Convergent subsequences and passage to the limit._ We can extract a
subsequence (not relabeled) such that $\mathtt{c}_{n}\to\mathtt{c}$ and
${\mathcal{U}}_{n}\rightharpoonup{\mathcal{U}}$ weakly in
${\mathrm{H}}^{1}_{\mathrm{loc}}({\mathbb{R}})$. We also fix the translations
in such a way that ${\mathcal{U}}_{n}(0)=u^{\gamma_{n}}_{\mathrm{m}}\to
u^{\gamma}_{\mathrm{m}}$. Weak convergence in
${\mathrm{H}}^{1}_{\text{loc}}({\mathbb{R}})$ implies uniform convergence on
any compact interval $K\Subset{\mathbb{R}}$. Hence, we have
${\mathcal{U}}_{n}(\xi)\to{\mathcal{U}}(\xi)$ for all $\xi\in{\mathbb{R}}$ and
${\mathcal{U}}(0)=u^{\gamma}_{\mathrm{m}}$ in particular. Of course, the
uniform bounds for ${\mathcal{U}}_{n}$ imply
${\mathcal{U}}\in{\mathrm{C}}^{1,\text{Lip}}({\mathbb{R}})$ with
$\|{\mathcal{U}}^{\prime\prime}\|_{{\mathrm{L}}^{\infty}}\leq C_{2}$.
Moreover, since all ${\mathcal{U}}_{n}$ are nondecreasing, we have
${\mathcal{U}}^{\prime}(\xi)\geq 0$ as well.

Testing the equation (2.7a) with
$\Phi\in{\mathrm{C}}^{\infty}_{\mathrm{c}}({\mathbb{R}})$ and integrating over
the arbitrarily chosen compact subset $K\Subset{\mathbb{R}}$ with
$\mathrm{sppt}\Phi\subset K$ yields

$\displaystyle\int_{K}-\mathtt{c}_{n}{\mathcal{U}}_{n}^{\prime}\Phi\;\\!\mathrm{d}\xi=\int_{K}\bigg{(}{-}D{\mathcal{U}}_{n}^{\prime}\Phi^{\prime}+\Big{(}F({\mathcal{U}}_{n})+\gamma_{n}\int_{0}^{\infty}\\!\Gamma(\tau){\mathcal{U}}_{n}(\cdot{+}\mathtt{v}_{n}\tau)\;\\!\mathrm{d}\tau\Big{)}\Phi\bigg{)}\;\\!\mathrm{d}\xi.$

Exploiting the continuity of $F$ and the locally uniform convergence
${\mathcal{U}}_{n}\to{\mathcal{U}}$, the limit $n\to\infty$ leads to

$\displaystyle\int_{K}-\mathtt{c}\,{\mathcal{U}}^{\prime}\Phi\;\\!\mathrm{d}\xi=\int_{K}\bigg{(}-D{\mathcal{U}}^{\prime}\Phi^{\prime}+\Big{(}F({\mathcal{U}})+\gamma\int_{0}^{\infty}\Gamma(\tau){\mathcal{U}}(\cdot{+}\mathtt{v}\tau)\;\\!\mathrm{d}\tau\Big{)}\Phi\;\\!\mathrm{d}\xi.$

Thus, the pair $(\mathtt{c},{\mathcal{U}})$ is a solution of (2.7a), which
satisfies ${\mathcal{U}}(0)=u^{\gamma}_{\mathrm{m}}$.

_Step 3: Nontriviality of ${\mathcal{U}}$._ We still need to show that
${\mathcal{U}}$ is not equal to the constant solution
$u^{\gamma}_{\mathrm{m}}$. Since we already know that ${\mathcal{U}}$ is
monotone, the limits $U_{\pm}=\lim_{\xi\to\pm\infty}{\mathcal{U}}(\xi)$ exist.
It is easy to see that these limits satisfy $F_{\gamma}(U_{\pm})=0$. Hence, it
suffices to show $U_{-}\lneqq u^{\gamma}_{\mathrm{m}}$ and $U_{\gamma}\gneqq
u^{\gamma}_{\mathrm{m}}$. For this, it is sufficient, to find a $\delta>0$
such that ${\mathcal{U}}^{\prime}_{n}(0)\geq\delta$, which implies
${\mathcal{U}}^{\prime}(0)\geq\delta>0$.

To show this, we consider the case $\mathtt{v}_{n}\leq 0$ and estimate
${\mathcal{U}}_{n}$ from above on ${]{-}\infty,0]}$, while the case
$\mathtt{v}_{n}\geq 0$ is treated analogously by estimating
${\mathcal{U}}_{n}$ from below on ${[0,\infty[}$. Because of
$\gamma_{n}\to\gamma$, we find $f_{*}>0$ such that
$F_{\gamma_{n}}(u)\leq-\frac{f_{*}}{u^{\gamma_{n}}_{\mathrm{m}}{-}u^{\gamma_{n}}_{-}}(u{-}u^{\gamma_{n}}_{-})(u^{\gamma_{n}}_{\mathrm{m}}-u)\leq
0$ holds for all $u\in[u^{\gamma_{n}}_{-},u^{\gamma_{n}}]$ and
$n\in{\mathbb{N}}$. Hence, using ${\mathcal{U}}^{\prime}_{n}\geq 0$ and
$\mathtt{v}_{n}\leq 0$ we obtain

$\displaystyle D{\mathcal{U}}^{\prime\prime}_{n}$
$\displaystyle=-\mathtt{c}_{n}{\mathcal{U}}^{\prime}_{n}-F({\mathcal{U}}^{\prime}_{n})-\gamma_{n}\int_{0}^{\infty}\Gamma(\tau){\mathcal{U}}_{n}(\xi{+}\mathtt{v}_{n}\tau)\;\\!\mathrm{d}\tau$
$\displaystyle\geq-|\mathtt{c}_{n}|{\mathcal{U}}^{\prime}_{n}-F_{\gamma_{n}}({\mathcal{U}}_{n})\geq-(|\mathtt{c}|{+}1){\mathcal{U}}^{\prime}_{n}+\frac{f_{*}}{u^{\gamma_{n}}_{\mathrm{m}}{-}u^{\gamma_{n}}_{-}}({\mathcal{U}}_{n}{-}u^{\gamma_{n}}_{-})\,(u^{\gamma_{n}}_{\mathrm{m}}{-}{\mathcal{U}}_{n}).$
(2.14)

With this, we can compare the curve
${\mathcal{C}}_{n}:\xi\mapsto({\mathcal{U}}_{n}(\xi),{\mathcal{U}}^{\prime}_{n}(\xi))$
in the phase plane for $(U,U^{\prime})$ with the curve $\widetilde{C}_{n}$
generated by the solution $\widetilde{U}_{n}$ of the ODE
$DU^{\prime\prime}=-(|\mathtt{c}|{+}1)U^{\prime}+\frac{f_{*}}{u^{\gamma_{n}}_{\mathrm{m}}{-}u^{\gamma_{n}}_{-}}(U{-}u^{\gamma_{n}}_{-})\,(u^{\gamma_{n}}_{\mathrm{m}}{-}U)$,
satisfying $\widetilde{U}^{\prime}_{n}\geq 0$ and
$(\widetilde{U}_{n}(\xi),\widetilde{U}^{\prime}_{n}(\xi))\to(u^{\gamma_{n}}_{-},0)$
for $\xi\to-\infty$.

The function ${\mathcal{U}}_{n}$ has the expansion
${\mathcal{U}}_{n}(\xi)=u^{\gamma}_{-}+d_{n}\mathrm{e}^{\lambda_{n}\xi}(1{+}o(1))$
for $\xi\to-\infty$ with $d_{n}>0$, where $\lambda_{n}$ is the positive root
of the characteristic equation
$D\lambda^{2}+\mathtt{c}_{n}\lambda+F^{\prime}(u^{\gamma_{n}}_{-})=0$. Hence,
the former curve ${\mathcal{C}}_{n}$ leaves the point $(u^{\gamma_{n}}_{-},0)$
to the right with positive slope $\lambda_{n}>0$, i.e.
$U^{\prime}=\lambda_{n}(U{-}u^{\gamma_{n}}_{-})+$h.o.t. Similarly, the curve
$\widetilde{C}_{n}$ generated by the solution $\widetilde{U}_{n}$ has the
expansion $U^{\prime}=\widetilde{\lambda}_{n}(U{-}u^{\gamma_{n}}_{-})+$h.o.t.,
where $\widetilde{\lambda}_{n}$ is the positive root of
$D\lambda^{2}-(|\mathtt{c}|{+}1)\lambda-f_{*}=0$. Because of
$\mathtt{c}_{n}\geq-(|\mathtt{c}|{+}1)$ and
$F^{\prime}(u^{\gamma_{n}}_{-})\leq-f_{*}<0$ we have
$\lambda_{n}>\widetilde{\lambda}_{n}>0$, which implies that the curve
${\mathcal{C}}_{n}$ lies above $\widetilde{C}_{n}$ in a neighborhood of
$(u^{\gamma_{n}}_{-},0)$.

Because of the comparison (2.14) we know that ${\mathcal{C}}_{n}$ must stay
above $\widetilde{C}_{n}$ until both curves hit the line
$U=u^{\gamma_{n}}_{\mathrm{m}}$. Thus, choosing $\widetilde{\xi}_{n}$ such
that
$\widetilde{U}_{n}(\widetilde{\xi}_{n})=u^{\gamma_{n}}_{\mathrm{m}}={\mathcal{U}}_{n}(0)$
and $\widetilde{U}_{n}(\xi)<u^{\gamma_{n}}_{\mathrm{m}}$ for
$\xi\leq\widetilde{\xi}_{n}$, we obtain
${\mathcal{U}}^{\prime}_{n}(0)\geq\widetilde{U}^{\prime}_{n}(\xi_{n})$, see
also Figure 2.2. By simple scaling we see that $\widetilde{U}_{n}$ has the
form

$\displaystyle\widetilde{U}_{n}(\xi)=u^{\gamma_{n}}_{-}+\big{(}u^{\gamma_{n}}_{\mathrm{m}}{-}u^{\gamma_{n}}_{-}\big{)}\,Y(\kappa(\xi{-}\xi_{n}))\
\text{ where }Y^{\prime\prime}+Y^{\prime}-\sigma Y(1{-}Y)=0,$
$\displaystyle\kappa=\frac{|\mathtt{c}|{+}1}{D},\quad\sigma=\frac{f_{*}\,D}{(|\mathtt{c}|{+}1)^{2}},\quad
Y(-\infty)=0,\quad Y(0)=1,\quad Y^{\prime}(\xi)>0.$

With this, we arrive at the desired result

${\mathcal{U}}^{\prime}_{n}(0)\geq\widetilde{U}^{\prime}_{n}(\xi_{n})=\big{(}u^{\gamma_{n}}_{\mathrm{m}}{-}u^{\gamma_{n}}_{-}\big{)}\,\kappa\,Y^{\prime}(0)\geq\frac{1}{2}\big{(}u^{\gamma}_{\mathrm{m}}{-}u^{\gamma}_{-}\big{)}\,\kappa\,Y^{\prime}(0)=:\delta>0.$

In summary we have shown that the limit pair $(\mathtt{c},{\mathcal{U}})$
satisfies the ODE (2.7a) as well as the boundary conditions (2.7b). Hence, by
the uniqueness in Proposition 2.4 we conclude
$\mathtt{c}=\mathtt{C}(\gamma,\mathtt{v})$, and the desired continuity of
$\mathtt{C}$ follows from $(\gamma_{n},\mathtt{v}_{n})\to(\gamma,\mathtt{v})$
and
$\mathtt{c}_{n}=\mathtt{C}(\gamma_{n},\mathtt{v}_{n})\to\mathtt{c}=\mathtt{C}(\gamma,\mathtt{v})$.
Indeed, the convergence along the chosen subsequence converts into convergence
of the full sequence, because the limit of any convergence subsequence of
$(\mathtt{c}_{n},{\mathcal{U}}_{n})$ is uniquely determined.

$u^{\gamma_{n}}_{-}$$u^{\gamma_{n}}_{\mathrm{m}}$$U$$U^{\prime}$${\mathcal{C}}_{n}$
Figure 2.2: Phase plane for $(U,U^{\prime})$ displaying the stream plot of the
ODE
$U^{\prime\prime}+\mathtt{c}\,U^{\prime}+f_{*}(U{-}u^{\gamma_{n}}_{-})(u^{\gamma_{n}}_{\mathrm{m}}{-}U)=0$
together with the (red) curve
${\mathcal{C}}_{n}:\xi\mapsto({\mathcal{U}}_{n}(\xi),{\mathcal{U}}^{\prime}_{n}(\xi))$
lying above the unstable manifold of $(u^{\gamma_{n}}_{-},0)$ (black curve).

###  2.5 Traveling fronts for the memory equation

According to Proposition 2.1, we obtain a traveling front for the memory
equation (2.1) by finding $\mathtt{v}\in{\mathbb{R}}$ such that
$\mathtt{v}=\mathtt{C}(\gamma,\mathtt{v})$. To guarantee existence of such
solutions, we derive suitable bounds and monotonicity properties for the wave
speed function $\mathtt{C}$, where again the comparison principle for sub- and
supersolutions is useful, see Propositions 2.2 and 2.3. We recall our choice
that all traveling fronts ${\mathcal{U}}_{\gamma,\mathtt{v}}$ satisfy
${\mathcal{U}}_{\gamma,\mathtt{v}}(-\infty)=u^{\gamma}_{-}\lneqq
u^{\gamma}_{+}={\mathcal{U}}_{\gamma,\mathtt{v}}(\infty)$.

######  Lemma 2.7 (Monotonicity of $\mathtt{C}$)

For all $\gamma,\widetilde{\gamma}\in[0,\gamma_{*}]$ and all
$\mathtt{v},\widetilde{\mathtt{v}}\in{\mathbb{R}}$, we have the following
implications:

$\displaystyle(a)\qquad\qquad\mathtt{v}\leq\widetilde{\mathtt{v}}$
$\displaystyle\Longrightarrow\quad\mathtt{C}(\gamma,\mathtt{v})$
$\displaystyle\geq\mathtt{C}(\gamma,\widetilde{\mathtt{v}});$ (2.15)
$\displaystyle(b)\ \gamma\leq\widetilde{\gamma}\text{ and
}\max\\{u^{\gamma}_{-},u^{\widetilde{\gamma}}_{-}\\}\geq 0$
$\displaystyle\Longrightarrow\quad\mathtt{C}(\gamma,\mathtt{v})$
$\displaystyle\geq\mathtt{C}(\widetilde{\gamma},\mathtt{v}).$ (2.16)

Proof. Throughout the proof we set
$u_{\gamma,\mathtt{v}}(t,x)={\mathcal{U}}\big{(}x{-}\mathtt{C}(\gamma,\mathtt{v})t\big{)}$
and insert one traveling front into the parabolic equation for the other
front, thus obtaining a super or a subsolution. Then, Proposition 2.3 provides
a comparison of the wave speeds.

_Ad (a)._ We insert $u_{\gamma,\widetilde{\mathtt{v}}}$ into the equation for
$u_{\gamma,\mathtt{v}}$ and obtain

$\dot{u}_{\gamma,\widetilde{\mathtt{v}}}{-}{\mathcal{A}}_{\gamma,\mathtt{v}}[u_{\gamma,\widetilde{\mathtt{v}}}]=\underbrace{\dot{u}_{\gamma,\widetilde{\mathtt{v}}}{-}{\mathcal{A}}_{\gamma,\widetilde{\mathtt{v}}}[u_{\gamma,\widetilde{\mathtt{v}}}]}_{=0}+\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma\int_{0}^{\infty}\\!\Gamma(\tau)\;\\!\big{(}u_{\gamma,\widetilde{\mathtt{v}}}(t,x{+}\widetilde{\mathtt{v}}\tau){-}u_{\gamma,\widetilde{\mathtt{v}}}(t,x{+}\mathtt{v}\tau)\big{)}\;\\!\mathrm{d}\tau.$
(2.17)

Using $\Gamma(\tau)\geq 0$,
${\mathcal{U}}^{\prime}_{\gamma,\widetilde{\mathtt{v}}}(\xi)\geq 0$, and
$\mathtt{v}\leq\widetilde{\mathtt{v}}$, we see that the integral is
nonnegative. Together with $\gamma\geq 0$, we find
$\dot{u}_{\gamma,\widetilde{\mathtt{v}}}-{\mathcal{A}}_{\gamma,\mathtt{v}}[u_{\gamma,\widetilde{\mathtt{v}}}]\geq
0$, i.e. $u_{\gamma,\widetilde{\mathtt{v}}}$ is a supersolution of
$\dot{u}={\mathcal{A}}_{\gamma,\mathtt{v}}[u]$, whereas
$u_{\gamma,\mathtt{v}}$ is a (sub)solution. Hence, Proposition 2.3 gives
(2.15).

_Ad (b)._ Assume $u^{\gamma}_{-}\geq 0$ such that
$u_{\gamma,\mathtt{v}}(t,x)\geq u^{\gamma}_{-}\geq 0$. We insert this solution
into the equation for $u_{\widetilde{\gamma},\mathtt{v}}$ and obtain

$\dot{u}_{\gamma,\mathtt{v}}{-}{\mathcal{A}}_{\widetilde{\gamma},\mathtt{v}}[u_{\gamma,\mathtt{v}}]=\underbrace{\dot{u}_{\gamma,\mathtt{v}}{-}{\mathcal{A}}_{\gamma,\mathtt{v}}[u_{\gamma,\mathtt{v}}]}_{=0}+\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\gamma{-}\widetilde{\gamma})\int_{0}^{\infty}\\!\Gamma(\tau)\;\\!u_{\gamma,\mathtt{v}}(t,x{+}\mathtt{v}\tau)\;\\!\mathrm{d}\tau.$
(2.18)

Since the integral is nonnegative, the assumption
$\gamma\leq\widetilde{\gamma}$ implies that $u_{\gamma,\mathtt{v}}$ is a
subsolution whereas $u_{\gamma,\widetilde{\mathtt{v}}}$ is a (super)solution.
Hence, (2.16) follows via Proposition 2.3.

If $u^{\gamma}_{-}<0$ we can rely on $u^{\gamma}_{-}\geq 0$. Interchanging the
role of $\gamma$ and $\widetilde{\gamma}$ in (2.18), we see that
$u_{\gamma,\widetilde{\mathtt{v}}}$ is a supersolution of
$\dot{u}={\mathcal{A}}_{\gamma,\mathtt{v}}[u]$ and (2.15) follows again.

Having collected all preliminary estimates, we can prove the existence of
traveling fronts for the original memory equation (2.1). Moreover, we are able
to give bounds on the corresponding wave speed.

######  Theorem 2.8 (Main result)

Let the Hypotheses (H1) and (H2) hold. Then, for all $\gamma\in[0,\gamma_{*}]$
the equation $\mathtt{c}=\mathtt{C}(\gamma,\mathtt{c})$ has a unique solution
$\mathtt{c}_{\gamma}$, and thus there exists a unique (up to translation),
asymptotically stable traveling-front solution
$u(t,x)=\overline{\mathcal{U}}_{\gamma}(x{+}\mathtt{c}_{\gamma}t)$ for the
memory equation (2.1), where $\overline{\mathcal{U}}_{\gamma}$ satisfies the
properties (2.7) and (2.8).

Proof. For a fixed $\gamma\in[0,\gamma_{*}]$ the function
$\mathtt{v}\mapsto\mathtt{C}(\gamma,\mathtt{v})$ is continuous and non-
increasing according to Lemma 2.6 and relation (2.15), respectively. Thus,
there is a unique solution to $\mathtt{v}=\mathtt{C}(\gamma,\mathtt{v})$,
which we call $\mathtt{c}_{\gamma}$. According to Proposition 2.4 there is a
unique (up to translation) traveling front
$\overline{\mathcal{U}}_{\gamma}:={\mathcal{U}}_{\gamma,\mathtt{c}_{\gamma}}$
for the auxiliary equation (2.3)with $\mathtt{v}=\mathtt{c}_{\gamma}$.

Because of
$\mathtt{v}=\mathtt{c}_{\gamma}=\mathtt{C}(\gamma,\mathtt{c}_{\gamma})$ we can
return to the memory equation using Proposition 2.1, and the result is
established.

Next we combine the available estimates and monotonicity of the wave speed
$\mathtt{C}(\gamma,\mathtt{v})$ to give bounds on $\mathtt{c}_{\gamma}$.
Before doing so, we recall the classical result that for the local equation,
i.e. $\mathtt{v}=0$, the sign of the wave speed $\mathtt{C}(\gamma,0)$ is
opposite to the sign of

${\mathcal{F}}(\gamma):=\int_{u^{\gamma}_{-}}^{u^{\gamma}_{+}}F_{\gamma}(u)\;\\!\mathrm{d}u.$

Indeed, to see this, we multiply (2.7a) by ${\mathcal{U}}^{\prime}$ and
integrate over ${\mathbb{R}}$ leading to

$-\mathtt{C}(\gamma,0)\int_{\mathbb{R}}{\mathcal{U}}^{\prime}_{\gamma,0}(\xi)^{2}\;\\!\mathrm{d}\xi=\lim_{R\to\infty}\Big{(}\frac{D}{2}{\mathcal{U}}^{\prime}_{\gamma,0}(\xi)^{2}\Big{)}\Big{|}_{-R}^{R}+\int_{\mathbb{R}}F_{\gamma}({\mathcal{U}}_{\gamma,0}(\xi)){\mathcal{U}}^{\prime}_{\gamma,0}(\xi)\;\\!\mathrm{d}\xi={\mathcal{F}}(\gamma).$

In particular, for ${\mathcal{F}}(\gamma)=0$, we obtain
$\mathtt{C}(\gamma,0)=0$, which means that a standing wave exists. Obviously
this implies $\mathtt{c}_{0}=0$, i.e. also the memory equation (2.1) has the
same standing wave solution as the local equation with nonlinearity
$F_{\gamma}$.

######  Corollary 2.9 (Bounds on the wave speed)

Let the assumptions (H1) and (H2) hold and let $F_{\gamma}$ satisfy (2.11),
then for all $\gamma\in[0,\gamma_{*}]$, we have the following bounds for the
speeds $\mathtt{c}_{\gamma}$ of the traveling front $\overline{\\\
}calU_{\gamma}$ for the memory equation (2.1):

$\displaystyle{\mathcal{F}}(\gamma)\leq 0$
$\displaystyle\quad\Longrightarrow\quad
0\leq\mathtt{c}_{\gamma}\leq\mathtt{C}(\gamma,0)\leq\Big{(}\frac{\Phi_{*}D}{u^{\gamma}_{+}{-}u^{\gamma}_{\mathrm{m}}}\Big{)}^{1/2},$
(2.19) $\displaystyle{\mathcal{F}}(\gamma)\geq 0$
$\displaystyle\quad\Longrightarrow\quad
0\geq\mathtt{c}_{\gamma}\geq\mathtt{C}(\gamma,0)\geq-\Big{(}\frac{\Phi^{*}D}{u^{\gamma}_{\mathrm{m}}{-}u^{\gamma}_{-}}\Big{)}^{1/2}.$

Proof. Since $\mathtt{v}\mapsto\mathtt{C}(\gamma,\mathtt{v})$ is non-
increasing and continuous, we easily see that the solution
$\mathtt{c}_{\gamma}$ of $\mathtt{v}=\mathtt{C}(\gamma,\mathtt{v})$ always
lies between $0$ (not included) and $\mathtt{C}(\gamma,0)$ (possibly
included). This provides the first two estimates in both lines of (2.19). The
last estimate (in both lines) is a direct consequence of (2.12) with
$\mathtt{v}=0$.

##  3 Derivation of memory equations from local PDEs

Equations with memory can be derived from local models if so-called internal
variables $w$ are eliminated. Changes in the main variable $u$ induce
instantaneous changes in $w$, but the internal dynamics of the $w$ system
leads to the delayed back-coupling from $w$. This effect stays local in space,
if the diffusion in the variable $w$ can be neglected.

###  3.1 Parabolic equation coupled to linear ODEs

For the modeling of pulse propagation in nerves, one often uses the coupling
between a parabolic equation and ODEs, see e.g. [Car77, Eqn. (0.1)] for the
Hodgkin-Huxley equation and [NAY62, Den91] for the FitzHugh–Nagumo-like
equations, which corresponds to the case $m_{\mathrm{w}}=1$ in the following
model:

$\dot{u}=Du_{xx}+F(u)+\sum_{i=1}^{m_{\mathrm{w}}}a_{i}w_{i},\qquad\dot{w}_{i}=-\lambda_{i}w_{i}+b_{i}u\
\text{ for }i=1,\ldots,m_{\mathrm{w}},$ (3.1)

where $a_{i},\ b_{i}$, and $\lambda_{i}$ are fixed real parameters with
$\lambda_{i}>0$. Clearly, the linear ODEs can be solved locally in space by
$w_{i}(t,x)=w_{i}(0,x)\mathrm{e}^{-\lambda_{i}t}+\int_{0}^{t}\mathrm{e}^{-\lambda_{i}(t{-}\tau)}b_{i}u(\tau,x)\;\\!\mathrm{d}\tau$.
Assuming that there is infinite history (which is compatible with our search
for traveling fronts that exist for all time) we may also write
$w_{i}(t,x)=\int_{0}^{\infty}\mathrm{e}^{-\lambda_{i}\tau}b_{i}u(t{-}\tau,x)\;\\!\mathrm{d}\tau$.
With this we obtain the memory equation (2.1) with the specific kernel

$\Gamma(\tau)=\frac{1}{\gamma}\sum_{i=1}^{m_{\mathrm{w}}}\mathrm{e}^{-\lambda_{i}\tau}a_{i}b_{i}\quad\text{
with }\gamma=\sum_{i=1}^{m_{\mathrm{w}}}\frac{a_{i}b_{i}}{\lambda_{i}}.$ (3.2)

Clearly, our theory is applicable also in cases where some of the products
$a_{i}b_{i}$ are negative, as long as $\Gamma$ is nonnegative, e.g.
$\mathrm{e}^{-2\tau}-2\mathrm{e}^{-3\tau}+\mathrm{e}^{-4\tau}=(\mathrm{e}^{-\tau}{-}\mathrm{e}^{-2\tau})^{2}$.
Thus, we need a positive feedback $a_{i}b_{i}$ for several species $i$, but we
may also allow for a negative feedback $a_{j}b_{j}$ for some components if
they are not too big.

For systems of the type (3.1) the existence of traveling waves, in particular
pulses, is studied in [GuR18] in detail, even in cases where Hypothesis (H2)
is violated, i.e. $\Gamma$ may change sign. The latter is necessary to handle
non-monotone fronts and pulses. Therefore, the assumptions in [GuR18] are more
general with respect to the types of traveling waves (e.g. pulses, non-
monotone fronts) under consideration.

However, when studying only traveling fronts, the assumptions of the present
article are much more general, since (H2) does not require any assumptions on
the (exponential) structure of $\Gamma$. In particular, we believe that our
approach may be generalized to nonlinear coupling terms as in Section 3.5,
whereas the approach in [GuR18] relies on the linear structure of the
equations for $w_{i}$. Moreover, the present approach allows to calculate
bounds on the wave speed, see Theorem 2.5 and Corollary 2.9.

###  3.2 The two-scale homogenization model

A significant body of work (cf. [HuZ95, Xin00, Hei01, BeH02, MSU07, CGW08,
BoM14]) considers the propagation in periodic media. If the period of the
oscillating coefficient is very small one can perform a homogenization and
consider traveling waves in the homogenized system. However, often the
diffusion coefficients of some of the species are very small as well, which
leads to a coupled system of the form

$\left.\begin{aligned}
\dot{v}&=\mathop{\mathrm{div}}\nolimits\big{(}{\mathbb{D}}_{\mathrm{v}}(\tfrac{1}{\varepsilon}x)\nabla
v\big{)}+f_{\mathrm{v}}(\tfrac{1}{\varepsilon}x,v,w)\\\
\dot{w}&=\mathop{\mathrm{div}}\nolimits\big{(}\varepsilon^{2}{\mathbb{D}}_{\mathrm{w}}(\tfrac{1}{\varepsilon}x)\nabla
w\big{)}+f_{\mathrm{w}}(\tfrac{1}{\varepsilon}x,v,w)\end{aligned}\right\\}\quad
t>0,\ x\in\Omega.$ (3.3)

Here the functions ${\mathbb{D}}_{\mathrm{v}}$, ${\mathbb{D}}_{\mathrm{w}}$,
$f_{\mathrm{v}}(\cdot,v,w)\in{\mathbb{R}}^{m_{\mathrm{v}}}$, and
$f_{\mathrm{w}}(\cdot,v,w)\in{\mathbb{R}}^{m_{\mathrm{w}}}$ are assumed to be
1-periodic in each component of $y=\frac{1}{\varepsilon}x\in{\mathbb{R}}^{d}$.

It is shown in [MRT14] that under suitable conditions on the diffusion
matrices ${\mathbb{D}}_{\mathrm{v}}$ and ${\mathbb{D}}_{\mathrm{w}}$ and the
reaction terms $f_{\mathrm{v}}$ and $f_{\mathrm{w}}$ the solutions
$(v_{\varepsilon},w_{\varepsilon})$ to the initial value problem converge in
the limit $\varepsilon\to 0$ to solutions $(V,W)$ of the following two-scale
model. For this we denote by
${\mathbb{T}}^{d}={\mathbb{R}}^{d}/{\mathbb{Z}}^{d}$ the $d$-dimensional torus
obtained by identifying the opposite sides of the unit cube. While the fast
diffusion of the variable $v_{\varepsilon}$ guarantees that the limit $V$ only
depends on the macroscopic variable $x\in\Omega$, the limit $W$ of the
solutions $w_{\varepsilon}$ is a two-scale function depending also on the
microscopic variable $y\in{\mathbb{T}}^{d}$:

$\displaystyle\dot{V}(t,x)$
$\displaystyle=\mathop{\mathrm{div}}\nolimits_{x}\big{(}{\mathbb{D}}_{\mathrm{v}}^{\text{eff}}\nabla_{x}V(t,x)\big{)}+\int_{{\mathbb{T}}^{d}}f_{\mathrm{v}}(y,V(t,x),W(t,x,y))\;\\!\mathrm{d}y$
$\displaystyle\text{in }\Omega,$ (3.4a) $\displaystyle\dot{W}(t,x,y)$
$\displaystyle=\mathop{\mathrm{div}}\nolimits_{y}\big{(}{\mathbb{D}}_{\mathrm{w}}(y)\nabla_{y}W(t,x,y)\big{)}+f_{\mathrm{w}}(y,V(t,x),W(t,x,y))$
$\displaystyle\text{in }\Omega{\times}{\mathbb{T}}^{d}.$ (3.4b)

Here ${\mathbb{D}}_{\mathrm{v}}^{\text{eff}}$ is the effective diffusion
tensor obtained by classical homogenization, see e.g. [BLP78]. The main point
in this theory is that it is not possible to replace the slowly diffusing
component $w^{\varepsilon}$ by its macroscopic average. We rather need to keep
track of the microscopic distribution of the $w^{\varepsilon}$ relative to the
underlying periodic microstructure. This is exactly done by the function $W$
depending on $x$ and $y$.

The original theory in [MRT14] and [Rei15] was developed for bounded Lipschitz
domains $\Omega\subset{\mathbb{R}}^{d}$. We show in Appendix A how the result
can be generalized to equations posed on the full space
$\Omega={\mathbb{R}}^{d}$, which is needed to treat traveling waves. For this
one introduces the weighted Lebesgue spaces

$\displaystyle{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d})$
$\displaystyle:=\Big{\\{}\>u\in{\mathrm{L}}^{2}_{\mathrm{loc}}({\mathbb{R}}^{d})\;\Big{|}\;\|u\|^{2}_{{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d})}:=\int_{{\mathbb{R}}^{d}}\varrho(x)|u(x)|^{2}\;\\!\mathrm{d}x<\infty\>\Big{\\}},$
(3.5)
$\displaystyle{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d}{\times}{\mathbb{T}}^{d})$
$\displaystyle:=\Big{\\{}\>U\in{\mathrm{L}}^{2}_{\mathrm{loc}}({\mathbb{R}}^{d}{\times}{\mathbb{T}}^{d})\;\Big{|}\;\|U\|_{{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d}\times{\mathbb{T}}^{d})}^{2}:=\int_{{\mathbb{R}}^{d}\times{\mathbb{T}}^{d}}\varrho(x)|U(x,y)|^{2}\;\\!\mathrm{d}x\;\\!\mathrm{d}y<\infty\>\Big{\\}},$

where for a radius $R>0$ we set $\varrho(x)=1/\cosh(|x|/R)$. With this and the
natural conditions on ${\mathbb{D}}_{\mathrm{v}}$,
${\mathbb{D}}_{\mathrm{w}}$, $f_{\mathrm{v}}$, and $f_{\mathrm{w}}$ we derive
the following result in Appendix A.

######  Theorem 3.1 (Two-scale homogenization)

Let $(v_{\varepsilon},w_{\varepsilon})_{\varepsilon>0}$ denote a sequence of
solutions to the initial value problem (3.3) on $\Omega={\mathbb{R}}^{d}$ with
initial conditions
$(v_{\varepsilon}^{0},w_{\varepsilon}^{0})=(v_{\varepsilon}(0,\cdot),w_{\varepsilon}(0,\cdot)$
that are bounded in
${\mathrm{L}}^{\infty}({\mathbb{R}}^{d})^{m_{\mathrm{v}}+m_{\mathrm{w}}}$ and
converge as follows:

$\displaystyle v_{\varepsilon}^{0}\to V^{0}\text{ in
}{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d})\quad\text{and}\quad
w_{\varepsilon}^{0}\operatorname{\xrightarrow{2s}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}W^{0}\text{
in }{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d}{\times}{\mathbb{T}}^{d}),$

then the solution $(v_{\varepsilon},w_{\varepsilon})$ converges to the
solution $(V,W)$ of the two-scale system (3.4),

$\displaystyle\begin{array}[]{ll}v_{\varepsilon}\rightharpoonup V\ \text{ in
}{\mathrm{L}}^{2}(0,T;{\mathrm{H}}^{1}_{\varrho}({\mathbb{R}}^{d})),&\ \nabla
v_{\varepsilon}\operatorname{\ext@arrow
0359{\arrowfill@\relbar\relbar\rightharpoonup}{}{2w}}\nabla
V{+}\nabla_{y}\widetilde{V}\text{ in
}{\mathrm{L}}^{2}(0,T;{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d}{\times}{\mathbb{T}}^{d})),\\\
w_{\varepsilon}\operatorname{\xrightarrow{2s}}W\text{ in
}{\mathrm{L}}^{2}(0,T;{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d}\times{\mathbb{T}}^{d})),&\varepsilon\nabla
w_{\varepsilon}\operatorname{\ext@arrow
0359{\arrowfill@\relbar\relbar\rightharpoonup}{}{2w}}\nabla_{y}W\qquad\text{
in
}{\mathrm{L}}^{2}(0,T;{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d}{\times}{\mathbb{T}}^{d})),\end{array}$

where
$\widetilde{V}\in{\mathrm{L}}^{2}([0,T]\times\Omega;{\mathrm{H}}^{1}({\mathbb{T}}^{d}))$
with $\int_{{\mathbb{T}}^{d}}\widetilde{V}\;\\!\mathrm{d}y=0$ denotes the
corrector function.

The major advantage of the two-scale model (3.4) is that it is again
homogeneous in the macroscopic spatial variable $x\in{\mathbb{R}}^{d}$, while
the periodic structure is restricted to the microscopic variable
$y\in{\mathbb{T}}^{d}$. Thus, we have a coupling that is local in
$x\in{\mathbb{R}}^{d}$ from $V(s,x)\in{\mathbb{R}}^{m_{\mathrm{v}}}$ to
$W(s,x,\cdot)\in{\mathrm{H}}^{1}({\mathbb{T}}^{d})^{m_{\mathrm{w}}}$. At later
times $t>s$ the internal parabolic evolution of $W(t,x,\cdot)$ via (3.4b)
leads to a delayed feedback of $V(s,x)$ for all $s<t$ of memory type.

In particular, it is possible to look for exact traveling waves for (3.4) in
the form

$(V(t,x),W(t,x,y))=({\mathcal{V}}(x{-}\mathtt{c}t),{\mathcal{W}}(x{-}\mathtt{c}t,y))\in{\mathbb{R}}^{m_{\mathrm{v}}+m_{\mathrm{w}}}.$

Transforming these two-scale solutions back into the one-scale form (called
folding in [MiT07, MRT14]) one obtains periodically oscillating traveling
waves, namely

$(\widetilde{v}_{\varepsilon}(t,x),\widetilde{w}_{\varepsilon}(t,x))=({\mathcal{V}}(x{-}\mathtt{c}t),{\mathcal{W}}(x{-}\mathtt{c}t,\frac{1}{\varepsilon}x))$

that provide the correct first-order approximation of the true solutions
$(v_{\varepsilon},w_{\varepsilon})$, see Figure 3.2 for a plot of such
solutions.

###  3.3 Reduction to a scalar equation

Here we give an explicit example for deriving a scalar memory equation from a
two-scale system with $(v,w)\in{\mathbb{R}}^{1}{\times}{\mathbb{R}}^{1}$. We
start by looking at a specific case for the full time-dependent system (3.4),
namely the FitzHugh-Nagumo case where $V$ and $W$ are scalar and are defined
on the real line $\Omega={\mathbb{R}}$. Moreover, we assume linear couplings
with $f_{\mathrm{v}}(y,V,W)=\Phi(y,V)+\alpha(y)W$ and
$f_{\mathrm{w}}(y,V,W)=-b(y)W+\beta(y)V$:

$\displaystyle\dot{V}(t,x)\quad$
$\displaystyle=D_{\mathrm{v}}V_{xx}(t,x)+\int_{{\mathbb{T}}}\Phi(y,V(t,x))\;\\!\mathrm{d}y+\int_{{\mathbb{T}}}\alpha(y)W(t,x,y)\;\\!\mathrm{d}y,$
(3.6a) $\displaystyle\dot{W}(t,x,y)$
$\displaystyle=\left({\mathbb{D}}_{w}(y)W_{y}(t,x,y)\right)_{y}-b(y)W(t,x,y)+\beta(y)V(t,x),$
(3.6b)

where ${\mathbb{T}}={\mathbb{T}}^{1}={\mathbb{R}}_{\\!/{\mathbb{Z}}}$. All
functions $\Phi$, $\alpha$, $\beta$ and $b$ are assumed to be continuous. In
general, the coupling parameters $\alpha(y)$ and $\beta(y)$ may change sign,
while the microscopic diffusion coefficient ${\mathbb{D}}_{\mathrm{w}}(y)$ and
the damping factor $b$ are assumed to be strictly positive. The solution
$\psi(t,y)$ of the linear equation
$\dot{\psi}=({\mathbb{D}}(y)\psi_{y})_{y}-b(y)\psi$, $\psi(0,\cdot)=\psi^{0}$
has the semigroup representation
$\psi(t,y)=\int_{\mathbb{T}}H(t,y,\widetilde{y})\psi^{0}(\widetilde{y})\;\\!\mathrm{d}\widetilde{y}$,
where the Greens function $H(t,y,\widetilde{y})$ is strictly positive for
$t>0$ by the maximum principle for linear parabolic equations, see e.g.
[Eva98, Thm. 12, p. 376].

By introducing the effective nonlinearity
$F(V):=\int_{\mathbb{T}}\Phi(y,V)\;\\!\mathrm{d}y$ and expressing $W$ as a
linear functional over the history of $V$ via (3.6b), we obtain a Nagumo
equation with memory kernel:

$\displaystyle\dot{V}(t,x)=V_{xx}(t,x)+F(V(t,x))+\int_{0}^{\infty}\\!\widehat{\Gamma}(\tau)V(t{-}\tau,x)\;\\!\mathrm{d}\tau$
(3.7) $\displaystyle\text{with
}\widehat{\Gamma}(\tau)=\int_{{\mathbb{T}}}\int_{{\mathbb{T}}}H(\tau,y,\widetilde{y})\alpha(\widetilde{y})\beta(y)\;\\!\mathrm{d}y\;\\!\mathrm{d}\widetilde{y}.$

Thus, using $H>0$ a sufficient condition for $\widehat{\Gamma}(\tau)\geq 0$ in
our Hypothesis (H2) is given by $\alpha(y),\beta(y)\geq 0$ for all
$y\in{\mathbb{T}}$ or vice versa $\alpha(y),\beta(y)\leq 0$.

######  Remark 3.2 (On the positivity of $\widehat{\Gamma}$)

The given conditions on $\alpha$ and $\beta$ are far from optimal. Indeed,
writing ${\mathbb{L}}\psi=-({\mathbb{D}}_{\mathrm{w}}\psi_{y})_{y}+b\psi$ we
find a complete orthonormal set $(\psi_{n})_{n\in{\mathbb{N}}}$ in
${\mathrm{L}}^{2}({\mathbb{T}})$ of injunctions, i.e.
${\mathbb{L}}\psi_{n}=\lambda_{n}\psi_{n}$ with
$0<\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n}\to\infty$.

Expanding the coupling coefficients
$\alpha(y)=\sum_{\mathbb{N}}a_{n}\psi_{n}(y)$ and
$\beta(y)=\sum_{\mathbb{N}}b_{n}\psi_{n}(y)$, we find
$\widehat{\Gamma}(\tau)=\sum_{n=1}^{\infty}\mathrm{e}^{-\lambda_{n}\tau}a_{n}b_{n}$,
and conclude that the property $a_{n}b_{n}\geq 0$ for all $n\in{\mathbb{N}}$
is sufficient. Hence, setting $\alpha(y)=\beta(y)=\psi_{n}(y)$ and noticing
that, by the Sturm-Liouville property, the function $\psi_{n}$ has
$\lfloor\frac{n-1}{2}\rfloor$ zeros, we have constructed an example for
$\alpha$ and $\beta$ that change sign and still satisfy
$\widehat{\Gamma}(\tau)\geq 0$.

However, it is not even necessary that all of the products $a_{n}b_{n}$ are
nonnegative, see the arguments after (3.2). In [GuR18] the explicit choices
$\alpha(y)=-\sum_{n=1}^{m_{*}}\psi_{n}(y)$ and
$\beta(y)=\sum_{n=1}^{m_{*}}\sigma_{n}\psi_{n}(y)$ were made, and different
signs for $\sigma_{n}$ are explicitly allowed.

###  3.4 A homogenization example

In the spirit of Remark 3.2 we consider the two-scale system (3.6) in a
specific example fulfilling all the assumptions of our theory. For the
operator ${\mathbb{L}}W:=-W_{yy}+W$ we have the eigenfunctions
$\psi_{2n}(y)=\sqrt{2}\sin(2\pi ny)$ with eigenvalues $\lambda_{2n}=1{+}(2\pi
n)^{2}$. We set

$\alpha(y)=\psi_{2}(y)+\psi_{4}(y)\quad\text{and}\quad\beta(y)=\psi_{2}(y)+10\,\psi_{4}(y)-\psi_{6}(y).$

Hence, Hypothesis (H2) is satisfied with
$\gamma=\int_{0}^{\infty}\widehat{\Gamma}(\tau)\;\\!\mathrm{d}\tau=1/\lambda_{2}+10/\lambda_{4}\approx
0.08763>0$. Notice that the coupling coefficients $\alpha(\cdot)$ and
$\beta(\cdot)$ do not change sign simultaneously. For the cubic function, we
choose $F(u)=-u(u{-}0.25)(u{-}1)$. For this choice of parameters, the two-
scale solution $(V,W)$ is depicted in Figure 3.1, where the microscopic
average of the component $W$ vanishes, since the constant eigenfunction
$\psi_{1}(y)=1$ is not activated because of $\langle\psi_{1},\beta\rangle=0$.
However, the periodic oscillations of $W$ on the micro-scale are captured by
the two-scale limit.

Figure 3.1: The front $(V,W)$ of the two-scale system (3.6) moves from right
to left in $x$. Left: $V$ and average
$\int_{\mathbb{T}}W(t,x,y)\;\\!\mathrm{d}y$. Right: the $W(t,x,y)$-component
in $(x,y)$-plane.

We can compare the solution $(V,W)$ of our two-scale limit system with the
solution $(v_{\varepsilon},w_{\varepsilon})$ of the original system (3.4) with
rapidly oscillating coefficients. In Figure 3.2 one can observe that the
$\varepsilon$-periodic coupling coefficients induce $\varepsilon$-periodic
oscillations of the solutions. Whereas the amplitude of the oscillations of
the component $w_{\varepsilon}$ is of order $O(1)$, the postive diffusion
$D_{\mathrm{v}}$ reduces the amplitude of the oscillations of the component
$v_{\varepsilon}$ to order $O(\varepsilon)$, such that it vanishes in the
limit $\varepsilon\to 0$. Notice that the component $w_{\varepsilon}$ also
changes sign. Overall, the effective behavior of the oscillating solution
$(v_{\varepsilon},w_{\varepsilon})$ is nicely captured by the two-scale limit
$(V,W)$.

Figure 3.2: Solutions $(v_{\varepsilon},w_{\varepsilon})$ of the original
system (3.4): left $\varepsilon{=}2.5$, right $\varepsilon{=}0.25$.

###  3.5 Possible generalizations

For mathematical conciseness we have decided to restrict ourselves to memory
terms with a simple linear structure. However, as our theory is many based on
the quite general work in [Che97] it is clear that the results can be
generalized in several ways.

First, the integral memory can be replaced by discrete time delays in the form
such that the equation reads

$\dot{u}(t,x)=Du_{xx}(t,x)+F(u(t,x))+\sum_{j=1}^{j_{*}}\gamma_{j}u(t{-}\tau_{j},x).$
(3.8)

Assuming $\gamma_{j}>0$ and setting $\gamma=\sum_{j=1}^{j_{*}}\gamma_{j}$, we
obtain the memory equation (2.1) with a
$\Gamma(\tau)=\sum_{j=1}^{j_{*}}\frac{\gamma_{j}}{\gamma}\,\delta_{\tau_{j}}(\tau)$.
We refer to [WuZ01] for a general approach to traveling waves in reaction-
diffusion equations involving linear and nonlinear delay terms.

Secondly, further generalizations can be obtained by looking at nonlinear
couplings to ODEs, thus generalizing the coupled system (3.1):

$\dot{u}=Du_{xx}+F(u)+\sum_{i=1}^{m_{\mathrm{w}}}A_{i}(u,w_{i}),\qquad\dot{w}_{i}=-\lambda_{i}w_{i}+B_{i}(u),$
(3.9)

where the coupling functions $A_{i}:{\mathbb{R}}^{2}\to{\mathbb{R}}$ and
$B_{i}:{\mathbb{R}}\to{\mathbb{R}}$ are assumed to be smooth functions with
derivatives satisfying $\partial_{u}A(u,w_{i})\leq 0$,
$\partial_{w_{i}}A_{i}(u,w_{i})\geq 0$, and $B^{\prime}_{i}(u)\geq 0$. Again
we can express $w_{i}(t,x)$ as memory functional depending on $u(s,x)$ for
$s\in{]{-}\infty,t[}$. Hence, (3.9) can be written in the form

$\displaystyle\dot{u}(t,x)=Du_{xx}(t,x)+F(u(t,x))+\sum_{i=1}^{m_{\mathrm{w}}}A_{i}\big{(}u(t,x),{\mathrm{I}}\Gamma_{i}{*}B_{i}(u(\cdot,x))(t)\big{)},$
$\displaystyle\text{where
}{\mathrm{I}}\Gamma_{i}{*}h(t)=\int_{0}^{\infty}\mathrm{e}^{-\lambda_{i}\tau}h(t{-}\tau)\;\\!\mathrm{d}\tau.$

Hence we have a memory in time, which is local in the space variable
$x\in{\mathbb{R}}$. As in Section 2.2 we can introduce an auxiliary wave speed
$\mathtt{v}$ and turn the memory terms into spatially nonlocal terms. Because
of our assumptions on $A_{i}$ and $B_{i}$ we are exactly in the setting of
[Che97, Eqn. (1.14)]. Thus, it is expected that the methods developed here can
be extended to such nonlinear couplings.

Thirdly, it seems possible to generalize the theory to handle multidimensional
traveling waves in a cylindrical domain $\Omega={\mathbb{R}}{\times}\Sigma$ as
in [Gar86], but now with memory terms.

$\displaystyle\dot{u}(t,x,y)$
$\displaystyle=D\big{(}u_{xx}(t,x,y)+\Delta_{y}u(t,x,y)\big{)}$
$\displaystyle\quad+F(y,u(t,x,y))+\gamma\int_{0}^{\infty}\Gamma(\tau)u(t{-}\tau,x,y)\;\\!\mathrm{d}\tau$
$\displaystyle\text{for }t>0,\ (x,y)\in{\mathbb{R}}{\times}\Sigma,$
$\displaystyle 0$ $\displaystyle=u(t,x,y)$ $\displaystyle\text{for }t>0,\
(x,y)\in{\mathbb{R}}{\times}\partial\Sigma.$

However, then the “bistability” Hypothesis (H1) has to be formulated in terms
of the elliptic problem $u\mapsto D\Delta_{y}u+F(u)+\gamma u$ in $\Sigma$,
which should have exactly three solutions
$u^{\gamma}_{\alpha}\in{\mathrm{H}}^{1}_{0}(\Sigma)$ with appropriate
properties.

##  4 Applications to the cubic case

In this section we study the classical FitzHugh-Nagumo system with a bistable
cubic nonlinearity, namely

$\dot{u}=u_{xx}+F(u)-\beta v,\qquad\dot{v}=-v+u,\quad F(u)=-u(u{-}a)(u{-}1),$
(4.1)

where $a\in{]0,1[}$. Concerning traveling fronts, this system is equivalent to
the memory equation

$\dot{u}(t,x)=u_{xx}(t,x)+F(u(t,x))-\beta\int_{0}^{\infty}\mathrm{e}^{-\tau}u(t{-}\tau,x)\;\\!\mathrm{d}\tau.$
(4.2)

In particular, we see that the previous parameter $\gamma$ is given by
$\gamma=-\beta$, such that our theory developed above only applies for
$\beta\leq 0$. However, in the numerical simulations documented below, we are
free to choose $\beta>0$ as well.

We first observe that the nonlinearity

$F_{-\beta}(u)=-u(u{-}a)(u{-}1)-\beta u$

has a bistable structure if and only if $\beta<(1{-}a)^{2}/4$ and then

$u^{-\beta}_{-}=0,\quad
u^{-\beta}_{\mathrm{m}}=\frac{1{+}a}{2}-\frac{1}{2}\sqrt{(1{-}a)^{2}-4\beta},\quad
u^{-\beta}_{+}=\frac{1{+}a}{2}+\frac{1}{2}\sqrt{(1{-}a)^{2}-4\beta}.$

In [McK70], the wave speed of the local model $\dot{u}=u_{xx}+F_{-\beta}(u)$
is calculated explicitly, which gives

$\mathtt{C}^{\mathrm{FN}}(-\beta,0):=\frac{1}{\sqrt{2}}\big{(}2u^{-\beta}_{\mathrm{m}}-u^{-\beta}_{+}\big{)}=\frac{1}{2\sqrt{2}}\Big{(}1{+}a-3\sqrt{(1{-}a)^{2}-4\beta}\>\Big{)}.$

In the case $\beta=0$ there is no coupling between the ODE an the PDE, hence
$\mathtt{c}_{0}=\mathtt{C}^{\mathrm{FN}}(0,0)=(2a{-}1)/{\sqrt{2}}$ is known.

In the case $a\in{]\frac{1}{2},1[}$, which we fix from now on, the function
$\beta\mapsto\mathtt{C}({-}\beta,0)$ changes sign at

$\beta_{0}(a)=\tfrac{1}{9}\big{(}2a^{2}{-}5a{+}2\big{)}\in{]{-}\tfrac{1}{9},0[}.$

With $\mathtt{C}^{\mathrm{FN}}(0,0)=(2a{-}1)/{\sqrt{2}}>0$, we obtain
$\mathtt{C}^{\mathrm{FN}}(-\beta,0)>0$ for
$\beta\in{]\beta_{0}(a),\frac{1}{4}(1{-}a)^{2}]}$ and
$\mathtt{C}^{\mathrm{FN}}(-\beta,0)<0$ for $\beta<\beta_{0}(a)$.

From this, we see that we can apply our existence theory in Theorem 2.8 and
obtain a unique traveling front for the FitzHugh-Nagumo (4.1), or equivalently
for the memory equation (4.2) for all $\beta\leq 0$. This front connects the
values $u^{-\beta}_{-}=0$ and $u^{-\beta}_{+}$, and using Corollary 2.9 it
travels with the speed $\mathtt{c}_{-\beta}$ satisfying the bound

$\displaystyle
0\leq\mathtt{c}_{-\beta}\leq\mathtt{C}^{\mathrm{FN}}({-}\beta,0)$
$\displaystyle\quad\text{for }\beta\in[\beta_{0}(a),0],$
$\displaystyle\mathtt{C}^{\mathrm{FN}}({-}\beta,0)\leq\mathtt{c}_{-\beta}\leq
0$ $\displaystyle\quad\text{for }\beta\leq\beta_{0}(a).$

For numerical simulations we choose $a=0.6$ determine $\mathtt{c}_{-\beta}$ on
the whole bistable regime $\beta\leq(1{-}a)^{2}/4=0.04$. For this case, we
have $\mathtt{c}_{0}=\mathtt{C}^{\mathrm{FN}}(0,0)=0.2/\sqrt{2}\approx 0.1414$
and

$\mathtt{C}^{\mathrm{FN}}(-\beta,0)=\frac{1}{2\sqrt{2}}\big{(}1.6-3\sqrt{0.4^{2}-4\beta}\>\big{)}\approx
0.566-2.12\sqrt{0.04{-}\beta}$

as upper or lower bound for $\mathtt{c}_{-\beta}$. Moreover, we know that
$\mathtt{C}^{\mathrm{FN}}({-}\beta,0)$ changes sign at
$\beta_{0}(0.6)=-0.28/9\approx-0.0311$.

To determine the speed of the traveling front numerically, we solve the
coupled system (4.1) starting from front-like initial data with the correct
plateau $(u,v)=(u^{-\beta}_{+},u^{-\beta}_{+})$ for large $x$ and
$(u,v)=(0,0)$ for small $x$. The solution rapidly stabilizes into the front
and the speed $\mathtt{c}_{-\beta}$ can be measured. The numerical simulations
nicely confirm the bounds on the wave speed $\mathtt{c}_{-\beta}$ in Corollary
2.9. For all $\beta<\beta_{0}(0.6)\approx-0.0311$, Figure 4.1 shows
$\mathtt{C}(-\beta,0)\leq\mathtt{c}_{-\beta}\leq 0$.

$\beta$$\mathtt{c}_{-\beta}$$\mathtt{C}^{\mathrm{FN}}(-\beta,0)$ Figure 4.1:
The region $\beta\in[-5,-0.03]$ was sampled with $\beta$-steps of size $0.01$
to find $\mathtt{c}_{-\beta}\in[\mathtt{C}^{\mathrm{FN}}(-\beta,0),0]$.

For $\beta\in[\beta_{0}(0.6),0]$ Figure 4.2 shows
$0\leq\mathtt{c}_{-\beta}\leq\mathtt{C}(-\beta,0)$ as predicted by the theory.
For positive $\beta\in{]0,0.04[}$ Theorem 2.8 does not apply, however, the
comparison argument in (2.17) predicts the reverse relation
$\mathtt{c}_{-\beta}\geq\mathtt{C}(-\beta,0)\geq 0$, which is confirmed in the
simulations, which also show that $\mathtt{c}_{-\beta}$ and
$\mathtt{C}(-\beta,0)$ only differ less than a percent in relative size for
all $\beta\in[-0.06,0.03]$. Moreover, the identities
$\mathtt{c}_{-\beta}=\mathtt{C}(-\beta,0)$ are confirmed for
$\beta=\beta_{0}(0.6)$ and $\beta=0$.

$\mathtt{c}_{-\beta}>\mathtt{C}^{\mathrm{FN}}(-\beta,0)$$\mathtt{c}_{-\beta}<\mathtt{C}^{\mathrm{FN}}(-\beta,0)$$\mathtt{c}_{-\beta}>\mathtt{C}^{\mathrm{FN}}(-\beta,0)$$\mathtt{c}_{-\beta}$$\mathtt{C}^{\mathrm{FN}}(-\beta,0)$$\beta$

Figure 4.2: The region $\beta\in[-0.06,0.04]$ was sampled with $\beta$-steps
of size $0.001$ to compare $\mathtt{c}_{-\beta}$ and $\mathtt{C}(-\beta,0)$ in
the upper figure. As the two graphs differ only little, the lower figure
displays $c_{-\beta}-\mathtt{C}(-\beta,0)$, where the sign changes at
$\beta=\beta_{0}(0.6)\approx-0.0311$ and $\beta=0$ are clearly visible.

##  Appendix A Two-scale convergence in weighted Sobolev spaces

The aim of this appendix is to explain how the result in Theorem 3.1 for
$\Omega={\mathbb{R}}^{d}$ can be derived by generalizing the theory for
bounded Lipschitz domains $\Omega\subset{\mathbb{R}}^{d}$ as developed in
[MRT14, Rei15]. To this end we introduce a few notations from two-scale
convergence and two-scale homogenization as developed in [Ngu89, All92, MiT07,
CDG08]. For arbitrary (bounded or unbounded) Lipschitz domains
$\Omega\subseteq{\mathbb{R}}^{d}$, we set

$\displaystyle{\mathcal{T}}_{\varepsilon}:{\mathrm{L}}^{2}(\Omega)\to{\mathrm{L}}^{2}({\mathbb{R}}^{d}{\times}{\mathbb{T}}^{d});\quad({\mathcal{T}}_{\varepsilon}u)(x,y):=u_{\mathrm{ex}}\left(\varepsilon\left[\tfrac{x}{\varepsilon}\right]+\varepsilon
y\right),$

where $u_{\mathrm{ex}}\in{\mathrm{L}}^{2}({\mathbb{R}}^{d})$ is obtained by
extension with $0$ on ${\mathbb{R}}^{d}\backslash\Omega$ and
$[z]\in{\mathbb{Z}}^{d}$ denotes the integer part for any
$z\in{\mathbb{R}}^{d}$. We define _weak and strong two-scale convergence_ via

$\displaystyle u_{\varepsilon}\operatorname{\ext@arrow
0359{\arrowfill@\relbar\relbar\rightharpoonup}{}{2w}}U\text{ in
}{\mathrm{L}}^{2}(\Omega{\times}{\mathbb{T}}^{d})$
$\displaystyle\quad:\overset{\text{Def.}}{\Longleftrightarrow}\quad{\mathcal{T}}_{\varepsilon}u_{\varepsilon}\rightharpoonup
U_{\mathrm{ex}}\text{ in
}{\mathrm{L}}^{2}({\mathbb{R}}^{d}{\times}{\mathbb{T}}^{d}),$ $\displaystyle
u_{\varepsilon}\operatorname{\xrightarrow{2s}}U\text{ in
}{\mathrm{L}}^{2}(\Omega{\times}{\mathbb{T}}^{d})$
$\displaystyle\quad:\overset{\text{Def.}}{\Longleftrightarrow}\quad{\mathcal{T}}_{\varepsilon}u_{\varepsilon}\to
U_{\mathrm{ex}}\text{ in
}{\mathrm{L}}^{2}({\mathbb{R}}^{d}{\times}{\mathbb{T}}^{d}).$

We briefly recall the mathematical setting for the $\varepsilon$-problem (3.3)
such that the two-scale system (3.4) is rigorously justified. The fourth order
diffusion tensors
${\mathbb{D}}_{i}\in\mathrm{Lin}({\mathbb{R}}^{d{\times}m_{i}};{\mathbb{R}}^{d{\times}m_{i}})$
are uniformly elliptic and bounded, i.e. there exist constants $M,\mu>0$ such
that

$\displaystyle{\mathbb{D}}_{\mathrm{i}}(y)\xi:\xi\geq\mu|\xi|^{2}\quad\text{and}\quad|{\mathbb{D}}_{\mathrm{i}}(y)\xi|\leq
M|\xi|\quad\text{for all }\xi\in{\mathbb{R}}^{d{\times}m_{\mathrm{i}}}\text{
and }{\mathrm{i}}\in\\{{\mathrm{v}},{\mathrm{w}}\\},$

and they are not necessarily symmetric. For the reaction terms we assume
boundedness
$f_{\mathrm{i}}(\cdot,v,w)\in{\mathrm{L}}^{\infty}({\mathbb{T}}^{d})$ and
global Lipschitz continuity, i.e. there exists $L>0$ such that for
$v_{1},v_{2}\in{\mathbb{R}}^{m_{\mathrm{v}}}$ and
$w_{1},w_{2}\in{\mathbb{R}}^{m_{\mathrm{w}}}$

$\displaystyle|f_{\mathrm{i}}(y,v_{1},w_{1})-f_{\mathrm{i}}(y,v_{2},w_{2})|\leq
L(|v_{1}{-}v_{2}|+|w_{1}{-}w_{2}|)\quad\text{for
}{\mathrm{i}}\in\\{{\mathrm{v}},{\mathrm{w}}\\}.$

In addition to the weighted Lebesgue spaces
${\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d})$ and
${\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d}{\times}{\mathbb{T}}^{d})$ from
(3.5), we also define the weighted Sobolev spaces
${\mathrm{H}}^{1}_{\varrho}({\mathbb{R}}^{d})$ and
${\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d};{\mathrm{H}}^{1}({\mathbb{T}}^{d}))$
via

$\displaystyle{\mathrm{H}}^{1}_{\varrho}({\mathbb{R}}^{d})$
$\displaystyle:=\Big{\\{}\>u\in{\mathrm{H}}^{1}_{\mathrm{loc}}({\mathbb{R}}^{d})\;\Big{|}\;\|u\|_{{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d})}+\|\nabla
u\|_{{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d})}<\infty\>\Big{\\}},$
$\displaystyle{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d};{\mathrm{H}}^{1}({\mathbb{T}}^{d}))$
$\displaystyle:=\Big{\\{}\>U\in{\mathrm{L}}^{2}_{\mathrm{loc}}({\mathbb{R}}^{d};{\mathrm{H}}^{1}({\mathbb{T}}^{d}))\;\Big{|}\;\|U\|_{{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d}{\times}{\mathbb{T}}^{d})}+\|\nabla_{y}U\|_{{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d}{\times}{\mathbb{T}}^{d})}<\infty\>\Big{\\}}.$

Our choice $\varrho(x)=1/\cosh(|x|/R)$ gives
$\varrho,\varrho^{-1}\in{\mathrm{L}}^{1}_{\mathrm{loc}}({\mathbb{R}}^{d})$,
which implies that the weighted Lebesgue and Sobolev spaces are Banach spaces,
see e.g. [GoU09, Thm. 1]. Moreover, introducing the scalar product
$(u,v)_{{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d})}:=(\sqrt{\varrho}u,\sqrt{\varrho}v)_{{\mathrm{L}}^{2}({\mathbb{R}}^{d})}$
yields that they are also separable Hilbert spaces. The proof of Theorem 3.1
follows along the lines of [MRT14, Thm. 4.1] and [Rei15, Thm. 2.1.1, Thm.
2.2.1] with the following modifications.

  * •

The definition of the periodic unfolding operator

$\displaystyle{\mathcal{T}}_{\varepsilon}:{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d})\to{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d}{\times}{\mathbb{T}}^{d});\quad({\mathcal{T}}_{\varepsilon}u)(x,y):=u\left(\varepsilon\left[\tfrac{x}{\varepsilon}\right]+\varepsilon
y\right)$

is immediate and the notion of two-scale convergence follows analogously.
Notice that the assumptions on the weight function imply the unfolding
estimate

$\displaystyle\|{\mathcal{T}}_{\varepsilon}\varrho-\varrho\|_{{\mathrm{L}}^{\infty}({\mathbb{R}}^{d}{\times}{\mathbb{T}}^{d})}\leq\varepsilon\sqrt{d}\|\nabla\varrho\|_{{\mathrm{L}}^{\infty}({\mathbb{R}}^{d})}\leq\varepsilon\sqrt{d}/R.$

Hence, we obtain the upper bound for unfolded functions
$u\in{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d})$

$\displaystyle\|{\mathcal{T}}_{\varepsilon}u\|_{{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d}{\times}{\mathbb{T}}^{d})}\leq
U_{\varepsilon}\|u\|_{{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d})}\quad\text{with}\quad
U_{\varepsilon}=\big{(}1-\varepsilon\sqrt{d}/R\big{)}^{-1/2}.$

  * •

The standard compactness results for two-scale convergence [Ngu89, All92,
MiT07] also hold for weighted spaces, since they are, in particular, separable
Banach spaces and the Banach–Alaoglu Theorem is applicable.

  * •

The definition of the gradient folding operator
$\mathcal{G}^{\mathrm{slow}}_{\varepsilon}$ (cf. [MRT14, Def. 3.5]) needs to
be adapted as follows: In the case of slow diffusion of order
$O(\varepsilon^{2})$, we define for
$W\in{\mathrm{L}}^{2}_{\varrho}({\mathbb{R}}^{d};{\mathrm{H}}^{1}({\mathbb{T}}^{d}))$
the one-scale function
$\mathcal{G}^{\mathrm{slow}}_{\varepsilon}W:=\hat{w}_{\varepsilon}\in{\mathrm{H}}^{1}_{\varrho}({\mathbb{R}}^{d})$,
which is given by the Lax–Milgram lemma as the unique solution of the elliptic
problem

$\displaystyle\int_{{\mathbb{R}}^{d}}\big{[}\sqrt{\varrho}\hat{w}_{\varepsilon}-\mathcal{F}_{\varepsilon}\left(\sqrt{\varrho}W\right)\big{]}\cdot\sqrt{\varrho}\varphi+\big{[}\varepsilon\sqrt{\varrho}\nabla\hat{w}_{\varepsilon}-\mathcal{F}_{\varepsilon}\left(\sqrt{\varrho}\nabla_{y}W\right)\big{]}:\varepsilon\sqrt{\varrho}\nabla\varphi\;\\!\mathrm{d}x=0$

for all $\varphi\in{\mathrm{H}}^{1}_{\varrho}({\mathbb{R}}^{d})$.

  * •

For the slowly diffusing species $w_{\varepsilon}$, we choose the test
function
$\varphi^{\mathrm{slow}}_{\varepsilon}=\varrho\left(w_{\varepsilon}{-}\mathcal{G}^{\mathrm{slow}}_{\varepsilon}W\right)$
and continue as in [MRT14, Rei15]. For species $v_{\varepsilon}$ undergoing
diffusion of order $O(1)$, we proceed analogously (cf. [Rei15, Def. 1.2.7])
and choose the test function
$\varphi^{\mathrm{stand}}_{\varepsilon}=\varrho\left(v_{\varepsilon}-\mathcal{G}^{\mathrm{stand}}_{\varepsilon}V\right)$.

With these modifications for $\Omega={\mathbb{R}}^{d}$, the proofs in [MRT14,
Thm. 4.1] and [Rei15, Thm. 2.1.1, Thm. 2.2.1] can be easily adapted, and the
proof of Theorem 3.1 is complete.

#### Acknowledgments.

The authors are grateful to Christian Kühn for illuminating discussions and to
Stefanie Schindler for many helpful comments. This research was supported by
Deutsche Forschungsgemeinschaft through the Collaborative Research Center SFB
910 _Control of self-organizing nonlinear systems_ (Project no. 163436311) via
the subproject A5 “Pattern formation in coupled parabolic systems”.

## References

  * [AcK15] F. Achleitner and C. Kuehn.  Traveling waves for a bistable equation with nonlocal diffusion.  Adv. Diff. Eqns., 20(9-10), 887–936, 2015.
  * [All92] G. Allaire.  Homogenization and two-scale convergence.  SIAM J. Math. Anal., 23, 1482–1518, 1992.
  * [BeH02] H. Berestycki and F. Hamel.  Front propagation in periodic excitable media.  Comm. Pure Appl. Math., 55(8), 949–1032, 2002.
  * [BLP78] A. Bensoussan, J.-L. Lions, and G. Papanicolaou.  Asymptotic analysis for periodic structures, volume 5 of Studies in Mathematics and its Applications.  North-Holland Publishing Co., Amsterdam, 1978.
  * [BoM14] A. Boden and K. Matthies.  Existence and homogenisation of travelling waves bifurcating from resonances of reaction-diffusion equations in periodic media.  J. Dynam. Differential Equations, 26(3), 405–459, 2014.
  * [Car77] G. A. Carpenter.  A geometric approach to singular perturbation problems with applications to nerve impulse equations.  J. Diff. Eqns., 23(3), 335–367, 1977.
  * [CDG08] D. Cioranescu, A. Damlamian, and G. Griso.  The periodic unfolding method in homogenization.  SIAM J. Math. Anal., 40(4), 1585–1620, 2008.
  * [CGW08] X. Chen, J.-S. Guo, and C.-C. Wu.  Traveling waves in discrete periodic media for bistable dynamics.  Arch. Rational Mech. Anal., 189(2), 189–236, 2008.
  * [Che97] X. Chen.  Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations.  Adv. Differential Equations, 2(1), 125–160, 1997.
  * [Den91] B. Deng.  The existence of infinitely many traveling front and back waves in the FitzHugh-Nagumo equations.  SIAM J. Math. Analysis, 22(6), 1631–1650, 1991.
  * [Eva98] L. C. Evans.  Partial differential equations, volume 19 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 1998.
  * [FiM77] P. C. Fife and J. B. McLeod.  The approach of solutions of nonlinear diffusion equations to travelling front solutions.  Arch. Rational Mech. Anal., 65, 335–361, 1977.
  * [Gar86] R. Gardner.  Existence of multidimensional travelling wave solutions of an initial-boundary value problem.  J. Diff. Eqns., 61, 335–379, 1986.
  * [GoU09] V. Gol’dshtein and A. Ukhlov.  Weighted Sobolev spaces and embedding theorems.  Trans. Amer. Math. Soc., 361(7), 3829–3850, 2009.
  * [GuH06] J.-S. Guo and F. Hamel.  Front propagation for discrete periodic monostable equations.  Math. Ann., 335(3), 489–525, 2006.
  * [GuR18] P. Gurevich and S. Reichelt.  Pulses in FitzHugh–Nagumo systems with rapidly oscillating coefficients.  Multiscale Model. Simul., 16(2), 833–856, 2018.
  * [Hei01] S. Heinze.  Wave solutions to reaction-diffusion systems in perforated domains.  Zeits. Analysis Anwend., 20(3), 661–676, 2001.
  * [HuZ95] W. Hudson and B. Zinner.  Existence of traveling waves for reaction diffusion equations of Fisher type in periodic media.  In Boundary value problems for functional-differential equations, pages 187–199. World Sci. Publ., River Edge, NJ, 1995.
  * [McK70] H. McKean Jr. Nagumo’s equation.  Advances in Math., 4(3), 1970.
  * [MiT07] A. Mielke and A. M. Timofte.  Two-scale homogenization for evolutionary variational inequalities via the energetic formulation.  SIAM J. Math. Analysis, 39(2), 642–668, 2007.
  * [MRT14] A. Mielke, S. Reichelt, and M. Thomas.  Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion.  Networks Heterg. Materials, 9(2), 353–382, 2014.
  * [MSU07] K. Matthies, G. Schneider, and H. Uecker.  Exponential averaging for traveling wave solutions in rapidly varying periodic media.  Math. Nachr., 280(4), 408–422, 2007.
  * [NAY62] J. Nagumo, S. Arimoto, and S. Yoshizawa.  An active pulse transmission line simulating nerve axon.  Proc. Inst. Radio Eng., 50(10), 2061–2070, 1962.
  * [Ngu89] G. Nguetseng.  A general convergence result for a functional related to the theory of homogenization.  SIAM J. Math. Anal., 20(3), 608–623, 1989.
  * [Rei15] S. Reichelt.  Two-scale homogenization of systems of nonlinear parabolic equations.  PhD thesis, Institut für Mathematik, Humboldt-Universität zu Berlin, November 2015.
  * [WuZ01] J. Wu and X. Zou.  Traveling wave fronts of reaction-diffusion systems with delay.  J. Dynam. Differential Equations, 13(3), 651–687, 2001.
  * [Xin00] J. Xin.  Front propagation in heterogeneous media.  SIAM Review, 42(2), 161–230, 2000.