# Slow and fast minimal speed traveling waves of the FKPP equation with
chemotaxis

Christopher Henderson111The University of Arizona, Department of Mathematics,
Tucson, AZ, USA
E-mail address<EMAIL_ADDRESS>
###### Abstract

We examine a general model for the Fisher-KPP (FKPP) equation with nonlocal
advection. The main interpretation of this model is as describing a diffusing
and logistically growing population that is also influenced by intraspecific
attraction or repulsion. For a particular choice of parameters, this
specializes to the Keller-Segel-Fisher equation for chemotaxis. Our interest
is in the effect of chemotaxis on the speed of traveling waves. We prove that
there is a threshold such that, when interactions are weaker and more
localized than this, chemotaxis, despite being non-trivial, does not influence
the speed of traveling waves; that is, the minimal speed traveling wave has
speed 2 as in the FKPP case. On the other hand, when the interaction is
repulsive, we show that the minimal traveling wave speed is arbitrarily large
in a certain asymptotic regime in which the interaction strength and length
scale tend to infinity.

##  1 Introduction and main results

#### The general setting and the main goal

The model we consider is

$u_{t}+(vu)_{x}=u_{xx}+u(1-u),\qquad\text{ in }(0,+\infty)\times\mathbb{R},$
(1.1)

where, for $\chi\in\mathbb{R}$, $\sigma>0$, and $K\in L^{1}(\mathbb{R})\cap
L^{\infty}(\mathbb{R})$, we define

$v=\chi\,K_{\sigma}*u\qquad\text{ and }\qquad
K_{\sigma}=\frac{1}{\sigma}K\left(\frac{\cdot}{\sigma}\right).$ (1.2)

The unknown, $u$, typically represents the population density of a species
that is experiencing an intraspecific aggregation or dispersion effect such as
chemotaxis. For simplicity, we always refer to this nonlocal effect as
chemotaxis, although the model could be used in other settings. The parameters
$\chi$ and $\sigma$ encode the strength and length-scale of the chemotaxis,
respectively, while the sign of $\chi$ determines if the effect is attractive
($\chi>0$) or repulsive ($\chi<0$).

Before making our assumptions precise, we discuss the main goal of this
article. The equation (1.1) and similar reaction-diffusion equations model the
invasion of species into a new environment. When $\chi\equiv 0$, it is well-
known that this invasion occurs with speed two. More precisely, the minimal
speed traveling wave has speed two (we define traveling waves below). We aim
to understand the effect of chemotaxis on this invasion speed; that is, we aim
to answer the question: does the localized, nonlinear chemotaxis term alter
speed of the minimal speed traveling wave, and, if it does, when and how is
the speed changed? While some results exist, they are either partial or
require very specific structure of $K$. We discuss this at the conclusion of
this section.

#### Assumptions on $K$, $\chi$, and $\sigma$

Throughout the paper, $K$ is a bounded odd function that is increasing except
at $x=0$; that is, it is increasing in $(-\infty,0)$ and in $(0,+\infty)$. We
also assume that $K$ does not change sign in $(-\infty,0)$ and in
$(0,+\infty)$. Without loss of generality, we assume that

$\|K\|_{L^{1}(\mathbb{R})}=1\quad\text{ and }\quad 2|K(0^{+})|=-2K(0^{+})=1,$
(1.3)

where $K(0^{+})=\lim_{x\searrow 0}K(x)$. Defining $K(0^{-})$ analogously,
notice that $2|K(0^{+})|=|K(0^{+})-K(0^{-})|$; that is, $2|K(0^{+})|$ is the
magnitude of the “jump” of $K$ across the origin.

Finally, we assume that

$K=\overline{K}^{\prime}\qquad\text{ for some }\overline{K}\text{ such that
}(1+|x|)\overline{K}\in L^{1}(\mathbb{R}).$ (1.4)

We make this assumption, but note that this is not necessary for all results
below.

######  Remark 1.1.

We give some examples of admissible choices of $K$ here.

  1. (i)

$K(x)=-\frac{\operatorname{sign}(x)}{2}e^{-|x|}$. This is the kernel related
to the Keller-Segel model for chemotaxis; that is,
$v=(\chi/\sqrt{\sigma})\bar{v}_{x}$ for $\bar{v}$ satisfying
$-\sigma^{2}\bar{v}_{xx}+\bar{v}=u$. This has been studied extensively, see
the discussion below.

  2. (ii)

$K(x)=-\frac{\operatorname{sign}(x)}{2}\mathds{1}_{[-1,1]}(x)$.

  3. (iii)

$K(x)=-\frac{\operatorname{sign}(x)}{2}\left(1+\frac{|x|}{k-1}\right)^{-k}$
for $k>2$.

  4. (iv)

$K(x)=-\frac{\operatorname{sign}(x)}{2}e^{-\frac{|x|^{\alpha}}{D_{\alpha}}}$
for any $\alpha\in(0,1)$, where
$D_{\alpha}=\left(\int_{0}^{\infty}e^{-|x|^{\alpha}}\,dx\right)^{-\alpha}$.

The choice of $K$ reflects modeling assumptions on intraspecific interactions.
The specific choice in Remark 1.1.(i) is due to an assumption that the
chemical signal secreted by each individual diffuses (quickly) according the
heat equation. More broadly, (1.1) can be considered as a general model in
which individuals, depending on the sign of $\chi$, either repel or attract
one another. One example of this phenomenon is herding, although many examples
exist. To further illustrate this, we discuss the choice Remark 1.1.(ii). This
corresponds to individuals that interact only when they are distance $\sigma$
or closer and could be due to, e.g., limited vision. Roughly, when $\chi<0$,
they “feel a pull” towards one another with strength $O(|\chi|)$ with a
multiplicative factor related to the fraction of the population within
distance $\sigma$. When $\chi>0$, they experience a “push” instead. In
general, aggregation-diffusion models arise both as gradient flows of a free
energy involving an interaction kernel and as limits of microscopic models;
see, e.g., the results, discussions, and references in [CDFLS, JabinWang].

Our interest in this article is, roughy, in the cases $|\chi|,|\chi|/\sigma\ll
1$ and $-\chi\gg 1$. As such, we make the standing assumption that

$\chi,\frac{\chi}{\sigma}<\frac{1}{2}.$ (1.5)

This simplifies some estimates but does not play a fundamental role in our
analysis.

#### Traveling waves and preliminary observations

Our main objects of study are the traveling wave solutions of (1.1), which we
define below. In particular, we are interested in the effect of the chemotaxis
term $v$ on the speed of the traveling wave.

######  Definition 1.2.

A traveling wave solution is a pair $(c,u)$ such that (i) $0<u\in
C^{2}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$, (ii) $c\geq 0$, (iii)
$\widetilde{u}(t,x):=u(x-ct)$ solves (1.1), and (iv)

$\lim_{x\to-\infty}u(x)=1,\quad\text{ and }\quad\lim_{x\to\infty}u(x)=0.$
(1.6)

We refer to $c$ as the speed and $u$ as the profile. We say that a traveling
wave $(c^{*},u^{*})$ is a minimal speed traveling wave if $c^{*}\leq c$ for
all traveling waves $(c,u)$.

For similar reaction-diffusion equations, one sometimes assumes that $u(x)$ is
simply uniformly positive as $x\to-\infty$. In fact, the convergence to one of
$u$ at $-\infty$ plays almost no role in our analysis; however, it simplifies
the statements of some intermediate lemmas to use the definition above.

For any traveling wave $u$, notice that

$-cu_{x}+(vu)_{x}=u_{xx}+u(1-u).$ (1.7)

This equation will be the basis for our analysis.

Using standard techniques, we can show that traveling waves exist.

######  Proposition 1.3.

Under the assumptions above, there exists a wave with speed

$c\leq 2\sqrt{1+\frac{|\chi|}{\sigma}}+\frac{|\chi|}{2}\leq
2+\frac{|\chi|}{\sigma}+\frac{|\chi|}{2}.$

Moreover, if $(c,u)$ is any traveling wave solution of (1.7), then $c\geq 2$.

We note that the lower bound on the traveling wave speed, that is, that
traveling waves cannot have a slower speed than $2$, follows directly from
previous results such as [HamelHenderson, Theorem 1.1], but we provide a
slightly different proof here.

#### Slow traveling waves

The first main result of the article is that, under a smallness condition on
$\chi\sigma^{2}$ and $\chi/\sigma$, the speed of the minimal speed traveling
wave is exactly $2$, which is the same as in the case where chemotaxis is not
present ($\chi=0$).

######  Theorem 1.4.

Under the assumptions above, there exists $\varepsilon_{0}>0$ such that, if
$|\chi|(\sigma^{-1}+\sigma^{2})<\varepsilon_{0}$, then there exists a
traveling wave solution $(c,u)$ of (1.1) such that $c=2$.

In view of the discussion of Proposition 1.3, the thrust of Theorem 1.4 is to
establish the upper bound on $c$. The standard method of establishing an upper
bound on the speed is to build supersolutions of $u$. There are two main
obstructions to this, both stemming from the fact that $v$ depends nonlocally
on $u$. First, (1.7) does not enjoy the maximum principle. Second, the
standard arguments use strongly imposed structure on the advection, such as
periodicity, which $v$ does not have.

In some cases, such difficulties can be sidestepped by ad hoc methods;
however, to our knowledge, such proofs use strongly the structure of the
advection [SalakoShenXue, ConstantinKiselevRyzhik]. These are discussed below
in the discussion of the history of this problem, and, in that discussion, we
describe why their methods cannot generalize to our setting.

We now describe the main idea of the proof of Theorem 1.4. We work on the
level of the “slab problem” on $[-a,a]$ for $a\gg 1$, as is typical in the
construction of traveling waves. As described above, we need only establish an
upper bound. To this end, we treat $v$ as a given function and construct a
supersolution to the linear equation

$-c\overline{u}_{x}+(v\overline{u})_{x}=\overline{u}_{xx}+\overline{u}(1-u).$

By removing an integrating factor involving $v$, we reduce this problem to
studying the principal eigenfunction and eigenvalue of

$-\varphi^{a}_{xx}-V\varphi^{a}=\lambda\varphi^{a},\qquad\text{where
}V=-u-(c-2)\left(1+\frac{c-2}{4}\right)+v\left(\frac{2c-v}{4}\right)+\frac{v_{x}}{2},$
(1.8)

on $[-a,a]$ with periodic boundary conditions. A significant advantage of
(1.8) is that it is self-adjoint and, thus, one may analyze $\lambda$ through
the Rayleigh quotient. There are two main steps in the analysis.

First, we establish the nonnegativity of $\lambda$. Heuristically, this occurs
because, due to the $u$ term, $V$ is negative everywhere except far to the
right, where it can take, at most, a small positive value (and that can only
occur if $c>2$); hence, a potential eigenfunction trying to “minimize” its
eigenvalue, must be large only on the far right. On the other hand, it must be
periodic, meaning that it will be large on the far left where $V$ is very
negative. We are unable to make this intuition rigorous. Instead, in order to
establish this rigorously, we develop a pair of functional inequalities
(3.17)-(3.18) allowing us to show that, if $c>2$,

$\int(\varphi_{x}^{a})^{2}-\int V(\varphi^{a})^{2}dx\geq 0,$

when $a$ is sufficiently large. This, after multiplying (1.8) by $\varphi^{a}$
and integrating by parts, yields the nonnegativity of $\lambda$.

This brings us to the second step. The fact that $\lambda\geq 0$ allows us to
construct a supersolution of (1.7) via $\varphi^{a}$. This supersolution is
only useful if we have good pointwise bounds on $\varphi^{a}$ that hold for
all $a$. We control the size of $\varphi^{a}$ by constructing barriers and
using the regularity imparted by (1.8). We use this to show that, if $c>2$,
$u(0)$ must be exponentially small in the parameter $a$, which contradicts the
fact that we have constructed $u$ to take a fixed positive value at $x=0$.

Theorem 1.4 is proved in Section 3.3.

#### Fast traveling waves

In view of Proposition 1.3 and Theorem 1.4, it is natural to wonder if
chemotaxis can speed up traveling waves. Our second main result establishes
this.

######  Theorem 1.5.

Fix any traveling wave solution $(c,u)$ of (1.1) in the sense of Definition
1.2. For any $\varepsilon\in(0,1)$, there exists $A_{\varepsilon}$ such that

$c\geq(1-\varepsilon)\frac{|\chi|}{2}$

whenever

$\sigma>A_{\varepsilon}\quad\text{ and
}\quad\frac{\sigma}{-\chi}>A_{\varepsilon}.$

The constant $A_{\varepsilon}$ depends only on $\varepsilon$ and $K$.

Theorem 1.5 is essentially sharp when $1\ll-\chi\ll\sigma$. Indeed, the lower
bound in Theorem 1.5 and the upper bound in Proposition 1.3 match up to
$o(\chi)$. Roughly, this shows that the minimal speed traveling wave has speed
$|\chi|/2+o(\chi)$.

We now discuss the idea of the proof. Roughly, the wider the front222Here, we
loosely think of the front as the region in which $u\not\approx 1,0$. is in a
reaction-diffusion problem, the faster the wave should propagate. Intuitively,
this is due to a wider region in which the “reaction” (in this case, the
population reproducing) occurs. Mathematically, one can see this by
integrating (1.7) to obtain

$c=\int_{-\infty}^{\infty}u(1-u)\,dx,$

and noticing that $u(1-u)$ is $\approx 0$ away from the front. On the other
hand, if the front is narrow, one can show that $v\approx|\chi|/2$ everywhere
“near” the front. To see this, notice that, as $\sigma\gg 1$, roughly half the
convolution occurs where $u\approx 1$ while the other half the convolution
occurs where $u\approx 0$. In this case, the wave is “pushed” by $v$ at the
speed $\approx|\chi|/2$. The proof proceeds by quantifying this intuition in a
careful way.

In contrast to the usual FKPP equation, in which fronts are pulled (that is,
the speed is determined by the behavior far to the right where $u\approx 0$),
the intuition above shows that fronts are pushed when $-\chi>2$ and $\sigma$
is sufficiently large. Thus, the combination of Theorems 1.4 and 1.5 shows a
transition from pulled to pushed. To our knowledge the only similar setting in
which this is observed is the Burgers-FKPP equation considered in
[BramburgerHenderson], discussed below. In both cases, a “threshold” exists
for certain parameters under which the advection has no effect on the speed
(although the front profile is changed) and over which speed-up occurs.

#### History of the problem, related work, and further discussion of main
difficulties

The general model considered here was introduced by Hamel and the present
author in [HamelHenderson]. There, we considered instead the Cauchy problem,
where $u$ solves (1.1) with initial data $u_{0}$ that is compactly supported
(at least on the right), and showed that spreading speed is at least $2$. This
lower bound, which applies quite generally to a collection of models,
immediately yields a lower bound for the speed of traveling waves (see
Proposition 1.3). Additionally, we proved lower bounds on superlinear
propagation in the case when $\chi<0$ and $K\notin L^{1}$.

The majority of work to this point has been on the Keller-Segel model (see
Remark 1.1.(i)). While there is a large body of work regarding well-posedness
on finite domains (see, e.g., [TelloWinkler] and the literature citing it) and
on propagation for the Cauchy problem, we focus our discussion here to those
works investigating traveling waves and other propagation phenomena. To our
knowledge, the first work in this direction was due to Nadin, Perthame, and
Ryzhik [NadinPerthameRyzhik], who showed constructed traveling waves with
(unknown) speed between $2$ and $2+\chi/(1-\chi/\sigma)$, in our notation,
under the assumption that $0\leq\chi<\sqrt{\sigma}\min\\{1,\sqrt{\sigma}\\}$.
This was later upgraded to a sharp result by Salako, Shen, and Xue
[SalakoShenXue] under the additional assumptions333The model considered by
Salako, Shen, and Xue has parameters $\chi$, $a$, $b$, $\lambda$, and $\mu$;
however, these can be reduced to the two parameters $\chi$ and $\sigma$ used
above by scaling. that $\sigma<1$ and

$0\leq\left(1+\frac{\sigma^{1/4}-1}{2(\sigma^{1/4}+1)}\right)\frac{\chi}{\sigma}\leq
1.$

Their proof is based on the ad hoc construction of sub- and supersolutions and
uses strongly the structure of Keller-Segel model. A key point in the argument
is that $\overline{v}$ (see Remark 1.1) decays exponentially as $x\to\infty$.
In our setting, this is only possible if $K$ decays at least exponentially as
$x\to\infty$; hence, a new argument is needed in order to include, e.g., cases
(iii) and (iv) in Remark 1.1 among others. In some sense, their argument does
not illuminate the reason for the minimal speed traveling waves to be that of
the Fisher-KPP equation. This is a partial motivation for Theorem 1.4.
Specifically, our goal here is to develop general tools for proving and
increase our understanding of the lack of speed-up by advection.

We briefly make note of another branch of research investigating the coupling
of the Fisher-KPP equation with the parabolic Keller-Segel (as opposed to the
elliptic model discussed above). Here Salako and Shen [SalakoShenPP] have
established the minimal speed as 2; see also a very elegant, simple dynamical
systems argument of Bramburger [Bramburger]. While this setting is technically
not within the framework of the current manuscript, we believe the arguments
used here could be extended to such a case. Indeed, Salako and Shen’s argument
relies heavily on the representation formula of solution of the parabolic
Keller-Segel as a convolution in space and time. Finally, we mention that
there is an ongoing program in understanding the effect of different choices
of diffusion (see, e.g., [JWXY] and references therein).

We also mention the work of Calvez [Calvez], establishing traveling wave
solutions for a mesoscopic (kinetic) model for chemotaxis not involving the
logistic growth term $u(1-u)$. Like the traveling waves in Theorem 1.5, those
constructed by Calvez are driven by intraspecific interactions of the
population. The differences between our model and that of Calvez are quite
extreme and so the results, while quite similar in spirit, have very little
connection on a technical level.

From a broader perspective, the work here contributes to the ongoing
investigation of the influence of advection on front propagation in reaction-
diffusion equations. This is the other main motivation for this work. When the
advection is imposed, that is, independent of $u$, and has specific structure,
implicit formulas exist [BerestyckiHamel, Xin_Book, MajdaSouganidis,
EvansSouganidis]. These are often difficult to quantify, and, to our
knowledge, the only precise results are in certain asymptotic regimes such as
when the advection is large [ElSmailyKirsch, HamelZlatos, NolenXin1,
RyzhikZlatos, Zlatos] or small [NolenXin1, NolenXin5, HendersonSouganidis] or
the diffusion is small [ElSmaily]. We mention also the results for large
cellular flows with Hamilton-Jacobi models by Xin and Yu [XinYuG, XinYuHJ] and
the ABC flow by Xin, Yu, and Zlatos [XinYuZlatos].

There are two major differences between the current setting and the results
discussed in the previous paragraph: there the advection is imposed from the
outside and “uniform” over the spatial domain, whereas the advection here is
influenced by $u$ and is spatially localized near the front. In fact, those
two features are shared by many physically relevant settings (see also
[BerestyckiConstantinRyzhik, ConstantinKiselevRyzhik, CRRV] and the literature
citing it). Intimately tied to these features are two of the main
difficulties: (1) (1.1) does not enjoy a comparison principle, meaning that
techniques used in the settings described in the previous paragraph are not
available, and (2) the “advection” $v$, in our setting, does not enjoy the
specific structure that allowed precise calculations as in the works discussed
in the previous paragraph. To our knowledge, there are a dearth of results
like Theorem 1.5 in which precise estimates are established for nontrivial,
nonlinear effects coming from the advection. We note that similar nonlinear
effects on the propagation have been seen when the logistic term involves a
nonlocal, nonlinear term [ABBP, BHR_nonlocal, CHMTD, Penington], although the
effects and underlying mathematical analysis are quite different.

Above we mentioned [ConstantinKiselevRyzhik] briefly, but this demands further
discussion. In this work, Constantin, Kiselev, and Ryzhik study a system
composed of the Fisher-KPP equation for a quantity $T$, representing the
temperature of a fluid undergoing combustion, coupled to a fluid equation such
as the Navier-Stokes equation for a quantity $v$, representing the fluid
velocity, with the Boussinesq approximation involving $T$ in a two dimensional
cylinder. This models the situation in which a buoyancy force arising from
changes in $T$ induces a velocity field. Heuristically, this setting is quite
similar to the one considered in the current manuscript as it may be seen as a
nonlocal coupling of the solution of a reaction-diffusion equation with its
advection. Constantin, Kiselev, and Ryzhik show that, under a smallness
condition on the width of the cylinder and the strength of the coupling, the
only traveling waves are speed 2 planar waves (it should be noted that non-
planar waves exist in a variety of settings [BerestyckiConstantinRyzhik,
ConstantinLewickaRyzhik, Henderson_Boussinesq, Lewicka, LewickaMucha,
TexierPicardVolpert]). This is quite similar to our Theorem 1.4. Their simple,
elegant proof, however, hinges on the application of a Poincaré inequality on
the cross-section of the cylinder (hence, the importance of the smallness of
the cylinder) and is thus not applicable to the current setting. A substantial
difference, though, is that their proof hinges on showing that $v\equiv 0$,
while, in our setting, $v\not\equiv 0$ and, thus, the goal is to show a
trivial effect on the speed of a non-trivial advection.

One simplified, one-dimensional version of the reactive Boussinesq equation
discussed above is one in which the equation for the advection is replaced by
a forced Burgers equation [CRRV, BramburgerHenderson]. In particular, in
[BramburgerHenderson], Bramburger and the present author show a similar
threshold: depending on a parameter representing the strength of gravity
$\rho$, the minimum wave speed is $2$ if $\rho$ is below a certain critical
value and grows like $\mathcal{O}(\rho^{1/3})$ otherwise. Unfortunately, the
upper and lower bounds, while explicit, match only in their order; precisely,
we show a lower bound of $(3/2)^{1/3}\rho^{1/3}$ and an upper bound of
$\sqrt{3}\rho^{1/3}$ as $\rho\to\infty$. The proofs in [BramburgerHenderson]
are based on transforming the Burgers-FKPP system to a system of ODE and
constructing trapping regions. This approach is not possible in the present
setting as (1.7) cannot be converted into a system of ODE.

Finally, a recent thread of interesting work has begun on a related hyperbolic
model. Fu, Griette, and Magal [FuGrietteMagalWave] considered an inviscid
version of the Keller-Segel-Fisher equation with repulsive chemotaxis; that
is, (1.1) without a Laplacian, with the particular choice of $K$ given by
Remark 1.1.(i), and with $\chi<0$. To our knowledge, this work and the one of
Hamel and the present author [HamelHenderson] are the only works to notice the
effect of repulsive chemotaxis on propagation.

In [FuGrietteMagalWave], Fu, Griette, and Magal construct a discontinuous
traveling wave with speed of propagation
$c\in(|\chi|/(2+|\chi|/\sigma),|\chi|/2)$, in our choice of parameters, under
the assumption that $\chi/\sigma$ is positive and not too big; however, they
do not obtain a general lower bound on the speed for any traveling wave as we
do for the model considered here. We note that their bounds match ours in the
regime $1\ll\chi\ll\sigma$, which is not surprising because, as our proof
illuminates, propagation is entirely driven by the advection in this
asymptotic regime; that is, diffusion does not contribute to the propagation.
Their proof is based on the size of the jump discontinuity in the constructed
wave and does not apply to the present setting. We note that an analogous
threshold result for the asymptotic regime where
$|\chi|(\sigma^{-1}+\sigma^{2})\ll 1$ as in Theorem 1.4 is not possible in
their setting as it is a consequence of the interplay between chemotaxis and
diffusion. Fu, Griette, and Magal additionally investigate the well-posedness
of and convergence to $1$ of $u$ for the Cauchy problem. We mention also their
earlier works [FuGrietteMagal1, FuGrietteMagal2]. Recently, Griette, Magal,
and Zhao were able to construct continuous waves for the inviscid model as
well [GrietteMagalZhao].

We briefly mention the work of Crooks as well as that of Crooks and Mascia on
a class of models that depend nonlinearly on both $u$ and $u_{x}$, although
locally [Crooks, CrooksMascia]. Under some assumptions, they are able to
analyze the speed of the minimal wave in a precise way.

#### Notation

Throughout the paper, we use $C$ to denote a positive universal constant that
may change line-by-line. A universal constant is one that does not depend on
$\chi$, $\sigma$, or $\tau$. When constants have a dependence on one of these
quantities, we denote it with a subscript; for example, $C_{\chi}$ is a
constant that depends on $\chi$ but not on $\sigma$ or $\tau$.

When no confusion will arise, when using $L^{p}$ or other norms, we suppress
the dependence on the domain; for example, we write $\|u\|_{L^{\infty}}$
instead of $\|u\|_{L^{\infty}(\mathbb{R})}$.

#### Acknowledgements

The author is grateful to Alexander Kiselev and Francois Hamel for helpful
discussions early in the development of this work at the Banff workshop
“Interacting Particle Systems and Parabolic PDEs.” This work was partially
supported by NSF grants DMS-2003110 and DMS-2204615.

##  2 Preliminaries

We collect here some preliminary results on the size of $u$ and $v$ as well as
the monotonicity of $u$. These are used in the proof of Theorem 1.5. Since
analogous results (with exactly the same proof) hold for the slab problem used
in the proof of Theorem 1.4, we state and prove them here and omit the proofs
for the slab problem in the sequel.

######  Lemma 2.1 (Upper bound on $u$).

For any $c$, we have
$\|u\|_{L^{\infty}}\leq\max\left\\{1,\left(1-\chi/\sigma\right)^{-1}\right\\}$.

This lemma is a simple consequence of the maximum principle, using that
$\|v\|_{L^{\infty}}\leq\|u\|_{L^{\infty}}$ and that, for any $x$,

$v_{x}(x)=-\frac{\chi}{\sigma}u(x)+\chi\int_{0}^{\infty}(K_{\sigma})_{x}(y)(u(x-y)+u(x+y))\,dy.$
(2.1)

Hence, we omit the proof. We note that, in the above, there is a slight abuse
of notation: we do not make any assumptions on the regularity of $K_{\sigma}$,
so $(K_{\sigma})_{x}$ should be understood in the sense of Radon
measures444That is, $\mu=dK_{\sigma}$ is the measure on $[0,\infty)$ such that
$\mu((a,b))=K_{\sigma}(b)-K_{\sigma}(a)$ for any $0\leq a<b$..

We observe some preliminary bounds on $v$ and $v_{x}$ that are deduced
directly from the definition of $K$ and the maximum principle:

######  Lemma 2.2.

For any $c$, we have

$\begin{split}&\|v\|_{L^{\infty}}=|\chi|\,\|K_{\sigma}*u\|_{L^{\infty}}\leq\frac{|\chi|}{2}\|u\|_{L^{\infty}}\qquad\text{
and }\\\
&\|v_{x}\|_{L^{\infty}}=|\chi|\,\|(K_{\sigma}*u)_{x}\|_{L^{\infty}}\leq\frac{|\chi|}{\sigma}\,\|u\|_{L^{\infty}}.\end{split}$

Both inequalities in Lemma 2.2 follow from Young’s inequality for
convolutions. The proof of the first inequality uses explicitly the
cancellations due to the positivity properties of $K$ and $u$ in order to pick
up the extra factor of $1/2$. We omit the proof.

A useful tool is that $u$ is monotonic when it is small.

######  Lemma 2.3 (Monotonicity of $u$).

Suppose that $u$ is a traveling wave solution of (1.1) in the sense of
Definition 1.2. Then $u_{x}\leq 0$ whenever:

$\begin{split}&u<\frac{1}{1-\frac{\chi}{2\sigma}}=\frac{1}{1+\frac{|\chi|}{2\sigma}}\quad\text{if}\quad\chi\leq
0\qquad\text{ or }\qquad
u<\frac{1-\frac{2\chi}{\sigma}}{\left(1-\frac{\chi}{\sigma}\right)^{2}}\quad\text{if}\quad\chi>0.\end{split}$

###### Proof.

Fix any $\varepsilon>0$. Let $\overline{u}_{\rm mon}$ be
$(1-\chi/2\sigma)^{-1}$ if $\chi\leq 0$ and
$(1-2\chi/\sigma)/(1-\chi/\sigma)^{2}$ if $\chi>0$. We argue by contradiction,
assuming that there exists $x_{0}$ such that $u(x_{0})\leq\overline{u}_{\rm
mon}-\varepsilon$ and $u_{x}(x_{0})>0$. Define

$y_{0}=\min\left\\{y:u(y)\text{ is a local minimum and
}u(y)\leq\overline{u}_{\rm mon}-\varepsilon\right\\}.$

Since $u\to 1$ as $x\to-\infty$ and due to the existence of $x_{0}$, it
follows that $y_{0}$ is well-defined. We point out that $u(x)\geq u(y_{0})$
for all $x<y_{0}$ (if this were not true, we could find another point in the
set defining $y_{0}$ that is smaller than $y_{0}$).

We re-write (1.7) as

$u_{xx}-\left(v-c\right)u_{x}=-u\left(1-u-v_{x}\right).$

We claim that $1-u-v_{x}>0$ at $y_{0}$. If this were true, then, since $y_{0}$
is the location of a local minimum, we find

$0\leq u_{xx}-\left(v-c\right)u_{x}=-u\left(1-u-v_{x}\right)<0,$

which is a contradiction. Hence, due to this and the arbitrariness of
$\varepsilon$, the proof is finished if we can establish the positivity of
$1-u-v_{x}$.

We now establish this inequality. Using (2.1), we notice that, at $y_{0}$

$\begin{split}1-u-v_{x}&=1-u\left(1-\frac{\chi}{\sigma}\right)-\chi\int_{0}^{\infty}(K_{\sigma})_{y}(y)(u(y_{0}-y)+u(y_{0}+y))\,dy.\end{split}$
(2.2)

When $\chi\leq 0$, $-\chi(K_{\sigma})_{x}\geq 0$. Hence, using the choice of
$y_{0}$, we find

$\begin{split}-\chi\int_{0}^{\infty}(K_{\sigma})_{y}(y)(u(y_{0}-y)+u(y_{0}+y))\,dy&\geq-\chi\int_{0}^{\infty}(K_{\sigma})_{y}(y)u(y_{0}-y)\,dy\\\
&\geq-\chi
u(y_{0})\int_{0}^{\infty}(K_{\sigma})_{y}(y)\,dy=\frac{|\chi|}{2\sigma}u(y_{0}).\end{split}$

Using this along with the definition of $\overline{u}_{\rm mon}$ in (2.2), we
find, at $y_{0}$,

$1-u-v_{x}\geq 1-u\Big{(}1+\frac{|\chi|}{2\sigma}\Big{)}\geq
1-(\overline{u}_{\rm
mon}-\varepsilon)\Big{(}1+\frac{|\chi|}{2\sigma}\Big{)}>0.$

The proof is thus finished in this case.

When $\chi>0$, $-\chi(K_{\sigma})_{y}\leq 0$. Hence, by Lemma 2.1,

$-\chi\int_{0}^{\infty}(K_{\sigma})_{y}(y)(u(x-y)+u(x+y))\,dy\geq-2\|u\|_{L^{\infty}}\chi\int_{0}^{\infty}(K_{\sigma})_{y}(y)\,dy\geq-\frac{\chi}{\sigma}\frac{1}{1-\frac{\chi}{\sigma}}.$

Including this inequality in (2.2) and using the definition of
$\overline{u}_{\rm mon}$, we find

$1-u(x)-v_{x}(x)>1-(\overline{u}_{\rm
mon}-\varepsilon)\left(1-\frac{\chi}{\sigma}\right)-\frac{\chi}{\sigma}\frac{1}{1-\frac{\chi}{\sigma}}>0.$

In both cases, we have established that $1-u-v_{x}>0$, which completes the
proof. ∎

Finally we recall that $u$ converges to $1$ on the left when $\chi/\sigma$ is
appropriately bounded above. We cite [HamelHenderson] here as it works out-of-
the-box; however, the original result is [SalakoShenPE, Theorem C], which is
stated only for the Keller-Segel model but whose proof works analogously in
the more general case. We note that the proof of the result below requires
(1.5) and it may not be true without this assumption.

######  Lemma 2.4.

[HamelHenderson, Corollary 2.3] Suppose that $(c,u)$ is a traveling wave
solution of (1.1) in the sense that $\lim_{x\to\infty}u(x)=0$, $c>0$, and
$(c,u)$ satisfies (1.7). Then $\lim_{x\to-\infty}u(x)=1$.

##  3 Speed 2 traveling waves: Proposition 1.3 and Theorem 1.4

In this section, we construct traveling waves whose speed is unperturbed off
the case $\chi=0$ when $\chi(\sigma^{2}+\sigma^{-1})$ is sufficiently small.
We begin by constructing traveling waves in the usual fashion; that is,
building solutions to the problem on a finite domain (a “slab”) via the Leray-
Schauder fixed point theorem and then taking a limit as this approximating
domain tends to $\mathbb{R}$. This is the proof of Proposition 1.3.

After, we return to this slab problem to analyze the bounds on $c$, from which
we obtain the desired upper bound. This is the proof of Theorem 1.4. We
separate the precise bounds into another step for clarity since its proof is
quite technical and is independent of the rest of the construction.

###  3.1 The problem on a finite slab

Let $a>0$, $\tau\in[0,1]$, and $\theta\in(0,\theta_{0})$ for

$\theta_{0}<\min\Big{\\{}\frac{1}{100},\frac{1-\frac{2|\chi|}{\sigma}}{1+\frac{|\chi|}{\sigma}}\Big{\\}}$
(3.1)

that is chosen in the sequel. Throughout this section $\theta_{0}$ is a
constant decreased in order to encode the various statements “for $\theta$
sufficiently small” below (the notation is introduced in case the upper bound
$\theta_{0}$ of $\theta$ must be referenced). The parameter $\tau$ is for the
fixed point theorem referenced above.

Consider

$\begin{cases}-cu_{x}+\tau(\widetilde{v}u)_{x}=u_{xx}+u(1-u),\qquad&\text{ in
}(-a,a),\\\ u(-a)=1,\quad u(a)=0,\quad\max_{x\geq 0}u(x)=\theta,\end{cases}$
(3.2)

where we define $\widetilde{v}=\chi K_{\sigma}*\widetilde{u}$ with

$\widetilde{u}(x)=\begin{cases}1,\quad&\text{ if }x\leq-a,\\\ u(x),&\text{ if
}x\in(-a,a),\\\ 0,&\text{ if }x\geq a.\end{cases}$ (3.3)

It is apparent that were $u$ and $c$ to exist, both would depend on $a$ and
$\theta$. We suppress that dependence in the notation in this subsection. In
the sequel, however, we denote $u=u^{a}$ and $c=c^{a}$ in order to clarify the
presentation of the limit $a\to\infty$.

Our goal is to produce a solution to $(c,u)$ to (3.2) when $a$ is sufficiently
large and $\theta$ is sufficiently small. The first step is obtaining a priori
estimates on any solution $(c,u)$ of (3.2).

####  3.1.1 Bounds on $u$

It is clear that the maximum principle yields the same upper bound of $u$ in
Lemma 2.1 and the same monotonicity properties as Lemma 2.3. We collect this
in the following (recall (3.1)):

######  Lemma 3.1 (Upper bound on and monotonicity of $u$).

For all $a>0$ and $\theta\in(0,\theta_{0})$, we have

$\|u\|_{L^{\infty}}\leq\max\left\\{1,\left(1-\chi/\sigma\right)^{-1}\right\\}\qquad\text{and}\qquad
u_{x}\leq 0\ \text{ on }[0,a].$

In addition, a simple computation using the definition of $\widetilde{v}$
yields

$\|\widetilde{v}\|_{L^{\infty}}\leq\frac{|\chi|}{2}\max\\{1,(1-\chi/\sigma)^{-1}\\}\quad\text{
and
}\quad\|\widetilde{v}_{x}\|_{L^{\infty}}\leq\frac{|\chi|}{\sigma}\max\\{1,(1-\chi/\sigma)^{-1}\\}.$
(3.4)

See Lemma 2.2 for similar bounds.

Next we get a preliminary, suboptimal bound on the front speed that is
sufficient for the fixed point argument.

######  Lemma 3.2 (Upper bound on the speed).

There exists $\theta_{0}>0$ and $C>0$ such that if $\theta\in(0,\theta_{0})$
and $a$ is sufficiently large, depending only on $\theta$, then

$c\leq 2\sqrt{1+\frac{\tau|\chi|}{\sigma}}+\frac{\tau|\chi|}{2}\leq
2+\frac{\tau|\chi|}{\sigma}+\frac{\tau|\chi|}{2}.$

###### Proof.

Let $\psi_{A}(x)=Ae^{-\lambda x}$ for $\lambda>0$ to be chosen. Let
$A_{0}=\min\\{A\in\mathbb{R}:\psi_{A}\geq u\\}.$ First, notice that this is
well-defined since $A=e^{\lambda a}\|u\|_{L^{\infty}}$ is in the set, while no
negative $A$ are in the set due to the nonnegativity of $u$. In fact,
$A_{0}\geq\theta>0$ since

$u(0)=\theta\quad\text{ and }\quad\psi_{A}(0)=A.$

Finally, by continuity, there exists a point $x_{0}\in[-a,a]$ such that
$\psi_{A_{0}}(x_{0})=u(x_{0})$.

Since $\psi_{A_{0}}>0$ and $u(a)=0$, then $x_{0}\in[-a,a)$. In addition, if
$x_{0}=-a$, then $1=u(-a)=\psi_{A_{0}}(-a)=A_{0}e^{\lambda a}$. Hence,

$\theta=u(0)\leq\psi_{A_{0}}(0)=A_{0}=e^{-\lambda a}.$

This cannot happen if $a>\lambda^{-1}\log(1/\theta)$.

Thus, we have that $x_{0}\in(-a,a)$. Then $\psi_{A_{0}}-u$ has an interior
minimum at $x_{0}$ of zero, which yields

$u(x_{0})=\psi_{A_{0}}(x_{0}),\quad\psi_{A_{0},x}(x_{0})=u_{x}(x_{0}),\quad\text{
and }\quad\psi_{A_{0},xx}(x_{0})\geq u_{xx}(x_{0}).$

Using (3.2), these identities, (3.4), and (1.5), we find, at $x_{0}$,

$\begin{split}0&=u_{xx}+u(1-u)+cu_{x}-\tau(\widetilde{v}u)_{x}\leq\psi_{A_{0}}\left(\lambda^{2}+1-c\lambda+\lambda\tau\widetilde{v}-\tau\widetilde{v}_{x}\right)\\\
&\leq\psi_{A_{0}}\left(\lambda^{2}+1-c\lambda+\frac{\tau|\chi|\lambda}{2}+\tau\frac{|\chi|}{\sigma}\right).\end{split}$

We now choose $\lambda=(c-\tau|\chi|/2)/2$ to find, using the positivity of
$\psi_{A_{0}}$,

$\begin{split}0&\leq-\frac{(c-\tau|\chi|/2)^{2}}{4}+1+\tau\frac{|\chi|}{\sigma}.\end{split}$

This implies that $c\leq\tau|\chi|/2+2\sqrt{1+\tau|\chi|/\sigma}$, which,
after an application of Taylor’s theorem, concludes the proof. ∎

We now know that all coefficients in our equation are bounded in $L^{\infty}$,
which allows us the use of the Harnack inequality. We use this to derive an
estimate quantifying the fact that if $u$ is “small” somewhere then it is
small “nearby.”

######  Lemma 3.3 (Upper bound on spatial growth).

There exists a constant $C_{\chi,\chi/\sigma}$ such that

$u(x)\leq C_{\chi,\chi/\sigma}u(y)e^{C_{\chi,\chi/\sigma}|x-y|}\qquad\text{
for all }x,y\in[-a+1,a-1].$

###### Proof.

This follows by a simple iteration of the Harnack inequality $n\approx|x-y|$
times. ∎

######  Lemma 3.4 (Lower bound on the speed).

For all $\varepsilon>0$, there exists $\theta_{\varepsilon}$ and
$a_{\varepsilon}$ such that if $\theta\leq\theta_{\varepsilon}$ and
$a>a_{\varepsilon}$ then $c\geq 2-\varepsilon$.

###### Proof.

Here we simply construct a subsolution that “pushes” $u$ to speed $2$. We
argue by contradiction. Suppose that $\varepsilon>0$ and $c<2-\varepsilon$.

For any $A,\lambda>0$ and $R\in(0,a-1]$ to be chosen, let

$\underline{u}_{A}(x)=\frac{1}{A}e^{-\lambda
x}\cos\left(\frac{\pi}{2}\frac{2x-R}{R}\right)\qquad\text{ for
}x\in\left[0,R\right].$

Since $u>0$ on $[0,R]$, $\underline{u}_{A}<u$ on $[0,R]$ if $A$ is
sufficiently large. Let

$A_{0}=\inf\left\\{A>0:\underline{u}_{A}<u\text{ on
}\left[0,R\right]\right\\}.$

It is clear that $A_{0}$ is well-defined and that, by continuity, there exists
$x_{\text{\rm\tiny touch}}\in[0,R]$ such that
$\underline{u}_{A_{0}}(x_{\text{\rm\tiny touch}})=u(x_{\text{\rm\tiny
touch}})$. Since $u(0),u(R)>0$ and
$\underline{u}_{A_{0}}(0)=\underline{u}_{A_{0}}(R)=0$, we find
$x_{\text{\rm\tiny touch}}\in(0,R)$. Finally, by construction
$u-\underline{u}_{A_{0}}\geq 0$ on $[0,R]$ and has a minimum over $[0,R]$ of
$0$ at $x_{\text{\rm\tiny touch}}$.

For notational ease, we drop the $A_{0}$ notation and denote
$\underline{u}_{A_{0}}=\underline{u}$. Since $u-\underline{u}$ has a minimum
of $0$ at $x_{\text{\rm\tiny touch}}$, we find

$u_{x}(x_{\text{\rm\tiny touch}})=\underline{u}_{x}(x_{\text{\rm\tiny
touch}})\qquad\text{ and }\qquad u_{xx}(x_{\text{\rm\tiny
touch}})\geq\underline{u}_{xx}(x_{\text{\rm\tiny touch}}).$

Using these along with (3.2), we find, at $x_{\text{\rm\tiny touch}}$,

$\begin{split}0&\leq(u-\underline{u})_{xx}=-cu_{x}+\tau\left(\widetilde{v}u\right)_{x}-u(1-u)-\underline{u}_{xx}\\\
&=-c\underline{u}_{x}+\tau\widetilde{v}_{x}\underline{u}+\tau\widetilde{v}\underline{u}_{x}-\underline{u}(1-u)-\underline{u}_{xx}.\end{split}$
(3.5)

Recall that $u$ is decreasing on $[0,a]$, and, hence,
$\underline{u}_{x}(x_{\text{\rm\tiny touch}})=u_{x}(x_{\text{\rm\tiny
touch}})\leq 0$. In addition, $u(x_{\text{\rm\tiny touch}})\leq u(0)=\theta$.

We first estimate $\widetilde{v}$ and $\widetilde{v}_{x}$. Fix $L>0$ to be
chosen. We claim that there is a constant $C_{\chi,\sigma}$ such that

$|\widetilde{v}_{x}(x_{\text{\rm\tiny
touch}})|,|\widetilde{v}(x_{\text{\rm\tiny touch}})|\leq
C_{\chi,\sigma}K_{\sigma}(L)+C_{\chi,\sigma}\int_{L}^{\infty}|K_{\sigma}(x)|dx+C_{\chi,\sigma}\theta
e^{C_{\chi,\sigma}L}.$ (3.6)

We show the argument only for $\widetilde{v}$. The argument is similar for
$\widetilde{v}_{x}$, though additionally using (2.1). Indeed, decomposing the
integral and then using Lemma 3.3 on the close-to-$x_{\rm touch}$ portion, we
find

$\begin{split}|\widetilde{v}(x_{\rm
touch})|&\leq\int_{-L}^{L}|K_{\sigma}(y)|u(x_{\rm
touch}-y)dy+\left(\int_{-\infty}^{-L}+\int_{L}^{\infty}\right)|K_{\sigma}(y)|u(x_{\rm
touch}-y)dy\\\ &\leq C_{\chi,\chi/\sigma}\int_{-L}^{L}|K_{\sigma}(y)|u(x_{\rm
touch})e^{C_{\chi,\chi/\sigma}|y|}dy+\|u\|_{L^{\infty}}\left(\int_{-\infty}^{-L}+\int_{L}^{\infty}\right)|K_{\sigma}(y)|dy.\end{split}$

Then (3.6) is obtained by using the following: $K_{\sigma}$ is bounded and
odd, $u(x_{\rm touch})\leq\theta$, and $\|u\|_{L^{\infty}}$ is bounded due to
Lemma 3.1.

Returning to (3.6), choosing $L$ sufficiently large and $\theta$ sufficiently
small, we have $|\widetilde{v}_{x}(x_{\text{\rm\tiny
touch}})|,|\widetilde{v}(x_{\text{\rm\tiny touch}})|<\varepsilon^{4}$. Using
this, as well as the facts that $\tau\leq 1$, $c<2-\varepsilon$,
$\underline{u}_{x}\leq 0$, and $u\leq\theta$, we find

$0\leq-\left(2-\varepsilon+\varepsilon^{4}\right)\underline{u}_{x}-\underline{u}(1-\theta-\varepsilon^{4})-\underline{u}_{xx}.$

Let $y_{\text{\rm\tiny touch}}=\pi(2x_{\text{\rm\tiny touch}}-R)/(2R)$. Using
the definition of $\underline{u}$ to compute $\underline{u}_{x}$ and
$\underline{u}_{xx}$ and dividing through by $e^{-\lambda x_{\text{\rm\tiny
touch}}}/A$, we find

$\begin{split}&0\leq\left(2-\varepsilon+\varepsilon^{4}\right)\left(\frac{\pi}{R}\sin\left(y_{\text{\rm\tiny
touch}}\right)+\lambda\cos\left(y_{\text{\rm\tiny
touch}}\right)\right)-\cos(y_{\text{\rm\tiny
touch}})(1-\theta-\varepsilon^{4})\\\
&\qquad-\left(-\frac{\pi^{2}}{R^{2}}\cos(y_{\text{\rm\tiny
touch}})+2\lambda\frac{\pi}{R}\sin(y_{\text{\rm\tiny
touch}})+\lambda^{2}\cos(y_{\text{\rm\tiny touch}})\right)\\\
&=\cos(y_{\text{\rm\tiny
touch}})\left(\left(2-\varepsilon+\varepsilon^{4}\right)\lambda-1+\theta+\varepsilon^{4}-\lambda^{2}+\frac{\pi^{2}}{R^{2}}\right)+\sin(y_{\text{\rm\tiny
touch}})\left(\left(2-\varepsilon+\varepsilon^{4}-2\lambda\right)\frac{\pi}{R}\right).\end{split}$

To eliminate the $\sin$ term, whose sign is unknown, let
$\lambda=1-\varepsilon/2+\varepsilon^{4}/2$. This yields

$0\leq\cos(y_{\text{\rm\tiny
touch}})\left(\frac{1}{4}\left(2-\varepsilon+\varepsilon^{4}\right)^{2}-1+\theta+\varepsilon^{4}+\frac{\pi^{2}}{R^{2}}\right).$

Further increasing $R$ (and, thus, $a$) and decreasing $\theta$ and
$\varepsilon$, if necessary, the coefficient of $\cos$ is negative. In
addition, as $x_{\text{\rm\tiny touch}}\in(0,R)$, $y_{\text{\rm\tiny
touch}}\in(-\pi/2,\pi/2)$, and, hence, $\cos(y_{\text{\rm\tiny touch}})>0$. We
conclude that the the right hand side above is negative, which is a
contradiction. The proof is finished. ∎

####  3.1.2 Existence of a solution on the slab

We now use Leray-Schauder degree theory to establish the following
proposition. See [Nirenberg_book] for a review of standard results in this
theory that are used below.

######  Proposition 3.5.

There exists $\theta_{0}>0$ such that, for any $\theta\in(0,\theta_{0})$ and
any $a$ sufficiently large, there exists a solution $(c,u)$ of (3.2) with
$\tau=1$. Further, $u\geq\theta$ on $[-a,0]$.

###### Proof.

We first point out the following bound on any solution $(c,u)$ of (3.2). By
Lemmas 3.1, 3.4 and 3.2 and elliptic regularity theory, we have that, for any
$\alpha\in(0,1)$, if $\theta$ is sufficiently small and $a$ sufficiently large
(depending only on $\theta$), there exists a constant $C_{0}>0$, depending on
$\chi$ and $\sigma$ but not on $\tau$, such that

$\frac{1}{C_{0}}<c\quad\text{ and }\quad c+\|u\|_{C^{2,\alpha}}\leq C_{0}.$
(3.7)

Let $\mathcal{B}=\\{(c,u)\in[0,2C_{0}]\times
C^{1,\alpha}:\|u\|_{C^{1,\alpha}}\leq 2C_{0},u\geq 0\\}$. Consider the
operator:

$S_{\tau}:\mathcal{B}\to C^{1,\alpha}\times\mathbb{R}$

such that $S_{\tau}(c,u)=(c+\theta-\max_{x\geq 0}u(x),\bar{u})$, where
$\bar{u}$ is the unique solution of the linear problem

$\begin{cases}-c\bar{u}_{x}+\tau(\chi(K_{\sigma}*\widetilde{u})\bar{u})_{x}=\bar{u}_{xx}+u(1-u)\quad&\text{
for all }x\in(-a,a),\\\ \bar{u}(-a)=1\quad\bar{u}(a)=0\end{cases}$

(recall the definition of $\widetilde{u}$ from (3.3)). It is important to note
that $\widetilde{u}$ is defined in terms of $u$, not $\bar{u}$, so that the
above equation is linear in $\bar{u}$.

At this point we note that the existence of a solution of (3.2) is equivalent
to the existence of a fixed point of $S_{1}$. We now establish the existence
of such a fixed point.

Standard elliptic regularity theory provides bounds on the $C^{2,\alpha}$ norm
of $\bar{u}$ depending only on the $C^{1,\alpha}$ norm of $u$ and $a$. Hence,
we have that $S_{\tau}$ is a compact operator. Moreover, if $a$ is
sufficiently large, any fixed point of $S_{\tau}$ is an element of the
interior of $\mathcal{B}$, by (3.7). Hence, we have that

$\deg(\operatorname{Id}-S_{1},\mathcal{B},0)=\deg(\operatorname{Id}-S_{0},\mathcal{B},0).$

On the other hand, any fixed point of $S_{0}$ is a solution to the “slab”
problem for the Fisher-KPP equation. The uniqueness of such solutions is well-
known (and easy to establish). Hence,
$\deg(\operatorname{Id}-S_{0},\mathcal{B},0)=1$ or $-1$. In either case, we
see that $\deg(\operatorname{Id}-S_{1},\mathcal{B},0)\neq 0$, which
establishes the existence of a fixed point.

The last step is to establish the lower bound of $u$ on $[-a,0]$. To this end,
if $u(x_{0})<\theta$ for some $x_{0}\in[-a,0]$, then we may apply Lemma 2.3
(which can clearly be extended to the slab problem) to conclude that $u$ is
decreasing on $[x_{0},a]$. It follows that $u(0)\leq u(x_{0})<\theta$, which
contradicts the fact that, by construction, $u(0)=\theta$. We conclude that
$u\geq\theta$ on $[0,a]$, as claimed. Thus, the proof is complete. ∎

###  3.2 Taking $a\to\infty$: the proof of Proposition 1.3

######  Proposition 3.6.

There exists a traveling wave solution $(c,u)$, in the sense of Definition 1.2
with

$c\leq 2\sqrt{1+\frac{|\chi|}{\sigma}}+\frac{|\chi|}{2}\leq
2+\frac{|\chi|}{\sigma}+\frac{|\chi|}{2}.$

Furthermore, $u(-\infty)=1$.

###### Proof.

For clarity, we denote the solution constructed in (3.5) as $(c^{a},u^{a})$.
The bounds in (3.7) are uniform in $a$. Hence, for any
$\theta\in(0,\theta_{0})$, there exists a sequence $a_{n}\to\infty$ and
$(c,u)\in[0,2+|\chi|/2+|\chi|/\sigma]\times C^{2,\alpha}$ such that, as
$n\to\infty$, $c^{a_{n}}\to c$ and $u^{a_{n}}\to u$. Using (3.2), it is easy
to check that

$-cu_{x}+(vu)_{x}=u_{xx}+u(1-u)\qquad\text{ for all }x\in\mathbb{R},$ (3.8)

where $v$ is as in (1.2).

Next, we establish the behavior of $u$ as $x\to\infty$. By Lemma 2.3, let
$\delta=\lim_{x\to\infty}u(x)$. Let $u_{n}(x)=u(x+n)$. Since $u_{n}$ satisfies
(3.7), we may take a subsequence $n_{k}\to\infty$ such that
$u_{n_{k}}\to\delta$ locally uniformly in $C^{2}$. Using this, as well as the
fact that $u_{n}$ satisfies (3.8) with $v_{n}(x)=v(x+n)$, we find

$0=\lim_{n\to\infty}\left(-c(u_{n})_{x}+(v_{n}u_{n})_{x}-(u_{n})_{xx}-u_{n}(1-u_{n})\right)=\delta(1-\delta).$

Note that we used that, since $u_{n}\to\delta$ locally uniformly then
$v_{n}\to 0$ locally uniformly. As $\delta<\theta<1$, it follows that
$\delta=0$. Thus $\lim_{x\to\infty}u(x)=0.$

It follows immediately from Lemma 2.3 and the fact that $u(0)=\theta$ that
$\liminf_{x\to-\infty}u(x)\geq\theta>0$. The fact that this limit is equal to
one is due to Lemma 2.4. This concludes the proof. ∎

In addition, we can show that all traveling waves, including the one
constructed in Proposition 3.6 have speed at least 2. One could prove this in
the manner of [NadinPerthameRyzhik, Proposition 4.1] or by changing variables
to the time-dependent problem and applying [HamelHenderson, Theorem 1.1];
however, we provide another proof here in the interest of giving another
perspective.

######  Proposition 3.7.

Suppose that $u$ is a traveling wave solution of (1.7) as in Definition 1.2.
Then $c\geq 2$.

###### Proof.

The proof of this claim follows exactly as in the Lemma 3.4 with the following
modification. Instead of constructing the subsolution $\underline{u}$ on
$[0,R]$, one constructs the subsolution shifted by $L$ to be on $[L,L+R]$ for
$L\gg 1$. Then all $\theta$ in the proof, which are used to bound
$u(x_{\text{\rm\tiny touch}})$, become $\theta_{L}=\min_{[L,\infty)}u$. The
proof follows since $\lim_{L\to\infty}\theta_{L}=0$ by (1.6). ∎

Clearly the combination of Proposition 3.6 and Proposition 3.7 yield
Proposition 1.3.

###  3.3 Minimal speed traveling waves with speed $2$

For notational ease, we fix a sequence $a_{n}\to\infty$ along which
$(c_{n},u_{n})=(c^{a_{n}},u^{a_{n}})$ converges to $(c_{U},U)$, where we use
$(c_{U},U)$ to denote the traveling wave constructed in the previous section.
Since we mainly work with $u_{n}$ in this section, we suppress the dependence
on $n$; however, $u$ always refers to the approximate traveling wave on the
slab $[-a_{n},a_{n}]$ and $U$ refers to its locally uniform limit as
$n\to\infty$. We suppress the dependence on $\chi$ and $\sigma$ when no
confusion will arise; otherwise we denote such dependence as a super-script,
e.g. $c^{\chi,\sigma}$.

We now improve the bounds of Lemma 3.2 to show that $c\leq 2$ for all $a$
sufficiently large, from which we conclude that $c_{U}=2$ (see Proposition
1.3). It is clear that Theorem 1.4 follows directly from the Proposition 3.8
(below) after taking $a\to\infty$. We state and prove Proposition 3.8 here,
postponing the proof of the main technical lemma until Section 3.3.1.

######  Proposition 3.8.

Suppose $\theta<\theta_{0}$. There exists $\delta_{\theta}>0$, depending only
on $\theta$ and $K$ such that if
$|\chi|(\sigma^{2}+\sigma^{-1})\in(0,\delta_{\theta})$ and
$a>1/\delta_{\theta}$, then

$c\leq 2.$

###### Proof.

We proceed by contradiction, writing

$c=2+\varepsilon\quad\text{ for some
}\varepsilon\in(0,2|\chi|(1+\sigma^{-1})],$ (3.9)

where the upper bound follows from Lemma 3.2. Define $w:[-a,a]\to\mathbb{R}$
by

$u(x)=\exp\left\\{-\frac{c}{2}x+\frac{1}{2}\int_{0}^{x}\widetilde{v}(y)\,dy\right\\}w(x).$

Let $V(x)=1-(u+(c-\widetilde{v})^{2}/4-\widetilde{v}_{x}/2)$. From (3.2), we
find

$\begin{cases}-w_{xx}-wV=0\qquad\text{ on }(-a,a),\\\
w(-a)=\exp\left\\{-\frac{1}{2}ca-\frac{1}{2}\int_{0}^{-a}\widetilde{v}(y)\,dy\right\\},\quad
w(0)=\theta,\quad w(a)=0.\end{cases}$ (3.10)

At this point it is useful to notice a few key properties of $V$. First,

$V=-u-\varepsilon\left(1+\frac{\varepsilon}{4}\right)+\widetilde{v}\left(\frac{c}{2}-\frac{\widetilde{v}}{4}\right)+\frac{\widetilde{v}_{x}}{2}.$
(3.11)

The important thing is that, when $|\chi|(1+\sigma^{-1})$ is small enough,
$V<0$ except (perhaps) far to the right where it is small and decays to zero.
This heuristic should imply that the principal eigenvalue $\lambda$ of
$-\partial_{xx}-V$ should be positive. The corresponding principal
eigenfunction is then a supersolution of (3.10), which we use to derive a
contradiction.

We now make this argument precise. Let $(\lambda,\varphi)$ solve

$\begin{cases}-\varphi_{xx}-\varphi V=\lambda\varphi\qquad\text{ in
}(-a,a),\\\ 1=\varphi(-a)=\varphi(a),\quad\varphi>0.\end{cases}$ (3.12)

One way to construct a solution of (3.12) is via the Rayleigh quotient
representation of $\lambda$; that is,

$\lambda=\min_{\psi\in H_{\rm
per}^{1}([-a,a])}\frac{\int_{-a}^{a}(\psi_{x}^{2}-V\psi^{2})\,dx}{\int_{-a}^{a}\psi^{2}dx},$
(3.13)

where $H^{1}_{\rm per}([-a,a])$ is the closure of all $2a$ periodic $C^{1}$
functions under the $H^{1}([-a,a])$ norm. The existence and positivity of a
minimizer and the fact that a minimizer is, up to normalization, a solution of
(3.12) is classical.

We postpone the analysis of (3.12) momentarily, though we record the important
properties of $(\lambda,\varphi)$ in Lemma 3.9, below, which is proved in
Section 3.3.1.

######  Lemma 3.9.

For $|\chi|(\sigma^{2}+\sigma^{-1})$ and $\theta_{0}$ small enough,
$\lambda\geq 0$ and $\varphi(0)\leq\exp\\{a/2\\}$ for all $a$ sufficiently
large when $c=2+\varepsilon$ with $\varepsilon\geq 0$.

We now use Lemma 3.9 to conclude. Let $\varphi_{A}(x)=A\varphi(x)$. If $A$ is
sufficiently large, then $\varphi_{A}>w$ because $w$ is bounded and $\varphi$
is uniformly positive. Let $A_{0}=\inf\\{A>0:\varphi_{A}>w\\}$. Then we find
$x_{0}\in[-a,a]$ such that $\varphi_{A_{0}}(x_{0})=w(x_{0})$. Since $w>0$ on
$[-a,a)$ and $\varphi>0$, it follows that $A_{0}>0$. There are three cases.

Case one: $x_{0}=a$. This, clearly, cannot occur since
$\varphi_{A_{0}}(a)=A_{0}\varphi(a)>0=w(a)$.

Case two: $x_{0}\in(-a,a)$. First, note that, due to Lemma 3.9,

$-\varphi_{xx}-V\varphi=\lambda\varphi\geq 0.$

Hence $\varphi_{A_{0}}$ is a supersolution of (3.12). Recall that
$\varphi_{A_{0}}>w(a)$ and $\varphi_{A_{0}}\geq w$ on $[-a,a)$, by
construction. Applying the strong maximum principle, we conclude that
$\varphi_{A_{0}}>w$ on $(-a,a)$, which contradicts the fact that
$x_{0}\in(-a,a)$. Thus this case cannot occur.

Case three: $x_{0}=-a$. Then we have that

$A_{0}=\varphi_{A_{0}}(-a)=w(-a)=\exp\left\\{-\frac{1}{2}\left(ca-\int_{0}^{-a}\widetilde{v}(y)\,dy\right)\right\\}.$

In addition, using this identity for $A_{0}$, we find

$\theta=w(0)\leq\varphi_{A_{0}}(0)\leq
A_{0}\varphi(0)\leq\exp\left\\{-\frac{1}{2}\left(ca-\int_{0}^{-a}\widetilde{v}(y)\,dy\right)\right\\}\exp\left\\{\frac{a}{2}\right\\}.$
(3.14)

By assumption $c\geq 2$. In addition, if $|\chi|(1+\sigma^{-1})$ is
sufficiently small, then, by (3.4),

$\frac{1}{2}\int_{0}^{-a}|\widetilde{v}(y)|\,dy\leq\frac{a}{4}.$

Hence, the right hand side of (3.14) clearly tends to zero as $a\to\infty$.
This contradicts the fact that $\theta$ is positive and independent of $a$.

We have reached a contradiction in all cases, which implies that $c\leq 2$.
This concludes the proof. ∎

####  3.3.1 Nonnegativity of the eigenvalue and asymptotics of its
eigenfunction

The first step is establishing the exponential decay of $u$ for all $a$
sufficiently large and all $\chi$ and $\sigma$ such that
$|\chi|(1+\sigma^{-1})$ is sufficiently small.

######  Lemma 3.10 (Exponential decay of $u$).

There exists $\mu,\delta_{\theta}>0$ such that if
$|\chi|(1+\sigma^{-1})\leq\delta_{\theta}$ and $a>1/\delta_{\theta}$ and
$\theta<1/4$, then

$u(x)\leq\theta e^{-\mu x}\qquad\text{ for all }x\geq 0.$

The constant $\delta_{\theta}$ depends on $\theta$.

###### Proof.

Let $\rho=u_{x}/u$, and fix $\mu\in(0,1/4]$ a constant to be chosen. We prove
this lemma in two steps. First, we show that $\rho$ cannot have a maximum
greater than $-\mu$ on $(0,a)$. Since $\rho(a)=-\infty$, then
$\rho(x)\leq\max\\{-\mu,\rho(0)\\}$. Second, we show that $\rho(0)\leq-\mu$.
We conclude that $\rho(x)\leq-\mu$, which finishes the proof.

Step one: $\rho$ cannot have an interior max larger than $-\mu$. Suppose that
$x_{0}$ is the location of an interior maximum $M\geq-\mu$. Using that
$u\rho=u_{x}$ and $u\rho_{x}+u\rho^{2}=u_{xx}$ in (3.2) (with $\tau=1$) and
that $\rho=M$ and $\rho_{x}=0$ at $x_{0}$, we obtain

$0=(c-\widetilde{v})u_{x}+u_{xx}+u(1-u-\widetilde{v}_{x})=u\left((c-\widetilde{v})M+M^{2}+(1-u-\widetilde{v}_{x})\right).$

Hence,

$M=-\frac{c-\widetilde{v}\pm\sqrt{(c-\widetilde{v})^{2}-4(1-u-\widetilde{v}_{x})}}{2}\leq\frac{-\left(c-\widetilde{v}\right)+\sqrt{(c-\widetilde{v})^{2}-4(1-u-\widetilde{v}_{x})}}{2}.$
(3.15)

Recall from Proposition 1.3 that

$2\leq c\leq 2+\frac{|\chi|}{2}+\frac{|\chi|}{\sigma}.$ (3.16)

Hence, taking $|\chi|(1+\sigma^{-1})$ to zero, the right hand side above tends
to

$\frac{-2+\sqrt{4-4(1-u)}}{2}=-1+\sqrt{u(x_{0})}\leq-1+\sqrt{\theta}.$

As $\theta<1/4$ by assumption, this yields $M<-1/2<-1/4\leq-\mu$. Hence, if
$|\chi|(1+\sigma^{-1})$ is sufficiently small, (3.15) cannot hold. This
contradiction concludes the proof of this step.

Step two: $\rho(0)\leq-\mu$. We argue by contradiction: if the claim were not
true for all $a$ sufficiently large and $|\chi|(1+\sigma^{-1})$ sufficiently
small, then there would exist sequences $a_{n}$, $\chi_{n}$, and $\sigma_{n}$,
such that $a_{n}\to\infty$ and $|\chi_{n}|(1+\sigma_{n}^{-1})\to 0$ as
$n\to\infty$ such that $\rho^{\chi_{n},\sigma_{n}}(0)>-\mu$.

Recall (3.16). By compactness,
$(c^{\chi_{n},\sigma_{n}},u^{\chi_{n},\sigma_{n}})$ converges to the minimal
speed traveling wave solution $(2,U)$ of the Fisher-KPP equation

$-2U_{x}=U_{xx}+U(1-U),$

with the normalization $U(0)=\theta$, and $\rho^{\chi_{n},\sigma_{n}}(0)$
converges to $U_{x}(0)/U(0)$.

Recall, from, e.g., [Hamel, Theorem 1.3], the asymptotics of the Fisher-KPP
equation: if $\overline{U}$ is the solution to the Fisher-KPP equation with
the normalization $\overline{U}(0)=1/2$, then there exists $\kappa$ such that
$\overline{U}(x)/(\kappa xe^{-x})\to 1$ as $x\to\infty$. By the uniqueness of
traveling wave solutions for the Fisher-KPP equation, there exists
$x_{\theta}$, tending to $\infty$ as $\theta$ tends to zero, such that
$U(x)=\overline{U}(x+x_{\theta})$. Further, by the monotonicity of the
traveling wave for the Fisher-KPP equation, $x_{\theta}$ increases as $\theta$
decreases. Hence,

$\sup_{[0,\infty)}\frac{U_{x}}{U}=\sup_{[x_{\theta},\infty)}\frac{\overline{U}_{x}}{\overline{U}}\leq\sup_{[0,\infty)}\frac{\overline{U}_{x}}{\overline{U}}.$

The first inequality follows since $\theta<\theta_{0}<1/2$.

Thus, let

$\mu=-\frac{1}{4}\sup_{[0,\infty)}\frac{\overline{U}_{x}}{\overline{U}}.$

First, notice that $\overline{U}_{x}/\overline{U}\to-1$ as $x\to\infty$, so
$\mu\leq 1/4$, which is consistent with our choice in the previous step. On
the other hand, since $\overline{U}_{x}/\overline{U}$ is a continuous,
negative function on $[0,\infty)$ such that $\overline{U}_{x}(0)<0$ and
$\overline{U}_{x}/\overline{U}\to-1$ as $x\to\infty$, we find that $\mu>0$.

Choosing $|\chi|(1+\sigma^{-1})$ sufficiently small, depending on $\theta$, we
find

$-\mu<\rho(0)\leq\frac{1}{2}U(0)=\frac{1}{2}\sup_{[0,\infty)}\frac{U_{x}}{U}\leq-\mu.$

This is a contradiction, which concludes the proof. ∎

The next step is to establish a Poincaré type inequality. This is the key
estimate in the proof of Lemma 3.9.

######  Lemma 3.11.

There exists a constant $C_{K}$, depending only on $K$, such that

$\displaystyle\int_{-a}^{a}\widetilde{v}_{x}f^{2}dx$
$\displaystyle\leq\frac{C_{K}|\chi|(1+\sigma^{2})}{\theta}\left(\int_{-a}^{a}uf^{2}dx+\int_{-a}^{a}|f_{x}|^{2}dx\right),$
(3.17) $\displaystyle\int_{-a}^{a}|\widetilde{v}|f^{2}dx$
$\displaystyle\leq\frac{C_{K}|\chi|(1+\sigma^{2})}{\theta}\left(\int_{-a}^{a}uf^{2}dx+\int_{-a}^{a}|f_{x}|^{2}dx\right)$
(3.18)

for all $f\in H_{\rm per}^{1}([-a,a])$.

We postpone the proof of this lemma momentarily and show its application in
the proof of Lemma 3.9.

###### Proof of Lemma 3.9.

Let $(\lambda,\varphi)$ solve (3.12). Then, using Lemma 3.11 and the form of
$V$ (see (3.11)), we find a universal constant $C$ such that

$\begin{split}-\int V\varphi^{2}dx&=\int
u\varphi^{2}dx+\int\varepsilon\left(1+\frac{\varepsilon}{4}\right)\varphi^{2}dx-\int\widetilde{v}\left(\frac{c}{2}-\frac{\widetilde{v}}{4}\right)\varphi^{2}dx-\int\frac{\widetilde{v}_{x}}{2}\varphi^{2}dx\\\
&\geq\int u\varphi^{2}dx-C\frac{C_{K}|\chi|(1+\sigma^{2})}{\theta}\left(\int
u\varphi^{2}dx+\int\varphi_{x}^{2}dx\right).\end{split}$ (3.19)

Multiplying (3.12) by $\varphi$ and integrating by parts yields

$\lambda\int\varphi^{2}dx=\int\varphi_{x}^{2}dx-\int V\varphi^{2}dx.$

This, along with (3.19), implies that if $|\chi|$ is sufficiently small, then

$\lambda\int\varphi^{2}dx\geq\int\varphi_{x}^{2}dx+\int
u\varphi^{2}dx-C\frac{C_{K}|\chi|(1+\sigma^{2})}{\theta}\left(\int
u\varphi^{2}dx+\int\varphi_{x}^{2}dx\right)\geq 0.$

Hence, up to decreasing $|\chi|$, $\lambda\geq 0$, which finishes the first
claim in Lemma 3.9.

Before we examine the bound on $\varphi(0)$, we require an upper bound on
$\lambda$. To this end, we construct a test function for (3.13); let

$\widetilde{\psi}(x)=\begin{cases}A\left(x-\frac{a}{2}\right)\qquad&\text{ if
}x\in\left(\frac{a}{2},\frac{3a}{4}\right),\\\
\frac{Aa}{4}-A\left(x-\frac{3a}{4}\right)\qquad&\text{ if
}x\in\left(\frac{3a}{4},a\right)\\\ 0&\text{ otherwise},\end{cases}$

where $A=(96/a^{3})^{1/2}$. See Figure 1 for a cartoon of $\widetilde{\psi}$.

\begin{overpic}[scale={.75}]{test_function.pdf} \put(2.0,0.8){$-a$}
\put(95.4,0.8){$a$} \put(84.0,13.0){\color[rgb]{0,0,1} $\widetilde{\psi}$}
\put(72.3,0.8){$\frac{a}{2}$} \put(83.4,0.8){$\frac{3a}{4}$}
\put(50.5,12.0){\color[rgb]{1,0,0} $\frac{Aa}{4}=\sqrt{\frac{6}{a}}$}
\end{overpic} Figure 1: A cartoon of $\widetilde{\psi}$, which is represented
by the dashed line.

We point out that

$\int\widetilde{\psi}^{2}\,dx=\frac{A^{2}a^{3}}{96}=1\quad\text{ and
}\quad\int\widetilde{\psi}_{x}^{2}\,dx=A^{2}\frac{a}{2}=\frac{48}{a}.$

Using Lemma 3.10 and (3.4), there is $C>0$ such that
$\max_{x\in[a/2,a]}(-V)\leq C|\chi|(1+\sigma^{-1})$ for $a$ sufficiently
large. Using $\widetilde{\psi}$ as a test function in (3.13) along with these
three estimates, yields

$\lambda\leq C|\chi|(1+\sigma^{-1}),$ (3.20)

when $a$ is chosen sufficiently large.

We now obtain the growth bound on $\varphi$. Let

$\delta=2\max_{x\in[0,a]}\left(-\lambda-V(x)\right)=2\max_{x\in[0,a]}\left(-\lambda+u+\varepsilon\left(1+\frac{\varepsilon}{4}\right)-\widetilde{v}\left(\frac{c}{2}-\frac{\widetilde{v}}{4}\right)-\frac{\widetilde{v}_{x}}{2}\right).$
(3.21)

Recall, from (3.4) that
$|\widetilde{v}|,|\widetilde{v}_{x}|\leq|\chi|(1+\sigma^{-1})$. Using these
and (3.20) while evaluating the right hand side of (3.21), we find

$\delta\geq
2u(0)-C|\chi|\left(1+\frac{1}{\sigma}\right)=2\theta-C|\chi|\left(1+\frac{1}{\sigma}\right).$

Hence, after decreasing $|\chi|(1+\sigma^{-1})$, we see that $\delta>0$.
Moreover, after using the monotonicity of $u$ (Lemma 2.3) as well as the bound
on $\varepsilon$ (3.9), we obtain a matching upper bound:

$\delta\leq
2\left(\max_{x\in[0,a]}u\right)+C|\chi|\left(1+\frac{1}{\sigma}\right)=2u(0)+C|\chi|\left(1+\frac{1}{\sigma}\right)=2\theta+C|\chi|\left(1+\frac{1}{\sigma}\right).$

It follows that there exists a universal constant $C$ such that

$|\delta-2\theta|<C|\chi|\left(1+\frac{1}{\sigma}\right).$

For the remainder of the proof, we assume that $\theta$ and
$|\chi|(1+\sigma^{-1})$ are sufficiently small so that $2\sqrt{\delta}<1/2$
and $\delta>\theta/2$ (recall that $\theta<\theta_{0}<1/100$).

We now claim that $\varphi(0)\leq e^{2\sqrt{\delta}a}$, which would complete
the proof. We proceed by contradiction, assuming that
$\varphi(0)>e^{2\sqrt{\delta}a}$. For any $A>0$, let

$\underline{\varphi}_{A}(x)=A\left(\left(e^{2\sqrt{\delta}a}-\frac{e^{3\sqrt{\delta}a}-1}{e^{\sqrt{\delta}a}-e^{-\sqrt{\delta}a}}\right)e^{\sqrt{\delta}x}+\left(\frac{e^{3\sqrt{\delta}a}-1}{e^{\sqrt{\delta}a}-e^{-\sqrt{\delta}a}}\right)e^{-\sqrt{\delta}x}\right).$

Importantly, we have that $\underline{\varphi}_{A}(a)=A$,
$\underline{\varphi}_{A}(0)=Ae^{2\sqrt{\delta}a}$, and

$-(\underline{\varphi}_{A})_{xx}=-\delta\underline{\varphi}_{A}<(\lambda+V)\underline{\varphi}_{A}.$

We let $A_{0}=\sup\\{A:\underline{\varphi}_{A}<\varphi$ on $[0,a]\\}$.
Continuity implies that $\varphi$ is positive and bounded away from 0 and,
hence, $A_{0}$ is well-defined and positive. By continuity, there exists
$x_{0}\in[0,a]$ such that
$\underline{\varphi}_{A_{0}}(x_{0})=\varphi(x_{0})>0$. There are three cases
to consider.

Case one: $x_{0}\in(0,a)$. Then, $x_{0}$ is the location of a minimum of
$\varphi-\underline{\varphi}_{A_{0}}$, which implies that
$\varphi_{xx}\geq\underline{\varphi}_{A_{0},xx}$ at $x_{0}$. Thus, at $x_{0}$,

$-\varphi_{xx}\leq-(\underline{\varphi}_{A_{0}})_{xx}<(\lambda+V)\underline{\varphi}_{A_{0}}=(\lambda+V)\varphi=-\varphi_{xx},$

which is a contradiction. Hence, $x_{0}\notin(0,a)$.

Case two: $x_{0}=0$. Then we have that

$\varphi(0)=\underline{\varphi}_{A_{0}}(0)=A_{0}e^{2\sqrt{\delta}a}.$

By assumption, $\varphi(0)>e^{2\sqrt{\delta}a}$. Hence, $A_{0}>1$. On the
other hand, by construction of $A_{0}$, we have that
$\varphi(a)\geq\underline{\varphi}_{A_{0}}(a)$. This yields a contradiction
because $\varphi(a)=1$ and $\underline{\varphi}_{A_{0}}(a)=A_{0}>1$.

Case three: $x_{0}=a$. This implies that

$1=\varphi(a)=\underline{\varphi}_{A_{0}}(a)=A_{0}.$

Since $\varphi(a)=\underline{\varphi}_{A_{0}}(a)$ and
$\varphi(x)\geq\underline{\varphi}_{A_{0}}(x)$ for all $x\in[0,a]$, we have
that

$\begin{split}\varphi_{x}(a)&\leq\underline{\varphi}_{A_{0},x}(a)=\sqrt{\delta}\left(e^{2\sqrt{\delta}a}-\frac{e^{3\sqrt{\delta}a}-1}{e^{\sqrt{\delta}a}-e^{-\sqrt{\delta}a}}\right)e^{\sqrt{\delta}a}-\sqrt{\delta}\left(\frac{e^{3\sqrt{\delta}a}-1}{e^{\sqrt{\delta}a}-e^{-\sqrt{\delta}a}}\right)e^{-\sqrt{\delta}a}\\\
&=\sqrt{\delta}\frac{-2e^{\sqrt{\delta}a}+1+e^{-2\sqrt{\delta}a}}{1-e^{-2\sqrt{\delta}a}}\leq-2\sqrt{\delta}e^{\sqrt{\delta}a}+\sqrt{\delta}\frac{1+e^{-2\sqrt{\delta}a}}{1-e^{-2\sqrt{\delta}a}}\leq-2\sqrt{\delta}e^{\sqrt{\delta}a}+1.\end{split}$
(3.22)

where the last line follows when $a$ is sufficiently large. The first equality
comes from the definition of $\underline{\varphi}_{A_{0}}$ and that $A_{0}=1$,
while the second and third follow from direct computations.

We now obtain a contradiction by showing that $\varphi_{x}(a)$ cannot take
such a large value. Before beginning we make a note about the regularity of
$\varphi_{x}$. Note that $\varphi_{xx}=-(\lambda+V)\varphi\in L^{2}_{\rm
per}([-a,a])$ and, hence, $\varphi_{x}$ is continuous and periodic on
$[-a,a]$. As we use below, one consequence of this is that
$\varphi_{x}(-a)=\varphi_{x}(a)$.

Assume that $a$ is sufficiently large that, via (3.22), $\varphi_{x}(a)<0$.
Let

$\bar{x}=\min\\{x\geq-a:\varphi_{x}(x)=\varphi_{x}(a)/2\\}.$

We show that $\bar{x}$ is well-defined. Since $\varphi$ is periodic,

$\int_{-a}^{a}\varphi_{x}\,dx=0.$

Hence, $\varphi_{x}$ must be nonnegative somewhere. Since $\varphi_{x}(a)<0$,
it follows that $\\{x:\varphi_{x}(x)=\varphi_{x}(a)/2\\}$ is nonempty, which
implies that $\bar{x}$ is well-defined.

We note that

$\varphi_{x}(x)\leq\frac{\varphi_{x}(a)}{2}\quad\text{ for all
}x\in[-a,\bar{x}].$ (3.23)

We see this as follows. Since $\varphi$ is periodic and $\varphi_{xx}$ is
bounded, then $\varphi_{x}$ is continuous and periodic on $[-a,a]$. This first
yields that

$\varphi_{x}(-a)=\varphi_{x}(a)<0.$ (3.24)

Using (3.24) along with the continuity of $\varphi_{x}$, the definition of
$\bar{x}$, and the intermediate value theorem, we deduce (3.23).

Next, we notice that there exists $\xi\in[-a,x_{0}]$ such that

$\varphi_{xx}(\xi)=\frac{\varphi_{x}(\bar{x})-\varphi_{x}(-a)}{\bar{x}-(-a)}=\frac{\frac{\varphi_{x}(a)}{2}-\varphi_{x}(a)}{\bar{x}+a}=\frac{|\varphi_{x}(a)|}{2(\bar{x}+a)}.$
(3.25)

In order to show that $\varphi_{xx}(\xi)$ is large, we obtain a smallness
bound on $a+x_{0}$ through the positivity of $\varphi$. Indeed, by (3.24) and
the mean value theorem,

$0\leq\varphi(\bar{x})\leq\varphi(-a)+\frac{\varphi_{x}(a)}{2}(\bar{x}-(-a))=1-\frac{|\varphi_{x}(a)|}{2}(\bar{x}+a),$

which implies that $\bar{x}+a\leq 2/|\varphi_{x}(a)|$. Plugging this into
(3.25) yields

$\varphi_{xx}(\xi)\geq\left(\frac{\varphi_{x}(a)}{2}\right)^{2}.$ (3.26)

In addition, by (3.12), (3.20), and the fact that $\varphi(x)\leq 1$ for all
$x\in[-a,\bar{x}]$ (which follows from (3.24) and (3.12))

$|\varphi_{xx}(\xi)|=|(V+\lambda)\varphi|\leq C(1+|\chi|(1+\sigma^{-1})).$
(3.27)

Putting together (3.22), (3.26), and (3.27), we find

$\left(-\sqrt{\delta}e^{\sqrt{\delta}a}+\frac{1}{2}\right)^{2}\leq
C(1+|\chi|(1+\sigma^{-1})).$

We have reached a contradiction since the right hand side is independent of
$a$ while the left hand side tends to infinity as $a\to\infty$. This concludes
the proof.

∎

Now we prove the functional inequalities that were the crux of the argument
that $\lambda$ is nonnegative.

###### Proof of Lemma 3.11.

We begin by reducing (3.17) to (3.18). Indeed, integrating by parts, we obtain

$\int_{-a}^{a}\widetilde{v}_{x}f^{2}\,dx=f(-a)^{2}\left[\widetilde{v}(a)-\widetilde{v}(-a)\right]-2\int_{-a}^{a}\widetilde{v}ff_{x}\,dx.$
(3.28)

Using the Sobolev inequality, we see that, for some $C>0$ that changes line-
by-line in the sequel but is independent of all parameters,

$f(-a)^{2}[\widetilde{v}(a)-\widetilde{v}(-a)]\leq
C\|\widetilde{v}\|_{L^{\infty}}\left(\int_{-a}^{-a+1}(f^{2}+f_{x}^{2})\,dx\right)$

and using Young’s inequality, we find

$-2\int_{-a}^{a}\widetilde{v}ff_{x}dx\leq\int_{-a}^{a}|\widetilde{v}|f^{2}dx+\|\widetilde{v}\|_{L^{\infty}}\int_{-a}^{a}f_{x}^{2}dx.$

Hence, (3.28) becomes

$\int_{-a}^{a}\widetilde{v}_{x}f^{2}\,dx\leq
C\|\widetilde{v}\|_{L^{\infty}}\int_{-a}^{-a+1}f^{2}\,dx+\|\widetilde{v}\|_{L^{\infty}}\int_{-a}^{a}|f_{x}|^{2}dx+\int_{-a}^{a}|\widetilde{v}|f^{2}dx.$

Recalling the bounds on $\widetilde{v}$ in (3.4) and that $u\geq\theta$ on
$[-a,0]$ due to Proposition 3.5, we find

$\begin{split}\int_{-a}^{a}\widetilde{v}_{x}f^{2}\,dx&\leq\frac{C|\chi|}{\theta}\int_{-a}^{-a+1}uf^{2}\,dx+C|\chi|\int_{-a}^{a}f_{x}^{2}dx+\int_{-a}^{a}|\widetilde{v}|f^{2}dx\\\
&\leq\frac{C|\chi|}{\theta}\left(\int_{-a}^{a}uf^{2}\,dx+\int_{-a}^{a}|f_{x}|^{2}\,dx\right)+\int_{-a}^{a}|\widetilde{v}|f^{2}\,dx.\end{split}$
(3.29)

It is now clear from (3.29) that (3.17) holds if (3.18) does. We now establish
(3.18). First, by Proposition 3.5, $u(x)\geq\theta$ for all $x\in[-a,0]$ so
that $|\widetilde{v}(x)|\leq C|\chi|u(x)/\theta$ for all $x\in[-a,0]$. Thus,

$\int_{-a}^{0}|\widetilde{v}|f^{2}\,dx\leq\frac{C|\chi|}{\theta}\int_{-a}^{0}uf^{2}\,dx.$
(3.30)

Next, we bound the integral on $[0,a]$. Indeed,

$\begin{split}\int_{0}^{a}|\widetilde{v}|f^{2}\,dx&=\int_{0}^{a}|\widetilde{v}|\left(\int_{0}^{x}f_{x}dy+f(0)\right)^{2}dx\leq
2\int_{0}^{a}|\widetilde{v}|\left(x\int_{0}^{x}f_{x}^{2}dy+f(0)^{2}\right)\,dx.\end{split}$
(3.31)

As above, we bound $f(0)$ using the Sobolev inequality and the fact that
$u\geq\theta$ on $[-1,0]$. Indeed:

$\begin{split}|f(0)|^{2}&\leq
C\int_{-1}^{0}f_{x}^{2}\,dx+C\int_{-1}^{0}f^{2}\,dx\leq
C\int_{-1}^{0}f_{x}^{2}\,dx+\frac{C}{\theta}\int_{-1}^{0}uf^{2}\,dx\\\ &\leq
C\int_{-a}^{a}f_{x}^{2}\,dx+\frac{C}{\theta}\int_{-a}^{a}uf^{2}\,dx.\end{split}$

Hence, (3.31) becomes

$\int_{0}^{a}|\widetilde{v}|f^{2}\,dx\leq
C\left(\int_{-a}^{a}|f_{x}|^{2}\,dx+\frac{C}{\theta}\int_{-a}^{a}uf^{2}\,dx\right)\int_{0}^{a}(1+x)|\widetilde{v}|\,dx.$

Clearly we have that

$\int_{0}^{a}(1+x)|\widetilde{v}|\,dx\leq
2\int_{0}^{a}x|\widetilde{v}|\,dx+\int_{0}^{1}|\widetilde{v}|\,dx\leq
2\int_{0}^{a}x|\widetilde{v}|\,dx+C|\chi|.$

As a result, the proof is finished if

$\int_{0}^{a}x|\widetilde{v}|\,dx\leq C|\chi|\sigma^{2}.$ (3.32)

We now establish this inequality. We begin by using the exact form of
$\widetilde{v}$ along with the decay of $u$ given in Lemma 3.10. Indeed,

$\begin{split}\frac{1}{|\chi|}\int_{0}^{a}x|\widetilde{v}|\,dx&\leq\int_{0}^{a}x\int_{0}^{\infty}|K_{\sigma}(y)||\widetilde{u}(x-y)-\widetilde{u}(x+y)|\,dydx\\\
&=\int_{0}^{a}x\left(\int_{0}^{x/2}+\int_{x/2}^{\infty}\right)|K_{\sigma}(y)||\widetilde{u}(x-y)-\widetilde{u}(x+y)|\,dydx\\\
&\leq\theta\int_{0}^{a}x\int_{0}^{x/2}|K_{\sigma}(y)|e^{-\mu(x-y)}\,dydx+2\int_{0}^{a}x\int_{x/2}^{\infty}|K_{\sigma}(y)|\,dydx.\end{split}$
(3.33)

In the last inequality, we used that $u$ is bounded above by $2$ (recall Lemma
3.1 and that $|\chi|/\sigma<1/2$, by assumption).

We bound the first term on the last line of (3.33). Re-writing the integral,
we find

$\begin{split}\int_{0}^{a}x\int_{0}^{x/2}|K_{\sigma}(y)|e^{-\mu(x-y)}\,dydx&=\int_{0}^{a/2}|K_{\sigma}(y)|\int_{2y}^{a}xe^{-\mu(x-y)}\,dxdy\\\
&\leq\int_{0}^{a/2}|K_{\sigma}(y)|\left(\frac{e^{-\mu y}(1+2\mu
y)}{\mu^{2}}\right)\,dy\leq C,\end{split}$ (3.34)

where the last inequality follows from the fact that
$\|K_{\sigma}\|_{L^{1}}=\|K\|_{L^{1}}=1$ (see (1.3)).

Next we bound the second term on the last line of (3.33). Using the fact that
$|K|=-K=-\overline{K}^{\prime}$ on $[0,\infty)$ with $(1+|x|)\overline{K}\in
L^{1}$ (see (1.4)), we find

$\begin{split}\int_{0}^{a}x&\int_{x/2}^{\infty}|K_{\sigma}(y)|\,dydx=-\int_{0}^{a}x\int_{x/2}^{\infty}\frac{1}{\sigma}\overline{K}^{\prime}\left(\frac{y}{\sigma}\right)\,dydx=-\int_{0}^{a}x\int_{x/2}^{\infty}\left(\overline{K}\left(\frac{y}{\sigma}\right)\right)_{y}dydx\\\
&=\int_{0}^{a}x\overline{K}\left(\frac{x}{2\sigma}\right)\,dx\leq
C\sigma^{2}.\end{split}$ (3.35)

The combination of (3.33), (3.34), and (3.35) establishes (3.32). Thus, the
proof is complete. ∎

##  4 Fast traveling waves: Theorem 1.5

We now establish large lower bounds on the traveling wave speed $c$ whenever
$\sigma$ is large, in absolute terms and relative to $-\chi$.

###### Proof of Theorem 1.5.

Fix $\varepsilon\in(0,1/2)$ and suppose that

$c<\frac{(1-\varepsilon)|\chi|}{2}.$ (4.1)

Recall that

$\overline{K}(x)=-\int_{x}^{\infty}K(y)\,dy=-\int_{x\sigma}^{\infty}K_{\sigma}(y)\,dy\quad\text{and}\quad\overline{K}(0)=\frac{1}{2}.$

The last equality follows from the symmetry of $K$ and (1.3). Now we define
the constants

$\theta=\frac{|\chi|}{2R}\quad\text{ and }\quad
R=\frac{1}{2}\overline{K}^{-1}\left(\frac{1-\frac{\varepsilon}{4}}{2}\right).$
(4.2)

Without loss of generality, we may restrict to the case where
$\varepsilon<1/10$ and is sufficiently small that $R\leq 1$. Next, choose
$A_{\varepsilon}$ such that, if $\sigma,\sigma/|\chi|\geq A_{\varepsilon}$,
then

$\frac{\theta}{\sigma}=\frac{|\chi|}{2\sigma R}<\varepsilon/4\qquad\text{ and
}\qquad\varepsilon>\frac{1}{R\sigma}.$ (4.3)

The proof proceeds by showing that either the front is “stretched” by the
advective term, which results in a large speed, or the advection is large on a
large interval. To this end, we define “ends” of the front

$\begin{split}&x_{1}=\max\\{y\in\mathbb{R}:u(y)=1-\theta\sigma^{-1}\\}\quad\text{
and}\\\ &x_{2}=\max\\{y\in\mathbb{R}:u(y)=\theta\sigma^{-1}\\},\end{split}$

which are well-defined due to Lemma 2.4. Due to Lemma 2.3, we see that $u$ is
decreasing on $[x_{1},\infty)$ because, by the choice of $\theta$,
$1-\theta/\sigma<(1-\chi/2\sigma)^{-1}$ (recall that $R\leq 1$). We conclude
that

$u(x)\in\begin{cases}\left[1-\frac{\theta}{\sigma},1\right)\qquad&\text{ if
}x\leq x_{1}\\\
\left[\frac{\theta}{\sigma},1-\frac{\theta}{\sigma}\right]\qquad&\text{ if
}x\in[x_{1},x_{2}]\\\ \left(0,\frac{\theta}{\sigma}\right]&\text{ if }x\geq
x_{2}.\end{cases}$ (4.4)

We now separate into two cases.

Case one: $x_{2}-x_{1}>R\sigma$. For any $L>0$, by integrating (1.7) from $-L$
to $L$, we find

$c[u(-L)-u(L)]+[v(L)u(L)-v(-L)u(-L)]=u_{x}(L)-u_{x}(-L)+\int_{-L}^{L}u(1-u)\,dx.$

The first term on the left and the last term on the right are the crucial
terms. Our goal is to show that all other terms tend to zero. First, since
$u(-L)\to 1$ and $u(L)\to 0$ as $L\to\infty$, it follows that

$c[u(-L)-u(L)]\to c\quad\text{ and }\quad v(\pm L)=\chi(K_{\sigma}*u)(\pm
L)\to 0\qquad\text{ as }L\to\infty.$

Using elliptic regularity theory, since $u$ tends to constants at $\pm\infty$,
then $u_{x}(\pm L)\to 0$ as $L\to\infty$. Putting all of these together, we
find

$\begin{split}c&=\lim_{L\to\infty}c[u(-L)-u(L)]\\\
&=\lim_{L\to\infty}\left(-[v(L)u(L)-v(-L)u(-L)]+u_{x}(L)-u_{x}(-L)+\int_{-L}^{L}u(1-u)\right)\\\
&=\lim_{L\to\infty}\int_{-L}^{L}u(1-u)\,dx=\int_{-\infty}^{\infty}u(1-u)\,dx.\end{split}$

By Lemma 2.1, we observe that $u(1-u)\geq 0$. Hence,

$c\geq\int_{x_{1}}^{x_{2}}u(1-u)\,dx.$

We now obtain a lower bound on the integral term. By (4.4) and basic calculus,
$u(1-u)\geq(\theta/\sigma)(1-\theta/\sigma)$ on $[x_{1},x_{2}]$. Thus, we find

$\begin{split}c&\geq\int_{x_{1}}^{x_{2}}u(1-u)\,dx\geq\left(1-\frac{\theta}{\sigma}\right)\frac{\theta}{\sigma}|x_{2}-x_{1}|\geq\left(1-\frac{\theta}{\sigma}\right)\theta
R\geq\left(1-\frac{\varepsilon}{4}\right)\frac{|\chi|}{2},\end{split}$

which contradicts (4.1). Hence, case one cannot occur.

Case two: $x_{2}-x_{1}\leq R\sigma$. In this case, we build an explicit sub-
solution in order to “push” the wave at a fast speed. The key step in doing so
is in estimating the effect of the advective term $v$ on an interval near the
front.

To obtain this estimate, we obtain a lower bound on $v$ near the front for any
$x\in[x_{2},x_{2}+R\sigma]$. First, we note that $u(x+y)\leq u(x-y)$ for any
$x\geq x_{2}$ and $y\geq 0$. Indeed, if $x-y>x_{2}$, this follows because $u$
is decreasing on $[x_{2},\infty]$. On the other hand, if $x-y\leq x_{2}$ then
$u(x-y)\geq\theta/\sigma$ (see (4.4)) and $u(x+y)\leq\theta/\sigma$ since
$x+y\geq x_{2}$.

As a consequence of this inequality, we find

$\begin{split}v(x)=\chi(K_{\sigma}*u)(x)&=\chi\int_{0}^{\infty}K_{\sigma}(y)[u(x-y)-u(x+y)]\,dy\\\
&\geq\chi\int_{x-x_{1}}^{\infty}K_{\sigma}(y)[u(x-y)-u(x+y)]\,dy\quad\text{
for }x\geq x_{2}.\end{split}$

Above we used that $\chi K_{\sigma}\geq 0$ on the domain of integration. Then,
using (4.4) yields

$v(x)\geq\chi\int_{x-x_{1}}^{\infty}K_{\sigma}(y)\left(1-\frac{2\theta}{\sigma}\right)\,dy.$

Finally, observe that $x-x_{1}=(x-x_{2})+(x_{2}-x_{1})\leq 2R\sigma$ and,
thus,

$\begin{split}v(x)&\geq\chi\int_{2R\sigma}^{\infty}K_{\sigma}(y)\left(1-\frac{2\theta}{\sigma}\right)\,dy=|\chi|\left(1-\frac{2\theta}{\sigma}\right)\overline{K}(2R).\end{split}$
(4.5)

We now use the choice of constants $\theta$ and $R$, (4.2) and (4.3). We
conclude that, for all $x\in[x_{2},x_{2}+R\sigma]$,

$v(x)\geq\frac{|\chi|}{2}\left(1-\frac{\varepsilon}{2}\right)^{2}\geq c.$
(4.6)

The second inequality above follows from (4.1).

Using (4.6), we define our sub-solution. For any $A>0$, let

$\underline{u}_{A}(x)=A\sin\left(\frac{\pi(x-x_{2})}{R\sigma}\right).$ (4.7)

Notice that $\underline{u}_{A}(x_{2})=\underline{u}_{A}(x_{2}+R\sigma)=0$.

By Definition 1.2, $u>0$. Then, by the compactness of $[x_{2},x_{2}+R\sigma]$
and the continuity of $u$ (due to elliptic regularity),
$\inf_{[x_{2},x_{2}+R\sigma]}u>0$. Hence, $\underline{u}_{A}<u$ on
$[x_{2},x_{2}+R\sigma]$ if $A$ is sufficiently small. We then “raise”
$\underline{u}_{A}$ up until it touches $u$; i.e., let

$A_{0}=\sup\\{A:\underline{u}_{A}(x)<u(x)\text{ for
}x\in[x_{2},x_{2}+R\sigma]\\}.$

Thus, there exists $x_{0}\in(x_{2},x_{2}+R\sigma)$ such that
$\underline{u}_{A_{0}}(x_{0})=u(x_{0})$. Define

$\varphi(x)=u(x)-\underline{u}_{A_{0}}(x).$

Note that $\varphi\geq 0$ on $[x_{2},x_{2}+R\sigma]$,
$\varphi(x_{2}),\varphi(x_{2}+L\sigma)>0$, and that $x_{0}$ is the location of
a zero minimum of $\varphi$, which yields

$\varphi(x_{0})=0,\quad\varphi_{x}(x_{0})=0,\quad\text{ and
}\quad\varphi_{xx}(x_{0})\geq 0.$ (4.8)

Also, notice that, since $x_{0}>x_{2}$, then
$u(x_{0})<\theta/\sigma<(1-\chi/\sigma)^{-1}$, which yields
$\underline{u}_{A_{0},x}(x_{0})=u_{x}(x_{0})<0$.

We now derive a differential inequality for $\varphi$. Indeed, using (1.7),
(4.8), and that $\varphi(x_{0})=0$, we find, at $x_{0}$,

$\begin{split}0\geq(v-c)\varphi_{x}-&\varphi_{xx}-(1-v_{x})\varphi=-u^{2}-(v-c)\underline{u}_{A_{0},x}+\underline{u}_{A_{0},xx}+\underline{u}_{A_{0}}(1-v_{x}).\end{split}$
(4.9)

By (4.6), $v-c\geq 0$ at $x_{0}$. In addition, by Lemma 2.3 and the choice of
$x_{2}$, $\underline{u}_{A_{0},x}(x_{0})=u_{x}(x_{0})<0$. Thus, (4.9) becomes,
at $x_{0}$,

$0\geq-u^{2}+\underline{u}_{A_{0},xx}+\underline{u}_{A_{0}}(1-v_{x}).$ (4.10)

Since $x_{0}>x_{2}$, $u(x_{0})<\theta/\sigma$. Also, recall that
$u(x_{0})=\underline{u}_{A_{0}}(x_{0})$ and, by (2.1),

$v_{x}(x_{0})=\frac{|\chi|}{\sigma}u(x_{0})-|\chi|\int_{0}^{\infty}K_{\sigma,x}(y)(u(x_{0}-y)+u(x_{0}+y))\,dy.$

Hence, $-u(x_{0})^{2}>-\underline{u}_{A_{0}}(x_{0})\theta/\sigma$ and
$-v_{x}(x_{0})>-|\chi|\theta/\sigma^{2}$. Using these two inequalities along
with (4.10) yields, at $x_{0}$,

$0>\underline{u}_{A_{0},xx}+\underline{u}_{A_{0}}\left(1-\frac{\theta}{\sigma}-\frac{\theta|\chi|}{\sigma^{2}}\right).$
(4.11)

Using the explicit form of $\underline{u}_{A_{0}}$ above,
$\underline{u}_{A_{0},xx}=-(\pi/R\sigma)^{2}\underline{u}_{A_{0}}$; hence,

$0>\underline{u}_{A_{0}}(x_{0})\left(1-\frac{\theta}{\sigma}-\frac{|\chi|\theta}{\sigma^{2}}-\frac{\pi^{2}}{R^{2}\sigma^{2}}\right).$

Recalling (4.2) and (4.3), we obtain

$0>\underline{u}_{A_{0}}(x_{0})\left(1-\frac{\theta}{\sigma}-\frac{|\chi|\theta}{\sigma^{2}}-\frac{\pi^{2}}{R^{2}\sigma^{2}}\right)\geq\underline{u}_{A_{0}}(x_{0})\left(1-\frac{\varepsilon}{4}-\frac{\varepsilon^{2}R}{8}-\pi^{2}\varepsilon^{2}\right)\geq
0,$

which is a contradiction (recall that $\varepsilon<1/10$ and $R\leq 1$). This
completes the proof. ∎

## References