# Pushed, pulled and pushmi-pullyu fronts of the Burgers-FKPP equation

Jing An111Max-Planck Institute for Mathematics in the Sciences, Inselstr. 22,
04103 Leipzig, Germany<EMAIL_ADDRESS>Christopher Henderson222Department
of Mathematics, University of Arizona, Tucson, AZ 85721, USA;
<EMAIL_ADDRESS>Lenya Ryzhik333Department of Mathematics,
Stanford University, Stanford, CA 94305, USA<EMAIL_ADDRESS>
###### Abstract

We consider the long time behavior of the solutions to the Burgers-FKPP
equation with advection of a strength $\beta\in{\mathbb{R}}$. This equation
exhibits a transition from pulled to pushed front behavior at $\beta_{c}=2$.
We prove convergence of the solutions to a traveling wave in a reference frame
centered at a position $m_{\beta}(t)$ and study the asymptotics of the front
location $m_{\beta}(t)$. When $\beta<2$, it has the same form as for the
standard Fisher-KPP equation established by Bramson [9, 10]:
$m_{\beta}(t)=2t-(3/2)\log(t)+x_{\infty}+o(1)$ as $t\to+\infty$. This form is
typical of pulled fronts. When $\beta>2$, the front is located at the position
$m_{\beta}(t)=c_{*}(\beta)t+x_{\infty}+o(1)$ with
$c_{*}(\beta)=\beta/2+2/\beta$, which is the typical form of pushed fronts.
However, at the critical value $\beta_{c}=2$, the expansion changes to
$m_{\beta}(t)=2t-(1/2)\log(t)+x_{\infty}+o(1)$, reflecting the “pushmi-pullyu”
nature of the front. The arguments for $\beta<2$ rely on a new weighted Hopf-
Cole transform that allows to control the advection term, when combined with
additional steepness comparison arguments. The case $\beta>2$ relies on
standard pushed front techniques. The proof in the case $\beta=\beta_{c}$ is
much more intricate and involves arguments not usually encountered in the
study of the Bramson correction. It relies on a somewhat hidden viscous
conservation law structure of the Burgers-FKPP equation at $\beta_{c}=2$ and
utilizes a dissipation inequality, which comes from a relative entropy type
computation, together with a weighted Nash inequality involving dynamically
changing weights.

##  1 Introduction

We consider the long time behavior of the solutions to the Burgers-FKPP
equation

$\displaystyle u_{t}+\beta
uu_{x}=u_{xx}+u-u^{2},~{}~{}t>0,~{}x\in{\mathbb{R}}.$ (1.1)

Here, $\beta\in\mathbb{R}$ is a parameter that measures the strength of the
advection effect. The relevance of this type of nonlinear advection-reaction-
diffusion model in biological and chemical applications is discussed in
Chapter 13.4 of [38]. We refer the interested reader to this book, as well as
the references therein, rather than attempting to replicate the discussion
here.

Our main interest is in the study of the transition from the “pulled” to
“pushed” nature of the Burgers-FKPP equation that happens at $\beta_{c}=2$ and
its effect on the long time behavior of the solutions. In order to motivate
the basic questions addressed in the present paper, and before introducing
this transition phenomenon, we recall the well-known results for the classical
Fisher-KPP equation as well as the notions of “pulled” and “pushed” fronts.

#### The classical results for the Fisher-KPP equation

When $\beta=0$, the Burgers-FKPP equation (1.1) reduces to the classical
Fisher-KPP equation

$u_{t}=u_{xx}+u-u^{2},$ (1.2)

that dates back to the seminal work of Fisher [18] and Kolmogorov, Petrovskii
and Piskunov [30], and has been studied extensively since. This equation
arises in numerous applications in the physical and biological sciences, also
discussed in [38]. Beyond this, it has been used to study the fine properties
of branching Brownian motion after McKean [36] discovered its connection to
(1.2); see, for instance, [1, 2, 11, 12, 39] and references therein.

Both of the original papers [18] and [30] showed that (1.2) admits traveling
wave solutions of the form $u(t,x)=U_{c}(x-ct)$ for all $c\geq c_{*}=2$, which
necessarily satisfy the ODE

$-cU_{c}^{\prime}=U_{c}^{\prime\prime}+U_{c}-U_{c}^{2},~{}~{}~{}U_{c}(-\infty)=1,~{}~{}~{}\text{and}~{}~{}~{}U_{c}(+\infty)=0.$
(1.3)

We denote by $U_{*}(x)$ the traveling wave moving with the minimal speed
$c_{*}=2$. Given $c\geq c_{*}$, solutions to (1.3) are unique up to
translation in $x$. One may fix a normalization for $U_{*}(x)$, for example,
by requiring that $U_{*}(0)=1/2$.

It was observed already in [18, 30] (albeit at a different level of
mathematical rigor) that the solution to (1.2) with an initial condition that
is a step function

$\displaystyle u(0,x)=\mathds{1}(x\leq 0):=\begin{cases}1,~{}~{}~{}&\text{if
}x\leq 0,\\\ \displaystyle 0,&\text{if }x>0,\end{cases}$ (1.4)

converges in shape to a traveling wave: there exists a reference frame $m(t)$
so that

$u(t,x+m(t))\to U_{*}(x),~{}~{}\hbox{ as $t\to+\infty$, uniformly in
$x\in{\mathbb{R}}$.}$ (1.5)

It was also argued informally in [18] and proved in [30] that

$m(t)=2t+o(t)\hbox{ as $t\to+\infty$.}$ (1.6)

We refer to $m(t)$ as the location of the front at time $t>0$ since, roughly,
it separates the regions $\\{u(t,x)\approx 1\\}$ for $x\ll m(t)$ and
$\\{u(t,x)\approx 0\\}$ for $x\gg m(t)$.

This result was refined in the pioneering works by Bramson [9, 10], who used
probabilistic techniques and the connection of (1.2) to branching Brownian
motion to analyze the front position $m(t)$. In particular, Bramson showed
that there exists a constant $x_{\infty}$ that depends on the initial
condition $u_{\rm in}(x)$ for (1.2), so that

$m(t)=2t-\frac{3}{2}\log t+x_{\infty}+o(1)\hbox{ as $t\to+\infty$.}$ (1.7)

This asymptotic expansion holds as long as the initial conditions $u_{\rm
in}(x)$ decay sufficiently fast to zero as $x\to+\infty$. We also mention the
related work by Uchiyama [48] and Lau [31], which are based on PDE techniques,
and more recent refinements and alternative proofs of Bramson’s result in [7,
24, 26, 40, 41, 42], that use both probabilistic and PDE methods. Very
recently, spectral techniques have been applied to study the “Bramson shift”
in [3, 4, 5], including some problems that do not obey the comparison
principle.

In particular, it was shown in [24, 41] that convergence in (1.5) is only
algebraic in time:

$|u(t,x+m(t))-U_{*}(x)|\leq\frac{C(1+|x|)e^{-x}}{\sqrt{t}},~{}~{}~{}~{}x>0.$
(1.8)

The more precise results in [24] show that the convergence rate can not be
improved to an exponential in time rate: even convergence in shape to a
traveling wave does not hold beyond the order $O(t^{-1})$. The algebraic rate
of convergence of the solution to (1.2) to a shift of a traveling wave is
closely related to the fact that FKPP fronts are “pulled”– that is, the long
time behavior of the solutions is governed by the behavior far ahead of the
front, where $u$ is small. Hence, the problem is not compact in a certain
sense, making the algebraic-in-time (rather than exponential) rate of
convergence natural.

The “pulled” behavior should be contrasted with the class of equations of the
form similar to (1.2):

$v_{t}=v_{xx}+f(v)$ (1.9)

but with solutions that behave as “pushed fronts” – that is, the long time
behavior of the solutions is governed by the behavior at the front, where $u$
is neither small nor approximately $1$. An example of such $f$ is a bistable
nonlinearity of the form $f(v)=v(1-v)(\theta-v)$, with some $\theta\in(0,1)$.
Unlike the Fisher-KPP equation, in the pushed cases solutions to the initial
value problem for (1.9) with a rapidly decaying initial condition
$v(0,x)=v_{\rm{in}}(x)$ converge to a shift of the traveling wave $U_{f}(x)$
for (1.9) exponentially fast in time: there exists $\omega>0$ such that

$|v(t,x+m_{f}(t))-U_{f}(x)|\leq Ce^{-\omega t}.$ (1.10)

Moreover, the front location has the asymptotics

$m_{f}(t)=c_{f}t+x_{\infty},~{}~{}\hbox{as $t\to+\infty$},$ (1.11)

without any logarithmic in time correction. The rate of convergence in (1.10)
is exponential in time precisely because the fronts are “pushed,” so that the
long time behavior is determined by what happens in a compact region around
the front, and not by the tail behavior as $x\to+\infty$. We refer to the
monograph [34] and extensive presentations in [16, 47] for, respectively, an
applied mathematics and a physics perspective on pushed and pulled fronts, and
to [21] for a recent mathematical analysis, among many other references.

#### The pulled to pushed fronts transition in the Burgers-FKPP equation

As we have mentioned, the long time behavior of the reaction-diffusion
equations that admit either pulled or pushed fronts is reasonably well-
understood, at least on an intuitive level. An interesting aspect of the
Burgers-FKPP equation (1.1) is that it exhibits a transition from the pulled
to pushed behavior at $\beta_{c}=2$.

The behavior of traveling waves for (1.1) already illustrates the change in
behavior at $\beta_{c}=2$. For a given $\beta\in{\mathbb{R}}$, the Burgers-
FKPP equation (1.1) admits traveling wave solutions for all $c\geq
c_{*}(\beta)$, with the minimal speed

$\displaystyle c_{*}=\begin{cases}2,~{}~{}~{}&\text{if }\beta\leq 2,\\\
\displaystyle\frac{\beta}{2}+\frac{2}{\beta},&\text{if }\beta\geq
2.\end{cases}$ (1.12)

The minimal speed traveling wave $\phi_{\beta}$ satisfies

$-c_{*}\phi_{\beta}^{\prime}+\beta\phi_{\beta}\phi_{\beta}^{\prime}=\phi_{\beta}^{\prime\prime}+\phi_{\beta}-\phi_{\beta}^{2},~{}~{}~{}~{}\phi_{\beta}(-\infty)=1,~{}~{}~{}~{}\text{and}~{}~{}~{}~{}\phi_{\beta}(+\infty)=0.$
(1.13)

Once again, the solution to (1.13) is unique only up to a translation in $x$.
We fix the normalization by requiring that $\phi_{\beta}(0)=1/2$. It happens
that the traveling wave profile for $\beta\geq 2$ is explicit. Indeed, one can
check by direct computation that

$\phi_{\beta}(x)=\frac{1}{1+e^{\beta x/2}},~{}~{}\hbox{ for $\beta\geq 2$.}$
(1.14)

On the other hand, when $\beta<2$, the profile of the minimal speed traveling
wave is, to the best of our knowledge, not explicit, and the asymptotics of
$\phi_{\beta}$ as $x\to+\infty$ are no longer purely exponential, being given
by

$\phi_{\beta}(x)\sim(Ax+B)e^{-x},~{}~{}\hbox{ as $x\to+\infty$, for
$\beta<2$,}$ (1.15)

with some $A>0$ and $B\in{\mathbb{R}}$ that depend on $\beta$. This was shown,
for instance, in [38] by a phase plane analysis. It is discussed further in
Appendix A.

As explained in Remark 1 of [21], a quantitative mathematical criterion for a
traveling wave profile $U_{c}(x)$, that moves with a speed $c\geq 0$, to be
pushed is that

$U_{c}(x)e^{cx/2}\in L^{2}({\mathbb{R}}).$ (1.16)

Otherwise, a traveling wave is pulled. We will see the motivation behind the
criterion (1.16) in the discussion of the long time behavior of the solutions
to (1.1) for $\beta>2$, which is contained in Section 7. According to this
classification, the Burgers-FKPP traveling waves are pushed for $\beta>2$, and
pulled for $\beta\leq 2$, as can be seen from (1.14) and (1.15).

While the case $\beta=2$ obviously fails the pushed front criterion (1.16), it
has an additional property distinguishing it from the case $\beta<2$. The
traveling wave, still given by

$\phi_{2}(x)=\frac{1}{1+e^{x}}$ (1.17)

due to (1.14), satisfies the weaker condition (cf. (1.16), recalling that
$c_{*}=2$ when $\beta=2$) that

$\phi_{2}(x)e^{x}=\frac{e^{x}}{1+e^{x}}\in L^{\infty}(\mathbb{R}).$ (1.18)

As we will see, this reflects a very different long time behavior of the
solutions to (1.1) at $\beta=2$ compared both to $\beta<2$ and $\beta>2$.
Borrowing the terminology of [32], we will refer to such “dual nature” fronts
at $\beta=\beta_{c}$ as “pushmi-pullyu” fronts.

#### The large time behavior of the solutions

We now describe the main results of this paper on the large time behavior of
the solutions to the Burgers-FKPP equation (1.1) with $\beta\neq 0$ and
rapidly decaying initial conditions, generalizing the results on the standard
Fisher-KPP equation (1.2) discussed above. The main new feature is the
aforementioned transition from the pulled to pushed behavior at $\beta_{c}=2$.
The analysis at $\beta=\beta_{c}$ turns out to be surprisingly delicate.

The study of the large time behavior in the present paper relies, in
particular, on the notion of steepness of the solution. While such arguments
date back to the original KPP paper [30], we were to a large extent motivated
by the definition in the recent paper of Giletti and Matano [23]. As we will
only need it for smooth functions, we can formulate their notion as follows.
Let us denote by ${\cal W}$ the class of smooth monotonically decreasing
functions $u(x)$, $x\in{\mathbb{R}}$, such that

$\lim_{x\to-\infty}u(x)=1,~{}~{}\lim_{x\to+\infty}u(x)=0.$ (1.19)

Given two functions $u_{1},u_{2}\in{\cal W}$, we say that $u_{1}$ is steeper
than $u_{2}$ if

$|u_{1}^{\prime}(u_{1}^{-1}(z))|>|u_{2}^{\prime}(u_{2}^{-1}(z))|,~{}~{}\hbox{for
all $z\in(0,1)$.}$ (1.20)

In other words, the graph of $u_{1}(x)$ is steeper than the graph of
$u_{2}(x)$ when compared at each fixed level $z\in(0,1)$, rather than at a
fixed point $x\in{\mathbb{R}}$. This notion is translation invariant; if
$u_{1}$ is steeper than $u_{2}$, it is also steeper than any translate
$u_{2}(\cdot+h)$, with a fixed $h\in{\mathbb{R}}$. Equation (1.1) has the
following important property.

######  Proposition 1.1.

Let $u_{1}(t,x)$ and $u_{2}(t,x)$ be the solutions to (1.1) with the
corresponding initial conditions $u_{10},u_{20}\in{\cal W}$. If $u_{10}$ is
steeper than $u_{20}$, then $u_{1}(t,\cdot)$ is steeper than $u_{2}(t,\cdot)$
for all $t>0$.

This result was essentially proved for the classical Fisher-KPP equation (1.2)
in the original KPP paper [30]. For the convenience of the reader, we present
the proof for the Burgers-FKPP case in Section 2. An interesting aspect here
is that while the steepness comparison is used throughout the paper, most of
its applications are not in the spirit of the elegant intersection number type
of arguments but to produce estimates. An exception is the proof of
Proposition 1.1 itself and its consequences discussed in Section 2.

The main result of the present paper is the following:

######  Theorem 1.1.

Let $u(t,x)$ be the solution to (1.1) with either (i) the initial condition
$u_{\rm in}(x)\in{\cal W}$ such that there exist $L_{1}\leq L_{2}$ so that
$u_{\rm in}(x)=1$ for all $x\leq L_{1}$ and $u_{\rm in}(x)=0$ for all $x\geq
L_{2}$, and $u_{\rm in}$ is steeper than the minimal speed traveling wave
$\phi_{\beta}$, or (ii) $u_{\rm in}(x)=\mathbbm{1}(x\leq 0)$. Then, for each
$\beta\leq 2$, there exists a constant $x_{\infty}$ that depends on $\beta$
and the initial condition $u_{\rm in}$ so that

$\displaystyle\lim_{t\to+\infty}u(t,x+m_{\beta}(t))=\phi_{\beta}(x),$ (1.21)

with the function $m_{\beta}(t)$ given by

$\displaystyle
m_{\beta}(t)=2t-\frac{3}{2}\log(t+1)-x_{\infty}+o(1),~{}\text{as}~{}t\to+\infty,$
(1.22)

if $\beta<2$, and, otherwise, by

$\displaystyle
m_{\beta=2}(t)=2t-\frac{1}{2}\log(t+1)-x_{\infty}+o(1),~{}\text{as}~{}t\to+\infty.$
(1.23)

For $\beta>2$, there exists $\omega>0$, which depends on $\beta$ but not on
$u_{\rm in}$, and $K>0$, which depends both on $\beta$ and $u_{\rm in}$, such
that

$\displaystyle\sup_{x\in\mathbb{R}}|u(t,x)-\phi_{\beta}(x-c_{*}t-x_{\infty})|<Ke^{-\omega
t}.$ (1.24)

This result reflects the different nature of the Burgers-FKPP fronts we have
discussed above for various values of $\beta\in{\mathbb{R}}$. For $\beta<2$,
the solution is pulled and the front location has the same asymptotics (1.22)
as (1.7) for the standard Fisher-KPP equation (1.2). For $\beta>2$, the
solution is pushed and the exponential-in-time convergence to the traveling
wave (1.24) agrees with what we have seen in (1.10) for pushed fronts. The new
asymptotics (1.23) for the “pushmi-pullyu” solutions at $\beta=\beta_{c}$ is
different from both of these cases.

One may ask if the asymptotics (1.22) and (1.23) can be refined, as was done
in [6, 7, 16, 24, 41] for the standard Fisher-KPP equation (1.2). It seems
that this is possible using the fascinating, if formal, technique of [7].
Arguing in the manner of [7], it turns out that the $o(1)$ terms in the
expansion of $m_{\beta}(t)$ are exactly the same as in the Fisher-KPP case
when $\beta<2$; however, they are different when $\beta=2$. In this “pushmi-
pullyu” case, one finds

$\displaystyle
m_{2}(t)=2t-\frac{1}{2}\log(t)-x_{\infty}-\frac{\sqrt{\pi}}{2\sqrt{t}}+\frac{(1-\log
2)}{4}\frac{\log t}{t}+o\Big{(}\frac{\log t}{t}\Big{)}.$ (1.25)

The first terms in this expansion above agree with the formal computation in
[16], performed by a completely different method, where it was computed up to
the order $O(t^{-1/2})$. While the error in [16] is given as $O(1/t)$, and the
$O(\log t/t)$ correction seems to be missing, it is tempting to conjecture
that (1.25) is universal for the “pushmi-pullyu” situations, such as, for
instance, the reaction-diffusion equation (1.28) below. The application of the
formal technique of [7] to derive (1.25) is contained in Section 8.

We should mention that the convergence of the solution to the Burgers-FKPP
equation (1.1) to a traveling wave was studied using the matched asymptotic
expansions in [33]. Their formal results agree with Theorem 1.1 in the cases
$\beta<2$ and $\beta>2$ but unfortunately overlook some of the details in the
case $\beta=2$, leading to an incorrect prediction. On the other hand, as we
have mentioned, the matched asymptotics analysis in [16, 34] predicts the
shift $(1/2)\log t$ for the “pushmi-pullyu” transition situations, such as
(1.28) below, mimicking, in a certain sense, the Burgers-FKPP equation at
$\beta=2$.

#### Connection to other pulled to pushed transition problems

The Burgers-FKPP equation at $\beta=2$ is not the only one example of a
“pushmi-pullyu” front. A well known instance is a particular generalized
Fisher nonlinearity considered in [25]:

$u_{t}=u_{xx}+u(1-u)(1+au).$ (1.26)

Here, the situation is qualitatively and even quantitatively extremely similar
to the Burgers-FKPP picture: traveling waves exist for all $c\geq c_{*}(a)$,
with $c_{*}(a)=2$ for all $0\leq a\leq 2$, and

$c_{*}(a)=\sqrt{\frac{a}{2}}+\sqrt{\frac{2}{a}},~{}~{}\hbox{for $a>2$,}$
(1.27)

as in (1.12). Even the traveling wave profile for $a=2$ is given exactly by
$\phi_{\beta=2}$ in (1.17).

A generalization of the above example of a pushed-pulled transition has been
discussed in [16]. The critical situation in that example is, after a simple
change of variables,

$u_{t}=u_{xx}+u(1-u^{n})(1+(n+1)u^{n}).$ (1.28)

As shown in [16], for any $n\in{\mathbb{N}}$, equation (1.28) exhibits the
“pushmi-pullyu” behavior, of which (1.26) with $a=2$ is one instance with
$n=1$. The minimal speed for (1.28) is $c_{*}=2$, independent of
$n\in{\mathbb{N}}$, and the corresponding traveling wave profile is explicit

$\phi(x)=\big{(}1+e^{nx}\big{)}^{-1/n},$ (1.29)

with the same purely exponential decay as in (1.17).

Numerous other examples, including the pushed-pulled transition for systems of
reaction-diffusion equations arising in chemistry and biology are discussed in
Section 3.13 of [47]. We also mention the repulsive Keller-Segel-FKPP equation
that was recently used to model the spread of a population in which
individuals reproduce and diffuse, influenced by a preference for low
population density regions with strength $|\chi|$. This is seen, for instance,
in slime molds [35] and bacteria [50]. Origins of some other similar
chemotaxis models have been discussed in [19, 20]. The pushed-pulled
transition at the level of traveling waves has been shown in [28]: there
exists a threshold $\chi_{0}>0$ such that, when $|\chi|<\chi_{0}$, the fronts
are pulled, while, as $\chi\to-\infty$, fronts are pushed.

A matched asymptotic analysis in [16, 34] predicts the results as in Theorem
1.1 to hold for the long time behavior of the solutions of equations at the
pushed-pulled transition, such as (1.28). However, to the best of our
knowledge, there are no rigorous results with this precision in any of such
critical cases. The best result in this direction seems to be the very recent
paper [22], which shows that when $a=2$, the level sets of the solutions to
(1.26) are located at a position

$X(t)=2t-\frac{1}{2}\log t+o(\log t),~{}~{}\hbox{ as $t\to+\infty$.}$

As discussed in detail in [4] and [5], the threshold cases that separate the
pulled and pushed fronts seem to be also outside the scope of the currently
available spectral methods.

#### Comments on the proofs

Let us comment on the strategy of the proof of the three cases in Theorem 1.1,
in the order of increasing difficulty and intricacy. The case $\beta>2$ falls
into the category of pushed fronts, and the proof follows the classical
strategy of [43, 44, 45], with appropriate modifications.

For $\beta<2$ in Theorem 1.1, we use an extension of the arguments in [26, 40]
for the Fisher-KPP equation, approximating the dynamics by the Dirichlet
problem for the linear heat equation on a half line. The Burgers drift term
causes a difficulty, since the linearization strategy used, for instance, in
[24, 27, 26, 40, 41] taking out the spatial exponential decay in the solution
seems insufficient for $0<\beta<2$. To overcome this, we pass to the moving
frame $x\to x-2t$, setting

$\widehat{u}(t,x)=u(t,x+2t),$ (1.30)

and introduce a weighted Hopf-Cole transform

$\displaystyle
v(t,x)=\exp\Big{(}x+\frac{\beta}{2}\int_{x}^{\infty}\widehat{u}(t,y)dy\Big{)}\widehat{u}(t,x),$
(1.31)

combining the standard Hopf-Cole transform for the heat equation and the
exponential weight used in the standard Fisher-KPP arguments. It turns out
that if the initial condition $u_{\rm in}$ for (1.1) is steeper than the
traveling wave $\phi_{\beta}$, then $v(t,x)$ satisfies a differential
inequality

$v_{t}\leq v_{xx},$ (1.32)

as in Proposition 3.1 below. We stress that the steepness comparison of the
initial condition to a traveling wave plays a crucial role in showing that
$v(t,x)$ satisfies the differential inequality (1.32).

With (1.32) in hand, we are able to construct upper and lower barriers in the
self-similar variables for the linearized equation for $v(t,x)$ on the half
line, and then the convergence in the tail implies the convergence in the bulk
due to the pulled-front nature of the dynamics, as in [40]. Interestingly,
this last step also utilizes the assumption that the initial condition, and
hence the solution, is steeper than the minimal speed traveling wave, in an
explicit quantitative way. Qualitatively, the case $\beta<2$ is similar to the
standard Fisher-KPP equation, and the weighted Hopf-Cole transform gives a
tool to see that. However, the repeated use of the steepness comparison is
something new in this argument for Burgers-FKPP equation.

The weighted Hopf-Cole transform also indicates the technical reason for the
transition at $\beta=2$: while it is easy to see from (1.31) that $v(t,x)\to
0$ as $x\to-\infty$ for $\beta<2$ (recall that $\widehat{u}\leq 1$), in the
case $\beta=2$ the function $v(t,x)$ approaches a positive constant as
$x\to-\infty$. This modifies the boundary condition for the linearized problem
for the upper and lower barriers in the self-similar variables, and ultimately
leads to the change in the logarithmic shift from $(3/2)\log t$ to $(1/2)\log
t$ at $\beta=2$.

Let us now discuss the ingredients of the the proof of Theorem 1.1 in the
critical case $\beta=2$, which is remarkably different from the approach for
the standard Fisher-KPP equation. This analysis is probably the most novel
part of the present paper. The first key observation is that when $\beta=2$,
the Burgers-FKPP equation (1.1) has a special structure: the function

$p(t,x)=e^{x}\widehat{u}(t,x),$ (1.33)

satisfies a spatially inhomogeneous conservation law:

$\displaystyle p_{t}+(e^{-x}p^{2})_{x}=p_{xx}.$ (1.34)

Here, $\widehat{u}(t,x)$ is defined in (1.30). An immediate consequence of
(1.34) is a conservation law for the exponential moment of $\widehat{u}(t,x)$:

$\int e^{x}\widehat{u}(t,x)dx=\int e^{x}u_{\rm in}(x)dx,~{}~{}\hbox{ for all
$t>0$.}$ (1.35)

This conservation law eventually leads to a lower bound for $m_{2}(t)$ of the
form

$m_{2}(t)\geq 2t-\frac{1}{2}\log t+O(1),~{}~{}\hbox{as $t\to+\infty$,}$ (1.36)

see the proof of Lemma 5.2 in Section 5.3 below.

A matching upper bound for $m_{2}(t)$ is related to the behavior of $p(t,x)$.
Note that, together with the explicit expression (1.17) for the profile
$\phi_{2}(x)$, the convergence to a traveling wave in shape in (1.21) yields,
roughly,

$p(t,0)\approx\exp\big{(}-(2t-m_{2}(t))\big{)}.$ (1.37)

Thus, an upper bound of the form

$m_{2}(t)\leq 2t-\frac{1}{2}\log t+O(1),~{}~{}\hbox{as $t\to+\infty$,}$ (1.38)

would follow from an $L^{\infty}$-bound on $p(t,x)$ of the form

$p(t,x)\leq\frac{C}{\sqrt{t}}.$ (1.39)

Such decay, while natural to expect in view of (1.34), is not automatic for
solutions of mass-conserving advection-diffusion equations, even if the
advection is bounded: see the end of Section 5.1 for simple examples of such
equations with solutions that do not decay in time.

The proof of (1.39), presented in Section 5, turns out to be rather intricate.
While (1.34) looks like a degenerate viscous conservation law, we were unable
to adapt the methods of [14] or [29] to (1.34) and instead take a different
approach. The first step is a relative entropy computation inspired by [15,
37] where it was used for linear advection-diffusion equations. An unusual
twist is that we compute the relative entropy not with respect to another
solution but to a super-solution to (1.34). This leads to a weighted
dissipation inequality for the function

$\varphi(t,x)=\frac{p(t,x)}{\rho(t,x)},\quad\text{ where
}\rho(t,x)={1-u(t,x+2t)},$ (1.40)

of the form

$\frac{d}{dt}\int\
\varphi^{2}(t,x)\rho(t,x)dx\leq-2\int\varphi_{x}^{2}(t,x)\rho(t,x)dx.$ (1.41)

The dissipation identity (1.41) is similar to that for the standard heat
equation, where it takes the form

$\frac{d}{dt}\int\ \varphi^{2}(t,x)dx\leq-2\int\varphi_{x}^{2}(t,x)dx,$ (1.42)

that is, as in (1.41) but without the weight $\rho(t,x)$. In the latter case,
(1.42) combined with the Nash inequality and a standard duality argument
directly leads to the temporal decay rate $t^{-d/2}$ in ${\mathbb{R}}^{d}$.
Here, the time-dependent weight $\rho(t,x)$ that appears in (1.41) is
degenerate as $x\to-\infty$, so the standard Nash inequality can not be used.
Instead, we obtain a Nash-type inequality for weighted spaces for a certain
class of degenerate weights: see Proposition 5.9 below. The weights need to
satisfy certain quantitative assumptions, and we need to verify that the
dynamics do not take the weight $\rho(t,x)$, defined in (1.40), out of the
class of the admissible weights or make the constants in the weighted Nash
inequality in Proposition 5.9 degenerate as $t\to+\infty$. Applying the
weighted Nash inequality in (1.41), leads to the appropriate decay of
$\varphi(t,x)$ in a weighted $L^{2}$-space. However, the nature of the Nash
inequality leads to an extra delay in time, before the decay sets in, that
depends on the initial condition. This, among other issues, prevents us from
using the duality argument to establish the weighted $L^{\infty}$-decay of
$\varphi(t,x)$ from the $L^{2}$-estimate. Instead, it comes from additional ad
hoc arguments, first establishing the bound at a large time $t^{\prime}$ less
than, but comparable to, $t$ and then extending it to the final time $t$. The
decay of $p(t,x)$ follows from that of $\varphi(t,x)$ and bounds on the
weights.

An extra technical complication is that the function $\rho(t,x)$ appears in
the denominator in the definition (1.40) of the function $\varphi(t,x)$ but
vanishes as $x\to-\infty$. As a result, the $L^{2}$-norm of $\varphi(t,x)$ is
actually infinite for a large class of interesting initial conditions and
extra approximations have to be used to deal with this issue. At this stage,
the assumption that $u_{\rm in}(x)=1$ for $x\leq L_{1}$ is actually not a
simplification but a complication that can not be avoided if one wants to
include the Heaviside function into the class of admissible initial
conditions. The need to control the behavior in the back of the front is
another reflection of the “pushmi-pullyu” nature of the solution.

#### Organization of the paper

This paper is organized as follows. Section 2 uses the steepness comparison
arguments to prove Proposition 1.1 and convergence of the solution to a
traveling wave in shape. Section 3 describes the weighted Hopf-Cole transform
leading to the differential inequality (1.32). The proof of Theorem 1.1 for
$\beta<2$ is contained in Section 4. Section 5 is devoted to the decay
estimates of the solutions to the inhomogeneous viscous conservation law
(1.34) that appears in the case $\beta=2$. It is here that we prove the
aforementioned $L^{\infty}$-decay of the solutions to (1.34). Section 6 uses
these results to prove Theorem 1.1 for $\beta=2$, bootstrapping the bounds
(1.36) and (1.38) to the precise asymptotics (1.23). The case $\beta>2$ is
considered in Section 7. Section 8 uses the techniques of [7] to obtain, by
formal arguments, further corrections to the front location asymptotics given
in Theorem 1.1 for $\beta\leq 2$. The result for $\beta<2$ is identical to the
standard Fisher-KPP equation but is different for $\beta=2$. Finally, Appendix
A contains some basic facts about the traveling waves for the Burgers-FKPP
equation.

Acknowledgment. JA was supported by Joe Oliger Fellowship. CH was partially
supported by NSF grant DMS-2003110. LR was partially supported by NSF grant
DMS-1910023, and ONR grant N00014-17-1-2145. We are grateful to John Leach for
a very friendly and helpful discussion of the results in [33] and [34].

##  2 Convergence to a traveling wave in shape

In this section, as a preliminary step to the proof of Theorem 1.1, we use a
strategy inspired by the original KPP paper [30] to show convergence in shape
of a solution to the Burgers-FKPP equation to a traveling wave. As the first
step, we prove Proposition 1.1.

#### The proof of Proposition 1.1

Let $u_{1}(t,x)$ and $u_{2}(t,x)$ be the solutions to (1.1) with the
respective initial conditions $u_{10},u_{20}\in{\cal W}$ such that $u_{10}$ is
steeper than $u_{20}$. First, we note that since the initial conditions are
decreasing, both $u_{1}(t,x)$ and $u_{2}(t,x)$ are decreasing and have the
left and right limits as in (1.19), so that both $u_{1}(t,\cdot)$ and
$u_{2}(t,\cdot)$ lie in $\cal W$.

To show that $u_{1}(t,\cdot)$ is steeper than $u_{2}(t,\cdot)$ for any $t>0$,
consider the functions

$w(t,x;k_{0})=u_{1}(t,x)-u_{2}(t,x+k_{0}),~{}~{}q(t,x)=u_{1}(t,x)+u_{2}(t,x+k_{0}),$

for a fixed $k_{0}\in{\mathbb{R}}$. The function $w(t,x;k_{0})$ satisfies

$w_{t}+\frac{\beta}{2}(qw)_{x}=w_{xx}+w-qw,$ (2.1)

with the initial condition

$w(0,x;k_{0})=u_{10}(x)-u_{20}(x+k_{0}).$ (2.2)

Since $u_{10}$ is steeper than $u_{20}$, it is also steeper than
$u_{20}(\cdot+k_{0})$. Therefore, there exists $x_{0}$ so that

$\hbox{$w(0,x;k_{0})>0$ for all $x<x_{0}$},$

and

$w(0,x;k_{0})<0$ for all $x>x_{0}$.

As $w(t,x;k_{0})$ is a solution to the parabolic equation (2.1), the strong
maximum principle implies that the function $w(t,x;k_{0})$ has exactly one
zero $y(t;k_{0})$ for all $t>0$, so that $w(t,x;k_{0})>0$ for all
$x<y(t;k_{0})$ and $w(t,x;k_{0})<0$ for all $x>y(t;k_{0})$, with
$y(0;k_{0})=x_{0}$. In addition, we have

$\partial_{x}u_{1}(t,y(t;k_{0}))<\partial_{x}u_{2}(t,y(t;k_{0})).$ (2.3)

Since this is true for all $k_{0}\in{\mathbb{R}}$, it follows that
$u_{1}(t,\cdot)$ is steeper than $u_{2}(t,\cdot)$. $\Box$

A standard approximation argument shows the following.

######  Corollary 2.1.

Let $v(t,x)$ and $u(t,x)$ be the solutions to (1.1) with the respective
initial conditions $v_{\rm in}\in{\cal W}$ and $u_{\rm
in}(x)=\mathbbm{1}(x\leq 0)$. Assume that $v_{\rm in}(x)$ is steeper than the
minimal speed traveling wave $\phi_{\beta}(x)$. Then, for any $t>0$ the
solution $v(t,\cdot)$ is steeper than $\phi_{\beta}$, and is less steep than
$u(t,x)$.

#### Convergence in shape

We now establish convergence of the solution in shape to a traveling wave.

######  Proposition 2.2.

Let $u(t,x)$ be the solution to (1.1) with the initial condition $u_{\rm
in}\in{\cal W}$ that is steeper than the minimal speed traveling wave
$\phi_{\beta}(x)$, or with $u_{\rm in}(x)=\mathbbm{1}(x\leq 0)$. Then, there
exists a function $m_{\beta}(t)$ such that $m_{\beta}^{\prime}(t)\to
c_{*}(\beta)$ as $t\to+\infty$ and

$u(t,x+m_{\beta}(t))\to\phi_{\beta}(x)~{}~{}\hbox{as $t\to+\infty$, uniformly
on ${\mathbb{R}}$.}$ (2.4)

Here, $\phi_{\beta}(x)$ is a solution to (1.13) with the minimal speed
$c_{*}=c_{*}(\beta)$.

Corollary 2.1 shows that it suffices to consider the solution $u(t,x)$ to
(1.1) with the initial condition $u(0,x)=\mathbbm{1}(x\leq 0)$. Note that for
any $\tau>0$, the function $u^{(\tau)}(t,x)=u(t+\tau,x)$ is the solution to
(1.1) with the initial condition $u^{(\tau)}(0,x)=u(\tau,x)$ that is less
steep than $u(0,x)$. It follows that for any $t>0$ and $\tau>0$ the function
$u(t,\cdot)$ is steeper than $u(t+\tau,\cdot)$. In addition, $u(t,\cdot)$ is
steeper than the minimal speed traveling wave $\phi_{\beta}(x)$ for all $t>0$.
Hence, if for each $v\in(0,1)$ and $t>0$, we let $x(t,v)$ be the unique point
such that $u(t,x(t,v))=v$, then, the function

$E(t,v)=u_{x}(t,x(t,v))\leq 0,$ (2.5)

is increasing in $t$ for all $v\in(0,1)$, and

$E(t,v)\leq\bar{E}(v):=\phi_{\beta}^{\prime}(\phi_{\beta}^{-1}(v)).$ (2.6)

Let now $m_{\beta}(t)$ be the position such that $u(t,m_{\beta}(t))=1/2$ for
all $t>0$, and consider the translate

$\tilde{u}(t,x)=u(t,x+m_{\beta}(t)),$

as well as the corresponding inverse $\xi(t,v)$ defined by
$\tilde{u}(t,\xi(t,v))=v$, for $0<v<1$. Observe that $\xi(t,1/2)=0$ for all
$t>0$. We see from (2.5)-(2.6) that the function $E(t,v)$ is negative and
increasing in time. Thus, it has a limit

$E(t,v)\to
E_{\infty}(v)\leq\bar{E}(v)=\phi_{\beta}^{\prime}(\phi_{\beta}^{-1}(v))<0,~{}~{}\hbox{as
$t\to+\infty$.}$ (2.7)

Hence

$\frac{\partial{\xi(t,v)}}{\partial{v}}=\frac{1}{E(t,v)}\to\frac{1}{E_{\infty}(v)},~{}~{}\hbox{as
$t\to+\infty$,}$

and

$\xi(t,v)=\int_{1/2}^{v}\frac{\partial{\xi(t,v^{\prime})}}{\partial{v^{\prime}}}dv^{\prime}\to\int_{1/2}^{v}\frac{dv^{\prime}}{E_{\infty}(v^{\prime})}:=\xi_{\infty}(v).$
(2.8)

As a consequence, the function $\tilde{u}(t,x)$ also converges uniformly on
compact sets to a limit $\tilde{u}_{\infty}(x)$:

$\tilde{u}(t,x)\to\tilde{u}_{\infty}(x)\hbox{ as $t\to+\infty$,}$ (2.9)

with $\tilde{u}_{\infty}(x)$ determined by

$\xi_{\infty}(\tilde{u}_{\infty}(x))=x.$ (2.10)

Moreover, due to (2.7), we have

$|\xi_{\infty}(v)|=\int_{1/2}^{v}\frac{dv^{\prime}}{|E_{\infty}(v^{\prime})|}\leq\int_{1/2}^{v}\frac{dv^{\prime}}{|\bar{E}(v^{\prime})|}:=\bar{\xi}(v).$
(2.11)

This yields the correct behavior of the limits $x\to\pm\infty$:

$\tilde{u}_{\infty}(-\infty)=1,~{}~{}\tilde{u}_{\infty}(+\infty)=0.$ (2.12)

Indeed, considering, for example the behavior $x\to+\infty$, we have

$+\infty=\lim_{x\to+\infty}x=\lim_{x\to+\infty}\xi_{\infty}(\tilde{u}_{\infty}(x))\leq\bar{\xi}(\tilde{u}_{\infty}(x)).$

and $\bar{\xi}^{-1}(\infty)=0$. The argument for $x\to-\infty$ is similar.

Furthermore, as $u(t,x)$ is strictly decreasing in $x$ and
$u_{x}(t,m_{\beta}(t))<0$, the function $m_{\beta}(t)$ is differentiable in
$t$. Hence, $\tilde{u}(t,x)$ satisfies

$\tilde{u}_{t}-\dot{m}_{\beta}(t)\tilde{u}_{x}+\beta\tilde{u}\tilde{u}_{x}=\tilde{u}_{xx}+\tilde{u}(1-\tilde{u}).$
(2.13)

Notice that

$\dot{m}_{\beta}(t)=-\frac{u_{t}(t,m_{\beta}(t))}{u_{x}(t,m_{\beta}(t))}=-\frac{u_{t}(t,m_{\beta}(t))}{\tilde{u}_{x}(t,0)}.$

By parabolic regularity theory, the numerator is bounded and, by the
convergence of $\tilde{u}$, the denominator is bounded away from zero. It
follows that $\dot{m}_{\beta}$ is bounded uniformly in $t$. Then, passing to
the limit $t\to+\infty$ and keeping the convergence (2.9) in mind we deduce
that there exists $c\in{\mathbb{R}}$ such that $\dot{m}_{\beta}(t)\to c$ as
$t\to+\infty$ and $\tilde{u}_{\infty}(x)$ satisfies

$-c\partial_{x}\tilde{u}_{\infty}+\beta\tilde{u}_{\infty}\partial_{x}\tilde{u}_{\infty}=\partial_{x}^{2}\tilde{u}_{\infty}+\tilde{u}_{\infty}(1-\tilde{u}_{\infty}),$
(2.14)

where we have switched to $\partial$ notation to avoid the awkward double
subscript.

From (2.14), we see that $\tilde{u}_{\infty}(x)$ is a traveling wave solution
to (1.1) moving with the speed $c$. It remains to show that $c=c_{*}(\beta)$.
The key point is that the steepness comparison argument above applies to any
traveling wave solution to

$-c\phi_{x}+\beta\phi\phi_{x}=\phi_{xx}+\phi-\phi^{2}.$ (2.15)

In other words, if we set

$E_{\phi}(v)=\phi^{\prime}(\phi^{-1}(v)),~{}~{}\hbox{ for $0<v<1$},$

then we know that

$E_{\infty}(v)\leq E_{\phi}(v),$

for any $\phi$ that satisfies (2.15) with some $c\geq c_{*}(\beta)$.
Therefore, the limit $\tilde{u}_{\infty}(x)$ is the traveling wave that is the
steepest among all traveling wave solutions. Lemma A.2 implies that
$\tilde{u}_{\infty}(x)=\phi_{\beta}(x)$ is the minimal speed traveling wave.
This finishes the proof of Proposition 2.2. $\Box$

##  3 The weighted Hopf-Cole transform

In this section, we discuss a weighted Hopf-Cole transform that will play a
key role in the analysis of the Burgers-FKPP equation for $\beta\leq 2$. Let
us recall that the standard Burgers equation

$u_{t}+\beta uu_{x}=u_{xx}$ (3.1)

can be linearized by means of the Hopf-Cole transform. Namely, if $u$ is a
solution to (3.1) then the function

$v(t,x)=\exp\Big{(}\frac{\beta}{2}\int_{x}^{+\infty}u(t,y)dy\Big{)}$ (3.2)

satisfies the heat equation

$v_{t}=v_{xx}.$ (3.3)

The second simple observation is that if $\widehat{u}(t,x)$ is the solution to
the standard Fisher-KPP equation in a frame moving with the speed $c_{*}=2$:

$\widehat{u}_{t}-2\widehat{u}_{x}=\widehat{u}_{xx}+\widehat{u}-\widehat{u}^{2},$
(3.4)

then the function

$v(t,x)=e^{x}\widehat{u}(t,x)$ (3.5)

satisfies

$v_{t}=v_{xx}-e^{-x}v^{2}.$ (3.6)

The nonlinear term in (3.6) is negligible for $x$ very large and positive but
plays the role of a large absorption for $x$ very negative. Therefore, the
solution to (3.6) should be well approximated by the solution of the heat
equation on a half-line $x>0$ with the Dirichlet boundary condition:

$\displaystyle v_{t}=v_{xx},~{}~{}x>0,$ (3.7) $\displaystyle v(t,0)=0.$

This simple idea is what is driving the convergence to a traveling proofs in
[24, 26, 40, 41].

The weighted Hopf-Cole transform that we discuss below allows us to adapt this
intuition to the Burgers-FKPP equation (1.1) with $\beta\leq 2$, and also
shows why the transition from pulled to pushed fronts happens at $\beta=2$.

We will consider the solution to (1.1) in the reference frame

$\tilde{u}(t,x)=u(t,x+m_{\beta}(t)),$ (3.8)

centered at

$m_{\beta}(t)=2t-\frac{r(\beta)}{2}\log(t+1).$ (3.9)

Here, we take

$r(\beta)=\left\\{\begin{matrix}3,~{}~{}\hbox{ if $\beta<2$},\cr
1,~{}~{}\hbox{ if $\beta=2$},\cr\end{matrix}\right.$ (3.10)

in accordance with the different behavior in Theorem 1.1 in these two cases.
In the above reference frame, (1.1) takes the form

$\tilde{u}_{t}-\Big{(}2-\frac{r(\beta)}{t+1}\Big{)}\tilde{u}_{x}+\beta\tilde{u}\tilde{u}_{x}=\tilde{u}_{xx}+\tilde{u}-\tilde{u}^{2}.$
(3.11)

Motivated by (3.2) and (3.5), we introduce the weighted Hopf-Cole transform

$\displaystyle
v(t,x)=\exp(\Gamma(t,x))\tilde{u}(t,x),~{}~{}\Gamma(t,x)=x+\frac{\beta}{2}\int_{x}^{+\infty}\tilde{u}(t,y)dy,$
(3.12)

that is a combination of (3.2) and (3.5). Note that, as long as $\beta<2$, the
function $v(t,x)$ tends to zero as $x\to-\infty$, and if $\beta=2$, then we
can only expect it to remain positive and bounded as $x\to-\infty$.
Furthermore, if $\beta>2$ then $v(t,x)$ should blow up as $x\to-\infty$. These
differences reflect the three different behaviors in Theorem 1.1.

The main result of this section is the following analogue of (3.6) in the
standard Fisher-KPP case. It will allow us to adapt an approximation similar
to the linear Dirichlet boundary problem (3.7) for $\beta<2$ in Section 4 and
with a different boundary condition for $\beta=2$ in Section 6. This will be
extremely important for the proof of Theorem 1.1 for $\beta\leq 2$.

######  Proposition 3.1.

Let $u(t,x)$ be the solution to (1.1) with $\beta\leq 2$ and the initial
condition $u(0,x)$ as in Theorem 1.1. Then, the function $v(t,x)$ defined in
(3.12) satisfies the differential inequality

$v_{t}-v_{xx}+\frac{r(\beta)}{2(t+1)}(v_{x}-v)\leq 0.$ (3.13)

As we will see in the proof, it is here, among other places, that the
steepness assumption on the initial condition $u(0,x)$ plays a crucial role,
together with propagation of steepness in Proposition 1.1.

#### Proof of Proposition 3.1

We claim that the function $v(t,x)$ satisfies an equation of the form

$v_{t}-v_{xx}+\frac{r(\beta)}{2(t+1)}(v_{x}-v)=-G(t,x;\tilde{u})v,$ (3.14)

where

$G(t,x;\tilde{u})=\tilde{u}(t,x)-\frac{\beta}{2}\int_{x}^{+\infty}\tilde{u}(t,y)(1-\tilde{u}(t,y))dy.$
(3.15)

Let us verify that (3.14) holds. We compute

$v_{t}=e^{\Gamma}\tilde{u}_{t}+\frac{\beta
v}{2}\int_{x}^{+\infty}\tilde{u}_{t}(t,y)dy\quad\text{ and }\quad
v_{x}=e^{\Gamma}\tilde{u}_{x}+\Big{(}1-\frac{\beta}{2}\tilde{u}\Big{)}v,$
(3.16)

so that

$\displaystyle v_{xx}$
$\displaystyle=e^{\Gamma}\tilde{u}_{xx}+\Big{(}1-\frac{\beta}{2}\tilde{u}\Big{)}e^{\Gamma}\tilde{u}_{x}-\frac{\beta}{2}\tilde{u}_{x}v+\Big{(}1-\frac{\beta}{2}\tilde{u}\Big{)}v_{x}$
(3.17)
$\displaystyle=e^{\Gamma}\tilde{u}_{xx}+2\Big{(}1-\frac{\beta}{2}\tilde{u}\Big{)}v_{x}-\Big{(}1-\frac{\beta}{2}\tilde{u}\Big{)}^{2}v-\frac{\beta}{2}\tilde{u}_{x}v.$

Using these identities in (3.11) gives

$\displaystyle v_{t}$ $\displaystyle-\frac{\beta
v}{2}\int_{x}^{+\infty}\tilde{u}_{t}(t,y)dy-\Big{(}2-\frac{r(\beta)}{2(t+1)}\Big{)}\Big{(}v_{x}-\Big{(}1-\frac{\beta}{2}\tilde{u}\Big{)}v\Big{)}+\beta
v\tilde{u}_{x}$ (3.18)
$\displaystyle=v_{xx}-2\Big{(}1-\frac{\beta}{2}\tilde{u}\Big{)}v_{x}+\Big{(}1-\frac{\beta}{2}\tilde{u}\Big{)}^{2}v+\frac{\beta}{2}v\tilde{u}_{x}+v-v\tilde{u}.$

To simplify this equation, we integrate (3.11) from $x$ to $\infty$ to get

$\displaystyle\int_{x}^{+\infty}\tilde{u}_{t}(t,y)dy=-\Big{(}2-\frac{r(\beta)}{2(t+1)}\Big{)}\tilde{u}+\frac{\beta}{2}\tilde{u}^{2}-\tilde{u}_{x}+\int_{x}^{+\infty}\tilde{u}(1-\tilde{u})dy.$
(3.19)

Substituting this back into (3.18) gives, after some algebra

$\displaystyle v_{t}$ $\displaystyle-
v_{xx}+\frac{r(\beta)}{2(t+1)}\big{(}v_{x}-v\big{)}+\beta v\tilde{u}_{x}$
(3.20) $\displaystyle=\frac{\beta
v}{2}\int_{x}^{+\infty}\tilde{u}(1-\tilde{u})dy+{\beta}\tilde{u}v_{x}+\frac{\beta^{2}}{2}\tilde{u}^{2}v-\beta\tilde{u}v-v\tilde{u}.$

Using the second identity in (3.16) in the right side gives

$\displaystyle v_{t}$ $\displaystyle-
v_{xx}+\frac{r(\beta)}{2(t+1)}\big{(}v_{x}-v\big{)}=\frac{\beta
v}{2}\int_{x}^{+\infty}\tilde{u}(1-\tilde{u})dy-v\tilde{u},$ (3.21)

which is exactly (3.14)-(3.15).

Here is the key observation.

######  Lemma 3.2.

If $\beta\leq 2$ and the initial condition $u(0,x)$ is as in Theorem 1.1, then
$G(t,x;\tilde{u})\geq 0$ for all $t>0$ and $x\in{\mathbb{R}}$.

Proof. The claim is trivially true for $\beta\leq 0$, so we only consider the
case $0<\beta\leq 2$. Let us first show that a traveling wave
$\phi_{\beta}(x)$ satisfies the inequality

$\frac{\beta}{2}\int_{x}^{+\infty}\phi_{\beta}(y)(1-\phi_{\beta}(y))dy\leq\phi_{\beta}(x),~{}~{}\hbox{
for all $x\in{\mathbb{R}}$.}$ (3.22)

We will prove (3.22) for $\beta<2$, and deduce the conclusion for $\beta=2$ by
continuity. Recall that for $\beta<2$ the traveling wave has the asymptotics
(A.24):

$\phi_{\beta}(x)\sim(Ax+B)e^{-x},~{}~{}\hbox{ as $x\to+\infty$,}$

with some constants $A$ and $B$. As $\beta<2$, it follows that there exists
$L>0$ so that (3.22) holds for all $x\geq L$, and we only need to prove this
inequality for $x<L$.

Next, we integrate the traveling wave equation, with $c_{*}(\beta)=2$:

$\displaystyle-$
$\displaystyle\phi_{\beta}^{\prime\prime}-2\phi_{\beta}^{\prime}+\beta\phi_{\beta}\phi_{\beta}^{\prime}=\phi_{\beta}-\phi_{\beta}^{2},$
(3.23) $\displaystyle\phi_{\beta}(-\infty)=1,~{}~{}\phi_{\beta}(+\infty)=0,$

from $x$ to $+\infty$ to get

$\displaystyle\phi_{\beta}^{\prime}(x)+2\phi_{\beta}(x)-\frac{\beta}{2}\phi_{\beta}^{2}(x)=\int_{x}^{+\infty}\phi_{\beta}(y)(1-\phi_{\beta}(y))dy.$
(3.24)

Passing to the limit $x\to-\infty$ in (3.24), and keeping in mind that
$\phi_{\beta}(-\infty)=1$, gives

$\displaystyle\frac{\beta}{2}\int_{-\infty}^{+\infty}\phi_{\beta}(y)(1-\phi_{\beta}(y))dy-\phi_{\beta}(-\infty)=-\frac{\beta^{2}}{4}+\beta-1=-(\frac{\beta}{2}-1)^{2}<0.$

Therefore, there exists $L_{1}<0$ so that (3.22) holds for all $x<L_{1}$. In
order to show that this inequality also holds for $L_{1}<x<L$, consider the
function

$G(x,\phi_{\beta})=\phi_{\beta}(x)-\frac{\beta}{2}\int_{x}^{+\infty}\phi_{\beta}(y)(1-\phi_{\beta}(y))dy.$
(3.25)

Note that

$G^{\prime}(x,\phi_{\beta})=\phi_{\beta}^{\prime}(x)+\frac{\beta}{2}\phi_{\beta}(1-\phi_{\beta}).$

Using (3.23), we obtain

$G^{\prime\prime}(x,\phi_{\beta})=\phi_{\beta}^{\prime\prime}+\frac{\beta}{2}\phi_{\beta}^{\prime}-\beta\phi_{\beta}\phi_{\beta}^{\prime}=\phi_{\beta}^{2}-\phi_{\beta}-2\phi_{\beta}^{\prime}+\frac{\beta}{2}\phi_{\beta}^{\prime}.$

Therefore, if $x_{0}$ is a critical point of $G(x,\phi_{\beta})$, then

$\phi_{\beta}^{\prime}(x_{0})=-\frac{\beta}{2}\phi_{\beta}(x_{0})(1-\phi_{\beta}(x_{0})),$

and

$G^{\prime\prime}(x_{0},\phi_{\beta})=\frac{2}{\beta}\phi_{\beta}^{\prime}-2\phi_{\beta}^{\prime}+\frac{\beta}{2}\phi_{\beta}^{\prime}=\Big{(}\frac{2}{\beta}+\frac{\beta}{2}-2\Big{)}\phi_{\beta}^{\prime}<0.$

Hence, the only possible critical points of $G(x,\phi_{\beta})$ are local
maxima. As $G(x)>0$ for $x>L$ and $x<L_{1}$ we deduce that
$G(x,\phi_{\beta})>0$ for all $x\in{\mathbb{R}}$, and (3.22) holds.

To finish the proof of Lemma 3.2, consider the integral

$I(t,x)=\int_{x}^{\infty}\tilde{u}(t,y)(1-\tilde{u}(t,y))dy.$ (3.26)

To write this integral differently, define $\tilde{x}$ by
$\tilde{u}(t,\tilde{x}(t,v))=v$ and note that

$\tilde{u}_{x}(t,\tilde{x}(t,v))=E(t,v),~{}~{}\hbox{ for $0<v<1$},$ (3.27)

with $E(t,v)$ as in (2.5). Then, making the change of variables $y\mapsto v$
via $y=\tilde{x}(t,v)$, we find

$\displaystyle I(t,x)$
$\displaystyle=\int_{x}^{\infty}\tilde{u}(t,y)(1-\tilde{u}(t,y))dy=\int_{0}^{\tilde{u}(t,x)}v(1-v)\frac{dv}{|E(t,v)|})\leq\int_{0}^{\tilde{u}(t,x)}v(1-v)\frac{dv}{|\bar{E}(v)|}$
(3.28)
$\displaystyle=\int_{\phi_{\beta}^{-1}(\tilde{u}(t,x))}^{\infty}\phi_{\beta}(y)(1-\phi_{\beta}(y))dy,$

with $\bar{E}(v)$ defined in (2.6). We used (2.6) in the inequality in the
first line in (3.28), employing the assumption that the initial condition is
steeper than the minimal speed traveling wave. Using (3.22) in (3.28) we see
that

$\displaystyle I(t,x)$
$\displaystyle\leq\int_{\phi_{\beta}^{-1}(\tilde{u}(t,x))}^{\infty}\phi_{\beta}(y)(1-\phi_{\beta}(y))dy\leq\frac{2}{\beta}\phi_{\beta}(\phi_{\beta}^{-1}(\tilde{u}(t,x)))=\frac{2}{\beta}\tilde{u}(t,x).$
(3.29)

We conclude that

$G(t,x;\tilde{u})=\tilde{u}(t,x)-\frac{\beta}{2}I(t,x)\geq 0,$

finishing the proof of Lemma 3.2, and hence that of Proposition 3.1 as well.
$\Box$

##  4 The proof of Theorem 1.1 for $\beta<2$

#### Outline of the proof

In this section, we prove Theorem 1.1 for $\beta<2$. The overall strategy of
the proof is similar to [40] which considers the classical Fisher-KPP equation
with $\beta=0$ but there are also non-trivial differences worth mentioning.

The first step is get control of the solution on the spatial scales $x\sim
O(\sqrt{t})$. This is done using the self-similar variables. The estimates are
precise enough to include the tail behavior of $\tilde{u}(t,x)$ on the
intermediate scales $x\sim O(t^{\gamma})$, with $\gamma\in(0,1/2)$ that are
between the traveling wave scale $x\sim O(1)$ and the diffusive scale $x\sim
O(\sqrt{t})$. Unlike in [40], in this step we rely crucially on the weighted
Hopf-Cole transform and Proposition 3.1 to construct upper and lower barriers
for the solution. The main estimate is the following:

######  Lemma 4.1.

For $\beta<2$, let $\tilde{u}(t,x)$ be the solution to (3.11) with the initial
condition as in Theorem 1.1. There exist $\alpha_{\infty}>0$ and
$\varepsilon_{0}>0$ so that, for any $0<\gamma<1/2$ and
$\varepsilon\in(0,\varepsilon_{0})$, there exists $T_{\varepsilon,\gamma}>0$
such that

$\displaystyle|\tilde{u}(t,x_{\gamma})-\alpha_{\infty}x_{\gamma}e^{-x_{\gamma}}|\leq{\varepsilon}x_{\gamma}e^{-x_{\gamma}}\qquad\text{
for all }t>T_{\varepsilon,\gamma},$ (4.1)

where $x_{\gamma}=(t+1)^{\gamma}$.

The second step is to use the pulled nature of the problem to show that the
control of $\tilde{u}$ given in Lemma 4.1 at $x=x_{\gamma}(t)$ induces
convergence to a traveling wave on the scales $x\sim O(1)$. Before discussing
the modifications required for $\beta\neq 0$, we recall the argument in [40].
It proceeds by constructing solutions $\tilde{u}_{\alpha}$ to the same
equation as satisfied by $\tilde{u}$ (which, in [40] is (3.11) with
$\beta=0$), considered on the half-line $(-\infty,(t+1)^{\gamma})$, with the
boundary condition

$\tilde{u}_{\alpha}(t,(t+1)^{\gamma})=\alpha x_{\gamma}e^{-x_{\gamma}},$

and with $\alpha=\alpha_{\infty}\pm\varepsilon$ (cf. [40, Section 4]). Then,
the analogue of (4.1) for $\beta=0$, which is Lemma 5.1 in [40], and the
comparison principle imply that

$\tilde{u}_{\alpha_{\infty}-\varepsilon}\leq\tilde{u}\leq\tilde{u}_{\alpha_{\infty}+\varepsilon}.$

One concludes after proving the convergence of $\tilde{u}_{\alpha}(t,x)$, as
$t\to+\infty$, to a suitable shift $\varphi_{\alpha}(x)$ of the traveling
wave, showing that the shifts for the waves
$\varphi_{\alpha_{\infty}\pm\varepsilon}$ are $O(\varepsilon)$ apart, and,
finally, letting $\varepsilon\to 0$.

Unfortunately, a direct attempt to do this in our setting fails for several
reasons. The convergence of $\tilde{u}_{\alpha}$ to $\varphi_{\alpha}$ in [40]
is achieved by the construction of an explicit super-solution for the equation
for the difference $e^{x}(\tilde{u}_{\alpha}-\varphi_{\alpha})$. However, the
Burgers term in our context leads to a growth term in this equation when
$\beta>0$ (see, in contrast, (4.63) when $\beta\leq 0$). We bypass this issue
by working with the weighted Hopf-Cole transforms of $\tilde{u}$ and the
traveling wave. One might be tempted to define $\tilde{u}_{\alpha}$ as above
and then take its weighted Hopf-Cole transform. However, there is no apparent
reason for $\tilde{u}_{\alpha}$ to be steeper than the traveling wave, meaning
that Lemma 3.2 does not apply, and we cannot deduce the key differential
inequality (3.13). To bypass this difficulty, we work at the level of the
weighted Hopf-Cole transform, defining $v_{\alpha}$ solving (3.14) treated as
a linear equation with the $\tilde{u}$ terms serving as given coefficients
(see (4.29)–(4.31) below), with a suitable boundary condition at
$x=(t+1)^{\gamma}$. Proceeding as in [40], we obtain an upper bound on
$v_{\alpha}$ given by a shift of the weighted Hopf-Cole transform of the
traveling wave and a decaying term. The lower bound is obtained using yet
another steepness comparison. Afterwards, an additional argument is needed to
upgrade this to the convergence of $\tilde{u}$ due to the fact that
$\tilde{u}$ and $v$ are connected in a nonlocal fashion.

Below, we first prove Lemma 4.1, then apply it to show closeness of the
weighted Hopf-Cole transforms of $\tilde{u}$ and the traveling wave
$\phi_{\beta}$ in Lemma 4.3, and, finally, deduce the closeness of $\tilde{u}$
and $\phi_{\beta}$ from this. For the sake of concreteness, we take
$\tilde{u}(0,x)=\mathbbm{1}(x\leq 0)$. The argument for general initial
conditions as in Theorem 1.1 is nearly verbatim.

### Analysis in the self-similar variables

We start with equations (3.14)-(3.15) and pass to self-similar variables: let

$\tau=\log(t+1),~{}~{}\eta=\frac{x}{\sqrt{t+1}}~{}~{}\text{and}~{}~{}\omega(\tau,\eta)=v\big{(}e^{\tau}-1,\eta
e^{\tau/2}\big{)}e^{-\tau/2}.$ (4.2)

In these variables, (3.14)-(3.15) becomes

$\displaystyle\omega_{\tau}+L\omega+\frac{3}{2}e^{-\tau/2}\omega_{\eta}=e^{\tau}\omega\Big{(}\frac{\beta}{2}\int_{\eta
e^{\tau/2}}^{\infty}\tilde{u}(1-\tilde{u})dy-\tilde{u}(e^{\tau}-1,\eta
e^{\tau/2})\Big{)},~{}~{}\tau>0,~{}\eta\in{\mathbb{R}},$ (4.3)

with the operator $L$ defined by

$\displaystyle
L\omega:=-\omega_{\eta\eta}-\frac{\eta}{2}\omega_{\eta}-\omega.$ (4.4)

Note that when $\beta=0$, which is the classical Fisher-KPP equation, (4.3)
reduces to a local equation

$\displaystyle\omega_{\tau}+L\omega+\frac{3}{2}e^{-\tau/2}\omega_{\eta}=-e^{3\tau/2-\eta\exp(\tau/2)}\omega^{2},~{}~{}\tau>0,~{}\eta\in{\mathbb{R}}.$
(4.5)

In that case, the nonlinear term in the right side is very small for $\eta>0$
but plays the role of a large absorption for $\eta<0$. This was used in [40]
to show that (4.5) is well-approximated by the linear problem

$\bar{\omega}_{\tau}+L\bar{\omega}+\frac{3}{2}e^{-\tau/2}\bar{\omega}_{\eta}=0,~{}~{}\eta>0,$
(4.6)

augmented with the Dirichlet boundary condition $\bar{\omega}(\tau,0)=0$. This
is main intuition behind the proof of the long-time asymptotics of $\omega$ in
[40].

We now collect the ingredients that would allow us to use the arguments of
[40]. First, Lemma 3.2 implies that, as long as the initial condition $u(0,x)$
is steeper than the minimal speed traveling wave, the right hand side of (4.3)
is non-positive. Hence, the solution to (4.6) is still a super-solution to
(4.3).

Second, the coefficient in the parenthesis in the right side of (4.3) can be
bounded below as follows: by Lemma 3.2, we have that

$\int_{\eta
e^{\tau/2}}^{\infty}\tilde{u}(1-\tilde{u})dy\leq\frac{2}{|\beta|}\tilde{u},$

leading to

$e^{\tau}\Big{(}\frac{\beta}{2}\int_{\eta
e^{\tau/2}}^{\infty}\tilde{u}(1-\tilde{u})dy-\tilde{u}\Big{)}\geq-2e^{\tau}\tilde{u}.$
(4.7)

It follows that the solution to

$\underline{\omega}_{\tau}+L\underline{\omega}+\frac{3}{2}e^{-\tau/2}\underline{\omega}_{\eta}=-2e^{\tau}\tilde{u}\underline{\omega}$
(4.8)

is a sub-solution to (4.3).

The third observation is that the right side of (4.3) is very small for
$\eta\gg e^{-\tau/2}$. To see this, we first show that, if $\beta<2$, then
there exist $A>1$ and $L>0$ so that

$u(t,x)\leq\bar{u}(t,x):=\frac{A}{1+e^{x-2t-L}},~{}~{}\hbox{ for all
$x\in{\mathbb{R}}$}.$ (4.9)

Indeed, the function $\bar{u}(t,x)$ satisfies (we set $L=0$ momentarily to
simplify the notation)

$\bar{u}_{t}=\frac{2Ae^{x-2t}}{(1+e^{x-2t})^{2}},~{}~{}~{}\bar{u}_{x}=-\frac{Ae^{x-2t}}{(1+e^{x-2t})^{2}},~{}~{}~{}\bar{u}_{xx}=-\frac{Ae^{x-2t}(1-e^{x-2t})}{(1+e^{x-2t})^{3}}.$

Hence, as long as $\beta<2$, we may choose $A\in(1,2/\beta)$ so that

$\displaystyle\bar{u}_{t}+\beta\bar{u}\bar{u}_{x}-\bar{u}_{xx}-\bar{u}(1-\bar{u})=A\frac{(2-\beta
A)e^{x-2t}}{(1+e^{x-2t})^{3}}\geq 0,$ (4.10)

With this choice of $A$, the function $\bar{u}(t,x)$ is a super-solution to
(1.1) for any $L\in{\mathbb{R}}$. Since $A>1$, we can choose $L>0$
sufficiently large so that $\mathbbm{1}(x\leq 0)\leq\bar{u}(0,x)$ for all
$x\in{\mathbb{R}}$. Then, the comparison principle for (1.1) implies that
(4.9) holds. As a consequence, we see that the right side of (4.3) can be
bounded by

$\tilde{u}(t,x)\leq\min\Big{\\{}1,\frac{A}{1+e^{x-({3}/{2})\log(t+1)-L}}\Big{\\}},$
(4.11)

whence

$\int_{x}^{\infty}\tilde{u}(t,y)dy\leq\begin{cases}C+\frac{3}{2}\log(t+1)-x\qquad&\text{
if }x\leq\frac{3}{2}\log(t+1)\\\ C(t+1)^{3/2}e^{-x}\qquad&\text{ if
}x\geq\frac{3}{2}\log(t+1).\end{cases}$ (4.12)

which are double-exponentially small on the scales
$\eta=O(e^{-(\frac{1}{2}-\gamma)\tau})$ for any $\gamma\in(0,1/2)$ (recall
that $x=\eta e^{\tau/2}$).

Finally, the asymptotics in (4.12) yield the approximate Dirichlet boundary
condition for $\omega$ at $\eta=-e^{(1/2-\gamma)\tau}$. Indeed, we have

$\begin{split}\omega(\tau,-e^{-(1/2-\gamma)\tau})&=e^{-\tau/2}\tilde{u}(e^{\tau}-1,-e^{\gamma\tau})\exp\left\\{-e^{\gamma\tau}+\frac{\beta}{2}\int_{-e^{\gamma\tau}}^{\infty}\tilde{u}dy\right\\}\\\
&\leq
C\exp\left\\{\tau-e^{\gamma\tau}\Big{(}1-\frac{\beta}{2}\Big{)}\right\\}.\end{split}$
(4.13)

In order to analyze the long-time behavior of $\omega$, the main point is the
following. The above arguments show that $\omega(\tau,\eta)$ should be well-
approximated by the solution to the linear problem (4.6) with the Dirichlet
boundary condition. The linear operator $L$, given by (4.4), is compact and
self-adjoint on $H^{1}_{0}(e^{\eta^{2}/4}d\eta;{\mathbb{R}}_{+})$. Its
spectrum consists of the eigenvalues $0,1,2,\dots$, and its principal
eigenfunction is $\eta e^{-\eta^{2}/4}$. Hence, the dominant behavior for
$\omega(\tau,\eta)$ as $\tau\to+\infty$ should be given by
$\alpha_{\infty}\eta e^{-\eta^{2}/4}$, for some $\alpha_{\infty}$ depending on
the initial data. This simple picture is complicated by the error terms in
(4.3) and the “not quite zero” boundary condition (4.13). However, they have
“fast” decay, so, with careful analysis, they can be suitably controlled. This
is the argument in a nutshell, even though the details of the proof are more
intricate.

In order to carry out this strategy, the authors of [40] require exactly the
four ingredients listed above: equations for the super- and sub-solutions
(given in our case by (4.6) and (4.8)), double-exponential decay of the
coefficients in the right side of (4.3) (see (4.11) and (4.12)), and the
approximate Dirichlet boundary condition, as in (4.13). Thus, the strategy of
that paper, which involves constructing successively more precise sub- and
super-solutions of $\omega$ using the spectral properties of $L$, can be
applied without alteration. As such, we state the following lemma giving the
asymptotics of $\omega$ and omit the details.

######  Lemma 4.2.

Let $\omega(\tau,\eta)$ be the solution of (4.3) on $\mathbb{R}$, with the
initial condition $\omega_{0}(\eta)$ such that $\omega_{0}(\eta)=0$ for all
$\eta>A$ for some $A>0$, and $\omega_{0}(\eta)=O(e^{(1-\beta/2)\eta})$ for
$\eta<0$. Then, for $\beta<2$, there exists a constant $\alpha_{\infty}>0$ and
functions $h$ and $R$ such that

$\displaystyle\omega(\tau,\eta)=(\alpha_{\infty}+h(\tau))\eta
e^{-\eta^{2}/4}+R(\tau,\eta)e^{-\eta^{2}/6},~{}~{}~{}~{}\eta\geq 0,$ (4.14)

and, for any $\gamma^{\prime}\in(0,1/2)$,

$\lim_{\tau\to\infty}h(\tau)=0\quad\text{ and }\quad|R(\tau,\eta)|\leq
C_{\gamma^{\prime}}e^{-(1/2-\gamma^{\prime})\tau}.$

#### The proof of Lemma 4.1

We now establish Lemma 4.1 using Lemma 4.2. Let us recall, from (3.12) and
(4.2), that

$\displaystyle
v(t,x)=\exp\Big{(}x+\frac{\beta}{2}\int_{x}^{\infty}\tilde{u}(t,y)dy\Big{)}\tilde{u}(t,x)=\sqrt{t+1}\omega\Big{(}\log(t+1),\frac{x}{\sqrt{t+1}}\Big{)}.$

It follows from (4.12) that we can take $T_{\varepsilon}$ sufficiently large,
so that when $t>T_{\varepsilon}$, we have

$\begin{split}&e^{-C(t+1)^{3/2}e^{-(t+1)^{\gamma}}}\sqrt{t+1}\omega\Big{(}\log(t+1),\frac{x_{\gamma}}{\sqrt{t+1}}\Big{)}\leq
e^{-\frac{\beta}{2}\int_{x_{\gamma}}^{\infty}\tilde{u}dy}v(t,x_{\gamma})=e^{x_{\gamma}}\tilde{u}(t,x_{\gamma})\end{split}$
(4.15)

and

$\begin{split}e^{x_{\gamma}}\tilde{u}(t,x_{\gamma})=e^{-\frac{\beta}{2}\int_{x_{\gamma}}^{\infty}\tilde{u}dy}v(t,x_{\gamma})\leq
e^{C(t+1)^{3/2}e^{-(t+1)^{\gamma}}}\sqrt{t+1}\omega\Big{(}\log(t+1),\frac{x_{\gamma}}{\sqrt{t+1}}\Big{)},\end{split}$
(4.16)

recalling that $x_{\gamma}=(t+1)^{\gamma}$. Using (4.15)-(4.16) first, and
then recalling Lemma 4.2 yields

$\displaystyle|e^{x_{\gamma}}\tilde{u}(t,x_{\gamma})-\alpha_{\infty}x_{\gamma}|$
(4.17)
$\displaystyle\leq\Big{|}e^{x_{\gamma}}\tilde{u}(t,x_{\gamma})-\sqrt{t+1}\omega\Big{(}\log(t+1),\frac{x_{\gamma}}{\sqrt{t+1}}\Big{)}\Big{|}+\sqrt{t+1}\omega\Big{(}\log(t+1),\frac{x_{\gamma}}{\sqrt{t+1}}\Big{)}\Big{|}e^{\frac{x_{\gamma}^{2}}{4(t+1)}}-1\Big{|}$
$\displaystyle\qquad\qquad+\Big{|}\sqrt{t+1}\omega\Big{(}\log(t+1),\frac{x_{\gamma}}{\sqrt{t+1}}\Big{)}e^{\frac{x_{\gamma}^{2}}{4(t+1)}}-\alpha_{\infty}x_{\gamma}\Big{|}$
$\displaystyle\leq
C\Big{(}(t+1)^{2}e^{-(t+1)^{\gamma}}+\frac{x_{\gamma}^{2}}{4(t+1)}\Big{)}\omega\Big{(}\log(t+1),\frac{x_{\gamma}}{\sqrt{t+1}}\Big{)}$
$\displaystyle\qquad\qquad+\Big{(}|h(\log(t+1))|+\frac{\sqrt{t+1}}{x_{\gamma}}|R(\tau,x_{\gamma}/\sqrt{t})|\Big{)}x_{\gamma}e^{-\frac{x_{\gamma}^{2}}{6(t+1)}}.$

Using that $h(\log(t+1))\to 0$ and choosing $\gamma^{\prime}\in(0,\gamma)$, we
bound the second term in the very right side above by

$\frac{\sqrt{t+1}}{x_{\gamma}}|R(\tau,x_{\gamma}/\sqrt{t})|\leq
C_{\gamma^{\prime}}(t+1)^{-(\gamma-\gamma^{\prime})}\to 0,\quad\text{ as
}~{}t\to\infty.$

To handle the first term in the right isee of (4.17), we simply notice that
$\omega$ is bounded due to Lemma 4.2 and the time dependent terms in front of
$\omega$ tend to zero. This gives (4.1). $\Box$

### From the scales $x\sim O(t^{\gamma})$ to $x\sim O(1)$

The next step is to pass from the control of the solution on the spatial
scales $x\sim O(t^{\gamma})$, provided by Lemma 4.1, to the spatial scales
$x\sim O(1)$ using the pulled nature of the Burgers-FKPP equation for
$\beta<2$.

The arguments are a bit different for $\beta\leq 0$ and $\beta\in(0,2)$. We
first discuss the latter case where the arguments deviate from [40], as we
have discussed in the outline of this section, and later explain why the case
$\beta<0$ is quite similar to what was done in [40] for $\beta=0$,

#### The case $0<\beta<2$

We are going to use an argument inspired by [40] but we will only apply it
after an application of the weighted Hopf-Cole transform, and the conclusion
is different as a result. We take a traveling wave $\phi_{\beta}(x)$, and
shift it into the moving frame:

$\varphi_{\alpha}(t,x)=\phi_{\beta}(x+\zeta_{\alpha}(t)),$ (4.18)

leaving the definition of $\zeta_{\alpha}(t)$ open for the moment. This
function satisfies

$\displaystyle\partial_{t}\varphi_{\alpha}-\partial_{x}^{2}\varphi_{\alpha}-\left(2-\frac{3}{2(t+1)}\right)\partial_{x}\varphi_{\alpha}+\beta\varphi_{\alpha}\partial_{x}\varphi_{\alpha}-\varphi_{\alpha}+\varphi_{\alpha}^{2}=\left(\frac{3}{2(t+1)}+\dot{\zeta}_{\alpha}\right)\partial_{x}\varphi_{\alpha}.$
(4.19)

Next, we define its Hopf-Cole transform as

$\psi_{\alpha}(t,x)=e^{\Gamma_{\alpha}(t,x)}\varphi_{\alpha}(t,x),~{}~{}~{}\Gamma_{\alpha}(t,x)=x+\frac{\beta}{2}\int_{x}^{\infty}\varphi_{\alpha}(t,y)dy,$
(4.20)

Noticing that

$\partial_{x}\varphi_{\alpha}(t,x)=\phi_{\beta}^{\prime}(x+\zeta_{\alpha}(t)),~{}~{}~{}\partial_{t}\varphi_{\alpha}(t,x)=\dot{\zeta}_{\alpha}(t)\phi_{\beta}^{\prime}(x+\zeta_{\alpha}(t))=\dot{\zeta}_{\alpha}(t)\partial_{x}\varphi_{\alpha}(t,x),$
(4.21)

we obtain an equation for the function $\psi$:

$\begin{split}&\partial_{t}\psi_{\alpha}-\partial_{x}^{2}\psi_{\alpha}+\frac{3}{2(t+1)}(\partial_{x}\psi_{\alpha}-\psi_{\alpha})-\frac{\beta\psi_{\alpha}}{2}\int_{x}^{\infty}\varphi_{\alpha}(1-\varphi_{\alpha})dy+\psi_{\alpha}\varphi_{\alpha}\\\
&=\frac{\beta\dot{\zeta_{\alpha}}e^{\Gamma_{\alpha}}\varphi_{\alpha}}{2}\Big{(}\int_{x}^{\infty}\partial_{x}\varphi_{\alpha}dy\Big{)}+e^{\Gamma_{\alpha}}\partial_{x}\varphi_{\alpha}\dot{\zeta_{\alpha}}\\\
&~{}~{}-e^{\Gamma_{\alpha}}\Big{(}-\frac{\beta}{2}\partial_{x}\varphi_{\alpha}\varphi_{\alpha}+\left(1-\frac{\beta}{2}\varphi_{\alpha}\right)^{2}\varphi_{\alpha}+2\left(1-\frac{\beta}{2}\varphi_{\alpha}\right)\partial_{x}\varphi_{\alpha}+\partial_{x}^{2}\varphi_{\alpha}\Big{)}\\\
&~{}~{}+\frac{3e^{\Gamma_{\alpha}}}{2(t+1)}\Big{(}\partial_{x}\varphi_{\alpha}-\frac{\beta}{2}\varphi_{\alpha}^{2}\Big{)}-\frac{\beta
e^{\Gamma_{\alpha}}\varphi_{\alpha}}{2}\int_{x}^{\infty}\varphi_{\alpha}(1-\varphi_{\alpha})dy+e^{\Gamma_{\alpha}}\varphi_{\alpha}^{2}\\\
&=e^{\Gamma_{\alpha}}\Big{(}\dot{\zeta_{\alpha}}+\frac{3}{2(t+1)}\Big{)}\Big{(}-\frac{\beta}{2}\varphi_{\alpha}^{2}+\partial_{x}\varphi_{\alpha}\Big{)}+\frac{\beta
e^{\Gamma_{\alpha}}\varphi_{\alpha}}{2}\Big{(}\partial_{x}\varphi_{\alpha}-\frac{\beta}{2}\varphi_{\alpha}^{2}+2\varphi_{\alpha}-\int_{x}^{\infty}\varphi_{\alpha}(1-\varphi_{\alpha})dy\Big{)}\\\
&=-e^{\Gamma_{\alpha}}\Big{(}\dot{\zeta_{\alpha}}+\frac{3}{2(t+1)}\Big{)}\Big{(}2\varphi_{\alpha}-\int_{x}^{\infty}\varphi_{\alpha}(1-\varphi_{\alpha})dy\Big{)}.\end{split}$
(4.22)

We used (3.24) in the last step above.

For each $\alpha\in(\alpha_{\infty}/2,2\alpha_{\infty})$, where
$\alpha_{\infty}>0$ is defined in Lemma 4.1, we fix $\zeta_{\alpha}(t)$ by the
normalization

$\psi_{\alpha}(t,(t+1)^{\gamma})=\alpha(t+1)^{\gamma}.$ (4.23)

We note that, differentiating (4.20) and using the asymptotics (A.24) of
$\phi_{\beta}$, it is easy to check that $\psi_{\alpha}$ is increasing in $x$
when $x\gg 1$. Hence, $\zeta_{\alpha}$ is well-defined for $t$ sufficiently
large. In the sequel, we work with $\alpha=\alpha_{\infty}\pm\varepsilon$, in
which case, $\psi_{\alpha}$ approximately matches with $v$ at
$x=(t+1)^{\gamma}$. Note a difference with [40]: the shift is determined by
the value of $\psi_{\alpha}(t,x)$ at the point $x=(t+1)^{\gamma}$, and not by
the value of $\varphi_{\alpha}(t,x)$ as in [40].

Next, using the asymptotics (A.24) of $\phi_{\beta}$, we find that

$\zeta_{\alpha}(t)=-\log(\alpha)-\frac{\log(\alpha)}{(t+1)^{\gamma}}+o(t^{-\gamma}),$
(4.24)

and

$|\dot{\zeta}_{\alpha}(t)|\leq\frac{C}{(t+1)^{1+\gamma}},$ (4.25)

where $C$ is independent of $\alpha$ over the interval
$(\alpha_{\infty}/2,2\alpha_{\infty})$. In addition, using (A.24) again, we
find that

$e^{\Gamma_{\alpha}}\Big{(}\varphi_{\alpha}+\int_{x}^{\infty}\varphi_{\alpha}(1-\varphi_{\alpha})dx\Big{)}\leq
C(|x|+1).$ (4.26)

In view of (LABEL:e.c6283), (4.25), and (4.26), we find, when
$|x|\leq(t+1)^{\gamma}$,

$\left|\partial_{t}\psi_{\alpha}-\partial_{x}^{2}\psi_{\alpha}+\frac{3}{2(t+1)}(\partial_{x}\psi_{\alpha}-\psi_{\alpha})-\frac{\beta\psi_{\alpha}}{2}\int_{x}^{\infty}\varphi_{\alpha}(1-\varphi_{\alpha})dy+\psi_{\alpha}\varphi_{\alpha}\right|\leq\frac{C}{(t+1)^{1-\gamma}}.$
(4.27)

Let us also recall that the Hopf-Cole transform $v(t,x)$ of the function
$\tilde{u}$, defined in (3.12), satisfies (3.14)-(3.15):

$v_{t}-v_{xx}+\frac{3}{2(t+1)}(v_{x}-v)=-\Big{(}\tilde{u}(t,x)-\frac{\beta}{2}\int_{x}^{+\infty}\tilde{u}(t,y)(1-\tilde{u}(t,y))dy\Big{)}v.$
(4.28)

For $\alpha\neq\alpha_{\infty}$, we define $v_{\alpha}(t,x)$ as the solution
to (4.28), thought of as a linear equation for $v$, with prescribed function
$\tilde{u}(t,x)$:

$\partial_{t}v_{\alpha}-\partial_{x}^{2}v_{\alpha}+\frac{3}{2(t+1)}(\partial_{x}v_{\alpha}-v_{\alpha})=-\Big{(}\tilde{u}(t,x)-\frac{\beta}{2}\int_{x}^{+\infty}\tilde{u}(t,y)(1-\tilde{u}(t,y))dy\Big{)}v_{\alpha}$
(4.29)

over $x<(t+1)^{\gamma}$ and $t>T_{\varepsilon}$ with the boundary condition

$v_{\alpha}(t,(t+1)^{\gamma})=\alpha(t+1)^{\gamma}$ (4.30)

and the initial condition

$v_{\alpha}(T_{\varepsilon},x)=v(T_{\varepsilon},x),$ (4.31)

where $T_{\varepsilon}$ is that of Lemma 4.1 with
$\varepsilon=|\alpha-\alpha_{\infty}|$.

We now make a couple of observations. First, note that the normalization
(4.23) and the boundary condition (4.30) imply that

$v_{\alpha}(t,(t+1)^{\gamma})=\psi_{\alpha}(t,(t+1)^{\gamma}),~{}~{}\hbox{ for
all $t>T_{\varepsilon}$ and $\varepsilon>0$.}$ (4.32)

Next, we claim that for any $\varepsilon>0$, we have

$v_{\alpha_{\infty}+\varepsilon}(t,x)>v(t,x)>v_{\alpha_{\infty}-\varepsilon}(t,x),~{}~{}\hbox{
for all $t\geq T_{\varepsilon}$ and $x<(t+1)^{\gamma}$.}$ (4.33)

This follows from the fact that $v$ and $v_{\alpha_{\infty}\pm\varepsilon}$
satisfy the same linear parabolic equation (4.28) and (4.29) and the same
initial condition (4.31), but take ordered values at the boundary. Indeed,
with the help of Lemma 4.1, we obtain

$\displaystyle v(t,(t+1))^{\gamma}$
$\displaystyle=\exp\Big{\\{}(t+1)^{\gamma}+\frac{\beta}{2}\int_{(t+1)^{\gamma}}^{\infty}\tilde{u}(t,y)dy\Big{\\}}\tilde{u}(t,(t+1)^{\gamma})$
(4.34)
$\displaystyle\leq(\alpha_{\infty}+\varepsilon/2)(t+1)^{\gamma}\exp\Big{\\{}\frac{\beta}{2}\int_{(t+1)^{\gamma}}^{\infty}\tilde{u}(t,y)dy\Big{\\}}$
$\displaystyle\leq\Big{(}\alpha_{\infty}+\varepsilon/2\Big{)}(t+1)^{\gamma}\exp\left\\{C(t+1)^{3/2}e^{-(t+1)^{\gamma}}\right\\}\leq
v_{\alpha_{\infty}+\varepsilon}(t,(t+1)^{\gamma}),$

for $t>T_{\varepsilon}$, up to increasing $T_{\varepsilon}$ to deal with the
different exponential factors in the second line above. A similar argument
gives $v_{\alpha_{\infty}-\varepsilon}(t,(t+1)^{\gamma})<v(t,(t+1)^{\gamma})$.
Thus, comparison yields (4.33), as claimed.

Finally, we observe that

$v_{\alpha_{\infty}+\varepsilon}(t,x)\leq\frac{\alpha_{\infty}+\varepsilon}{\alpha_{\infty}-\varepsilon}v_{\alpha_{\infty}-\varepsilon}(t,x)\qquad\text{
for all }t\geq T_{\varepsilon}\text{ and }x<(t+1)^{\gamma}.$ (4.35)

The above is due to a comparison argument using the fact that the function in
the right side of (4.35) solves the same linear equation as
$v_{\alpha_{\infty}+\varepsilon}$, with the same boundary condition, but with
larger initial data. The important consequence of (4.35), along with (4.33),
is that

$\begin{split}v_{\alpha_{\infty}+\varepsilon}(t,-(t+1)^{\gamma})&\leq\frac{\alpha_{\infty}+\varepsilon}{\alpha_{\infty}-\varepsilon}v_{\alpha_{\infty}-\varepsilon}(t,-(t+1)^{\gamma})\leq\frac{\alpha_{\infty}+\varepsilon}{\alpha_{\infty}-\varepsilon}v(t,-(t+1)^{\gamma})\\\
&=\frac{\alpha_{\infty}+\varepsilon}{\alpha_{\infty}-\varepsilon}\exp\Big{\\{}-(t+1)^{\gamma}+\frac{\beta}{2}\int_{-(t+1)^{\gamma}}^{\infty}\tilde{u}dy\Big{\\}}\tilde{u}\leq
Ce^{-\frac{1}{2}(1-\beta/2)(t+1)^{\gamma}}.\end{split}$ (4.36)

For the last inequality, we used (4.12).

Let us define, for any $\varepsilon\in(-1,1)$,

$s_{\varepsilon}(t,x)=v_{\alpha_{\infty}+\varepsilon}(t,x)-\psi_{\alpha_{\infty}+\varepsilon}(t,x),~{}~{}x<(t+1)^{\gamma},~{}~{}t>T_{\varepsilon}.$
(4.37)

The following bound will be crucial for us.

######  Lemma 4.3.

There exists $\lambda>0$, so that we have, for any $\varepsilon>0$:

$s_{\varepsilon}(t,x)\leq\frac{C}{(t+1)^{\lambda}}~{}~{}\hbox{ for all
$t>T_{\varepsilon}$ and $|x|\in(t+1)^{\gamma}$.}$ (4.38)

Proof. We temporarily abuse notation and denote $s_{\varepsilon}=s$ in this
proof. Using (4.27) and (4.29), we find, for $|x|\leq(t+1)^{\gamma}$,

$\displaystyle s_{t}$ $\displaystyle-
s_{xx}+\frac{3}{2(t+1)}(s_{x}-s)+s\Big{(}-\frac{\beta}{2}\int_{x}^{\infty}\tilde{u}(1-\tilde{u})dy+\tilde{u}\Big{)}$
(4.39)
$\displaystyle+\psi\Big{(}\tilde{u}-\varphi+\frac{\beta}{2}\Big{(}\int_{x}^{\infty}\varphi(1-\varphi)dy-\int_{x}^{\infty}\tilde{u}(1-\tilde{u})dy\Big{)}\Big{)}\leq\frac{C}{(t+1)^{1-\gamma}}.$

Using (4.36), we see that

$s(t,-(t+1)^{\gamma})\leq Ce^{-\frac{1}{2}(1-\beta/2)(t+1)^{\gamma}},$ (4.40)

and, by (4.32),

$s(t,(t+1)^{\gamma})=0.$ (4.41)

The inequality (4.40) is another reminder of the importance of the condition
$\beta<2$ here.

Lemma 3.2 tells us that the zero order coefficient in (4.39) is positive:

$G(t,x)=\tilde{u}(t,x)-\frac{\beta}{2}\int_{x}^{\infty}\tilde{u}(t,y)(1-\tilde{u}(t,y))dy\geq
0.$ (4.42)

In addition, we claim that

$(\tilde{u}-\varphi)+\frac{\beta}{2}\Big{(}\int_{x}^{\infty}\varphi(1-\varphi)dy-\int_{x}^{\infty}\tilde{u}(1-\tilde{u})dy\Big{)}\geq
0.$ (4.43)

Indeed, recalling from (4.21) that
$\varphi_{x}(t,x)=\phi_{\beta}^{\prime}(x+\zeta(t))$, and using the notation
$E$ and $\bar{E}$ set in (2.6), we find

$\displaystyle\int_{x}^{\infty}\varphi(1-\varphi)dy-\int_{x}^{\infty}\tilde{u}(1-\tilde{u})dy=\int_{0}^{\varphi(t,x)}z(1-z)\frac{dz}{|\bar{E}(z)|}-\int_{0}^{\tilde{u}(t,x)}z(1-z)\frac{dz}{|E(t,z)|}$
(4.44)
$\displaystyle\geq\int_{0}^{\varphi(t,x)}z(1-z)\frac{dz}{|\bar{E}(z)|}-\int_{0}^{\tilde{u}(t,x)}z(1-z)\frac{dz}{|\bar{E}(z)|}=-\int_{\varphi(t,x)}^{\tilde{u}(t,x)}z(1-z)\frac{dz}{|\bar{E}(z)|}.$

As we show in (A.23), we have

$\frac{z(1-z)}{|\bar{E}(z)|}\leq\frac{2}{\beta},$ (4.45)

and (4.43) follows from (4.44) and (4.45).

We deduce from (4.39), (4.42), and (4.43) that $s(t,x)$ satisfies a
differential inequality

$\displaystyle s_{t}$ $\displaystyle-
s_{xx}+\frac{3}{2(t+1)}(s_{x}-s)+G(t,x)s\leq\frac{C}{(t+1)^{1-\gamma}},$
(4.46)

with $G(t,x)\geq 0$ and boundary conditions (4.40)-(4.41). We can now argue as
in [40] that we can find $\lambda,\gamma,\varepsilon$ sufficiently small so
that, for

$\bar{s}(t,x)=\frac{1}{(t+1)^{\lambda}}\cos\Big{(}\frac{x}{(t+1)^{\gamma+\varepsilon}}\Big{)},$
(4.47)

we have

$\bar{s}_{t}-\bar{s}_{xx}+\frac{3}{2(t+1)}(\bar{s}_{x}-\bar{s})+G(t,x)\bar{s}\geq\frac{C}{(t+1)^{1-\gamma}}$
(4.48)

for $t\geq T$, up to possibly increasing $T$. The choice of $\gamma$ occurs in
this step. Hence, $\bar{s}$ is a super-solution of (4.39) and, up to
multiplying $\bar{s}$ by a large constant so that
$s(T,\cdot)\leq\bar{s}(T,\cdot)$, we have

$s(t,x)\leq\bar{s}(t,x)\qquad\text{ for all }t\geq T\text{ and
}|x|\leq(t+1)^{\gamma}.$

From this, (4.38) follows, finishing the proof of Lemma 4.3. $\Box$

By virtue of Lemma 4.3, we have now established that, for $t>T_{\varepsilon}$
and $|x|\leq(t+1)^{\gamma}$,

$v\leq
v_{\alpha_{\infty}+\varepsilon}=s_{\varepsilon}+\psi_{\alpha_{\infty}+\varepsilon}\leq\psi_{\alpha_{\infty}+\varepsilon}+\frac{C_{\varepsilon}}{(t+1)^{\lambda}}.$
(4.49)

The first inequality is due to (4.33) and the second is due to (4.38). We
point out that we do not have a more precise lower bound. This is another
place where the steepness comparison will play a crucial role.

Finally, note that the shifts corresponding to
$\psi_{\alpha_{\infty}\pm\varepsilon}$ satisfy, as in (4.24):

$\zeta_{\alpha_{\infty}\pm\varepsilon}(t)=-\log(\alpha_{\infty}\pm\varepsilon)+O(t^{-\gamma}).$
(4.50)

#### The end of the proof of Theorem 1.1 for $\beta<2$

#### The case $0<\beta<2$

The definitions (3.8) of $\tilde{u}$ and (4.18) of the difference
$\varphi_{\alpha_{\infty}}$, and the asymptotics (4.24) of $\zeta$ imply that
1.1 reduces to the uniform convergence of
$\tilde{u}-\varphi_{\alpha_{\infty}}$ to zero on ${\mathbb{R}}$. So far, we
have only shown a weak version of closeness for their weighted Hopf-Cole
transforms in (4.49). Indeed, notice that

$\psi_{\alpha_{\infty}+\varepsilon}(t,(t+1)^{\gamma})-v(t,(t+1)^{\gamma})=(\varepsilon+o(1))t^{\gamma},$
(4.51)

where $o(1)$ vanishes as $t\to\infty$. It follows that the established
inequalities are quite far apart due to the $t^{\gamma}$ factor. We handle
this issue now.

Before going to the proof, note that the uniform convergence to zero of
$\tilde{u}-\varphi_{\alpha_{\infty}}$ reduces to showing that, for any $L>0$,

$\lim_{t\to\infty}\sup_{x\in[-L,L]}|\tilde{u}(t,x)-\varphi_{\alpha_{\infty}}(t,x)|=0.$
(4.52)

This is sufficient due to the convergence in shape in Proposition 2.2.

Fix any $\varepsilon\in(0,\alpha_{\infty}/2)$. We note that, due to (4.50), we
have

$|\varphi_{\alpha_{\infty}}(t,x)-\varphi_{\alpha_{\infty}\pm\varepsilon}(t,x)|\leq
C\varepsilon\qquad\text{ for all $t$ sufficiently large and any $x$}.$ (4.53)

Hence, it suffices to establish upper bounds on
$\tilde{u}-\varphi_{\alpha_{\infty}+\varepsilon}$ and
$\varphi_{\alpha_{\infty}-\varepsilon}-\tilde{u}$.

We establish the latter first, as it is much simpler. Indeed, note that

$\begin{split}\varphi_{\alpha_{\infty}-\varepsilon}(t,(t+1)^{\gamma})&=\exp\Big{\\{}-(t+1)^{\gamma}-\frac{\beta}{2}\int_{(t+1)^{\gamma}}^{\infty}\varphi_{\alpha_{\infty}-\varepsilon}(t,y)dy\Big{\\}}\psi_{\alpha_{\infty}-\varepsilon}(t,(t+1)^{\gamma})\\\
&\leq(\alpha_{\infty}-\varepsilon)(t+1)^{\gamma}\exp\left(-(t+1)^{\gamma}\right).\end{split}$

Thus, due to Lemma 4.1, we have
$\varphi_{\alpha_{\infty}-\varepsilon}(t,(t+1)^{\gamma})<\tilde{u}(t,(t+1)^{\gamma})$.
Since $\tilde{u}$ is steeper than $\phi_{\beta}$ (recall 1.1), we get that

$\varphi_{\alpha_{\infty}-\varepsilon}(t,x)<\tilde{u}(t,x),~{}~{}~{}\text{for
all $t$ sufficiently large and $x<(t+1)^{\gamma}$},$ (4.54)

whence, for $t$ sufficiently large,

$\sup_{x\in[-L,L]}\left(\varphi_{\alpha_{\infty}-\varepsilon}(t,x)-\tilde{u}(t,x)\right)\leq
0.$ (4.55)

Next, we consider the much more involved upper bound on
$\tilde{u}-\varphi_{\alpha_{\infty}+\varepsilon}$. Fix any $x\in[-L,L]$ and
$t$ sufficiently large and write

$\begin{split}\tilde{u}(t,x)-&\varphi_{\alpha_{\infty}+\varepsilon}(t,x)=\exp\Big{\\{}-x-\frac{\beta}{2}\int_{x}^{\infty}\tilde{u}dy\Big{\\}}v(t,x)-\exp\Big{\\{}-x-\frac{\beta}{2}\int_{x}^{\infty}\varphi_{\alpha_{\infty}+\varepsilon}dy\Big{\\}}\psi_{\alpha_{\infty}+\varepsilon}(t,x)\\\
&=\exp\Big{\\{}-x-\frac{\beta}{2}\int_{x}^{\infty}\varphi_{\alpha_{\infty}+\varepsilon}dy\Big{\\}}(v(t,x)-\psi_{\alpha_{\infty}+\varepsilon}(t,x))\\\
&\qquad+e^{-x}v(t,x)\Big{[}\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{\infty}\tilde{u}dy\Big{\\}}-\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{(t+1)^{\gamma}}\tilde{u}dy\Big{\\}}\Big{]}\\\
&\qquad+e^{-x}v(t,x)\Big{[}\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{(t+1)^{\gamma}}\tilde{u}dy\Big{\\}}-\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{(t+1)^{\gamma}}\varphi_{\alpha_{\infty}+\varepsilon}dy\Big{\\}}\Big{]}\\\
&\qquad+e^{-x}v(t,x)\Big{[}\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{(t+1)^{\gamma}}\varphi_{\alpha_{\infty}+\varepsilon}dy\Big{\\}}-\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{\infty}\varphi_{\alpha_{\infty}+\varepsilon}dy\Big{\\}}\Big{]}\\\
&=:I_{1}+I_{2}+I_{3}+I_{4}.\end{split}$ (4.56)

As $\beta>0$, we immediately see that

$I_{2}\leq 0.$ (4.57)

Hence, we need only bound $I_{1}$, $I_{3}$, and $I_{4}$.

To handle the term $I_{1}$ in (4.56), we apply (4.49) to find, since
$\beta\in(0,2)$:

$\begin{split}I_{1}\leq\frac{C_{\varepsilon}}{(t+1)^{\lambda}}\exp\Big{\\{}-x-\frac{\beta}{2}\int_{x}^{\infty}\tilde{u}dy\Big{\\}}\leq\frac{C_{\varepsilon}}{(t+1)^{\lambda}}e^{L}.\end{split}$
(4.58)

The term $I_{4}$ as is the next simplest. Using (4.12), we see that

$e^{-x}v(t,x)\leq C(t+1)^{3/2}.$ (4.59)

Using (4.59) and also (A.24) to handle the tail integral over
$((t+1)^{\gamma},\infty)$, we obtain

$\begin{split}I_{4}&\leq
C(t+1)^{3/2}\Big{[}\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{\infty}\tilde{u}dy\Big{\\}}-\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{(t+1)^{\gamma}}\varphi_{\alpha_{\infty}+\varepsilon}dy\Big{\\}}\Big{]}\\\
&=C(t+1)^{3/2}\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{(t+1)^{\gamma}}\varphi_{\alpha_{\infty}+\varepsilon}dy\Big{\\}}\Big{[}1-\exp\Big{\\{}-\frac{\beta}{2}\int_{(t+1)^{\gamma}}^{\infty}\varphi_{\alpha_{\infty}+\varepsilon}dy\Big{\\}}\Big{]}\\\
&\leq
C(t+1)^{3/2}\Big{[}1-\exp\Big{\\{}-\frac{\beta}{2}\int_{(t+1)^{\gamma}}^{\infty}\varphi_{\alpha_{\infty}+\varepsilon}dy\Big{\\}}\Big{]}\leq
C(t+1)^{3/2}e^{-(t+1)^{\gamma}}.\end{split}$ (4.60)

We now handle $I_{3}$. If the bracketed term in the definition of $I_{3}$ is
non-positive, there is nothing to prove as $I_{3}\leq 0$. If the bracketed
term is positive, we write

$\begin{split}I_{3}&=e^{-x}[v(t,x)-\psi_{\alpha_{\infty}+\varepsilon}(t,x)]\Big{[}\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{(t+1)^{\gamma}}\tilde{u}dy\Big{\\}}-\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{(t+1)^{\gamma}}\varphi_{\alpha_{\infty}+\varepsilon}dy\Big{\\}}\Big{]}\\\
&\quad+e^{-x}\psi_{\alpha_{\infty}+\varepsilon}(t,x)\Big{[}\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{(t+1)^{\gamma}}\tilde{u}dy\Big{\\}}-\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{(t+1)^{\gamma}}\varphi_{\alpha_{\infty}+\varepsilon}dy\Big{\\}}\Big{]}:=I_{31}+I_{32}.\end{split}$

The reason for the extra step here is that we do not know a priori that
$e^{-x}v$ is bounded; however, we do know that
$e^{-x}\psi_{\alpha_{\infty}+\varepsilon}$ is.

For $I_{31}$, we use (4.49), the positivity of the bracketed term, and the
fact that the bracketed term is smaller than one (again recall that
$\beta\in(0,2)$) to find

$\begin{split}I_{31}\leq\frac{C_{\varepsilon}}{(t+1)^{\lambda}}e^{-x}\Big{[}\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{(t+1)^{\gamma}}\tilde{u}dy\Big{\\}}-\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{(t+1)^{\gamma}}\varphi_{\alpha_{\infty}+\varepsilon}dy\Big{\\}}\Big{]}\leq
e^{L}\frac{C_{\varepsilon}}{(t+1)^{\lambda}}.\end{split}$

For $I_{32}$, we first notice that

$e^{-x}\psi_{\alpha_{\infty}+\varepsilon}(t,x)\leq Ce^{L}.$

In addition, from (4.54), we see that

$\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{(t+1)^{\gamma}}\tilde{u}dy\Big{\\}}<\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{(t+1)^{\gamma}}\varphi_{\alpha_{\infty}-\varepsilon}dy\Big{\\}}.$

Thus, we find

$I_{32}\leq
Ce^{L}\Big{[}\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{(t+1)^{\gamma}}\varphi_{\alpha_{\infty}-\varepsilon}dy\Big{\\}}-\exp\Big{\\{}-\frac{\beta}{2}\int_{x}^{(t+1)^{\gamma}}\varphi_{\alpha_{\infty}+\varepsilon}dy\Big{\\}}\Big{]}.$

Taylor expanding the exponential and using the asymptotics (4.50) of
$\zeta_{\alpha_{\infty}\pm\varepsilon}$ and (A.24) of $\phi_{\beta}$, we see
that

$I_{32}\leq
C_{L}\int_{x}^{(t+1)^{\gamma}}(\varphi_{\alpha_{\infty}+\varepsilon}-\varphi_{\alpha_{\infty}-\varepsilon})dy\leq
C_{L}\varepsilon.$

We conclude that

$I_{3}\leq\frac{C_{\varepsilon}e^{L}}{(t+1)^{\lambda}}+C_{L}\varepsilon.$
(4.61)

Combining (4.56), (4.57), (4.58), (4.60), and (4.61), and increasing $t$
suitably, we obtain

$\sup_{x\in[-L,L]}\left(\tilde{u}(t,x)-\varphi_{\alpha_{\infty}+\varepsilon}\right)\leq
C_{L}\varepsilon.$ (4.62)

The claim (4.52) then follows from (4.53), (4.55), and (4.62). This concludes
the proof for $\beta\in(0,2)$.

#### The case $\beta\leq 0$

This case is essentially the same, except for some simplifications, so we only
highlight the changes that need to be made. The key estimate is to obtain an
upper bound on

$s_{\varepsilon}(t,x)=e^{x}(\tilde{u}(t,x)-\varphi_{\alpha_{\infty}+\varepsilon}).$

Now, $\varphi_{\alpha}$ is defined such that

$\varphi_{\alpha}(t,(t+1)^{\gamma})=\alpha(t+1)^{\gamma}e^{-(t+1)^{\gamma}-t^{2\gamma-1}/4}.$

In this case $\zeta_{\alpha}$ has the same asymptotics as in (4.50). Then
$s_{\varepsilon}$ satisfies

$\displaystyle s_{t}-s_{xx}$
$\displaystyle+\frac{3}{2(t+1)}(s_{x}-s)+\beta\tilde{u}(s_{x}-s)+s(\tilde{u}+\varphi_{\alpha_{\infty}+\varepsilon})$
(4.63) $\displaystyle=-\beta
s\partial_{x}\varphi_{\alpha_{\infty}+\varepsilon}-\partial_{x}\varphi_{\alpha_{\infty}+\varepsilon}\left(\dot{\zeta}_{\alpha_{\infty}+\varepsilon}+\frac{3}{2(t+1)}\right)\leq\frac{C}{(t+1)^{1-\gamma}}.$

The last inequality holds when $|x|\leq(t+1)^{\gamma}$ for the same reasons as
in (4.27) and due to the fact that $\beta\leq 0$ and
$\partial_{x}\varphi_{\alpha_{\infty}+\varepsilon}=\phi_{\beta}^{\prime}\leq
0$. Thus, the same upper bound as in Lemma 4.3 holds.

On the other hand, it is easy to check that
$\tilde{u}>\varphi_{\alpha_{\infty}-\varepsilon}$ on $x<(t+1)^{\gamma}$ for
$t$ sufficiently large. This is due to Lemma 4.1, which yields the correct
ordering at $x=(t+1)^{\gamma}$, and the fact that $\tilde{u}$ is steeper than
$\varphi_{\alpha_{\infty}-\varepsilon}$.

The combination of the above with a simpler version of the argument for the
case $\beta\in(0,2)$ yields the desired convergence. This finishes the proof
of 1.1 when $\beta<2$. $\Box$

##  5 The case $\beta=2$: bounds on the front location

We now turn to the case $\beta=2$, where the analysis is particularly
delicate. Let us define the shift $\mu(t)$ by

$u(t,2t-\mu(t))=\frac{1}{2}.$ (5.1)

Note that, due to the sign convention in (5.1), an upper bound on $\mu(t)$ is
a lower bound on the front location and vice versa. Our goal in this section
is to prove the following upper and lower bounds on $\mu(t)$.

######  Proposition 5.1.

Let $u(t,x)$ be the solution to the Burgers-FKPP equation (1.1), with the
initial condition as in Theorem 1.1, and let $\mu(t)$ be defined by (5.1).
There exists $m_{0}>0$ that depends on $u(0,x)$ so that

$\frac{1}{2}\log t-m_{0}\leq\mu(t)\leq\frac{1}{2}\log t+m_{0},~{}~{}\hbox{ for
all $t\geq 1$.}$ (5.2)

Later, in Section 6, we will improve these bounds to the precise asymptotics
of 1.1:

$\mu(t)=\frac{1}{2}\log t+x_{0}+o(1)\hbox{ as $t\to+\infty$.}$ (5.3)

Proposition 5.1 is, however, a crucial step in the proof of (5.3). Its proof
occupies almost all of the rest of this section and is the heart of this
paper. In order to explain the outline of the proof, we will need to do some
preliminary transformations, leading to Lemmas 5.2 and 5.4 below that imply
the conclusion of Proposition 5.1. We will discuss their proofs when we come
to their respective statements. As a technical comment, we mention that
without loss of generality we will, once again, take $u(0,x)=\mathbbm{1}(x\leq
0)$.

At the very end of this section, we use Proposition 5.1 to obtain a helpful
bound in an intermediate region in a short Section 5.6.

###  5.1 Outline of the proof of Proposition 5.1

####  5.1.1 An exponential moment

We first give a heuristic argument to explain the delay in the case $\beta=2$.
Consider the Burgers-FKPP equation with $\beta=2$, in the moving frame $x\to
x-2t+\mu(t)$, with an unknown shift $\mu(t)$:

$\displaystyle\tilde{u}_{t}-2\tilde{u}_{x}+\mu^{\prime}(t)\tilde{u}_{x}+2\tilde{u}\tilde{u}_{x}=\tilde{u}_{xx}+\tilde{u}(1-\tilde{u}).$
(5.4)

A simple computation shows that the exponential moment

$I(t)=\int\tilde{u}(t,x)e^{x}dx$ (5.5)

satisfies an ODE

$\displaystyle\frac{dI(t)}{dt}=\int(2\tilde{u}_{x}-\mu^{\prime}(t)\tilde{u}_{x}-2\tilde{u}\tilde{u}_{x}+\tilde{u}_{xx}+\tilde{u}(1-\tilde{u}))e^{x}dx=\mu^{\prime}(t)\int\tilde{u}e^{x}dx=\mu^{\prime}(t)I(t).$
(5.6)

Therefore, $I(t)$ evolves in time as

$\displaystyle I(t)=I(0)e^{\mu(t)},$ (5.7)

as long as we choose $\mu(0)=0$. This algebraic property is very specific for
$\beta=2$.

If $\mu(t)$ is the “correct frame”, in the sense that $\tilde{u}(t,x)$
converges to a traveling wave, we expect from the philosophy of the self-
similar variables that $\tilde{u}(t,x)$ has the asymptotics

$\displaystyle\tilde{u}(t,x)\sim\frac{1}{1+e^{x-x_{0}}}e^{-{x^{2}}/{(4t)}},~{}~{}\hbox{
for $t\gg 1$.}$ (5.8)

This indicates that if $I(t)$ is computed in the correct reference frame, then

$I(t)=\int
e^{x}\tilde{u}(t,x)dx\approx\int\frac{e^{x}}{1+e^{x-x_{0}}}e^{-{x^{2}}/{(4t)}}dx\sim
C_{0}\sqrt{t},~{}~{}\hbox{ for $t\gg 1$,}$ (5.9)

with an explicit constant $C_{0}$ that depends on $x_{0}$. For this to be
consistent with (5.7), we should have

$\mu(t)\sim\frac{1}{2}\log t+C_{1},~{}~{}~{}~{}\hbox{ for $t\gg 1$,}$ (5.10)

explaining the $(1/2)\log t$ shift for the front position. The rest of this
section and the following one is a justification of (5.10).

####  5.1.2 An inhomogeneous conservation law

In order to explain the outline of the proof of Proposition 5.1, we need to
introduce a change of variable related to the evolution of mass for the
exponential moment in (5.6). Let $u(t,x)$ be the solution to the Burgers-FKPP
equation (1.1) with the specific value $\beta=2$, and set

$\widehat{u}(t,x)=u(t,x+2t),$ (5.11)

which satisfies

$\displaystyle\widehat{u}_{t}-2\widehat{u}_{x}+2\widehat{u}\widehat{u}_{x}=\widehat{u}_{xx}+\widehat{u}(1-\widehat{u}),$
(5.12)

Note that we use a slightly different notation here for the shifted function
$\widehat{u}$ rather than $\tilde{u}$. This is to denote the difference in
shift: $\widehat{u}$ is shifted into the moving frame $2t$, while $\tilde{u}$
is shifted to the moving frame $2t-\mu(t)$ matching the front, as, for
instance, in the proof of 2.2.

The first key observation is that the function

$p(t,x)=e^{x}\widehat{u}(t,x),$ (5.13)

satisfies a viscous spatially inhomogeneous conservation law:

$p_{t}+\big{(}p^{2}e^{-x}\big{)}_{x}=p_{xx},$ (5.14)

with the initial condition

$p(0,x)=e^{x}\mathbbm{1}(x\leq 0).$ (5.15)

Recall that we assume, without loss of generality, that
$u(0,x)=\mathbbm{1}(x\leq 0)$. Notice that (5.14) conserves mass for $p(t,x)$:

$\int p(t,x)dx=\int p(0,x)dx=1.$ (5.16)

The normalization (5.1) in terms of $\widehat{u}(t,x)$ becomes

$\widehat{u}(t,-\mu(t))=\frac{1}{2},$ (5.17)

which translates into

$p(t,-\mu(t))=\frac{1}{2}e^{-\mu(t)}.$ (5.18)

A simple preliminary observation is that the solution to (5.14)-(5.15)
satisfies

$p(t,x)\leq e^{x},~{}~{}\hbox{ for all $t\geq 0$ and $x\in{\mathbb{R}}$,}$
(5.19)

simply because $\widehat{u}(t,x)\leq 1$. Note also that $p(x)=e^{x}$ and

$p(x)=\frac{e^{x}}{1+e^{(x-\xi)}}$ (5.20)

are exact solutions to (5.14), for any shift $\xi\in{\mathbb{R}}$.

The proof of Proposition 5.1 relies on the analysis of the solution to
(5.14)-(5.15) and proceeds in the following steps. First, we prove an upper
bound on $\mu(t)$.

######  Lemma 5.2.

There exists $K_{1}>0$ so that $\mu(t)$ be defined by (5.17) satisfies

$\mu(t)\leq\frac{1}{2}\log t+K_{1},~{}~{}\hbox{ for all $t\geq 1$.}$ (5.21)

Lemma 5.2 is proved in Section 5.3. Let us give a brief outline of the proof.
It is based on the conservation (5.16) of the total mass of $p(t,x)$, together
with estimating the mass of $p(t,x)$ separately in the regions
$\\{x<-\mu(t)\\}$, $\\{-\mu(t)<x<N\sqrt{t}\\}$ and $\\{x>N\sqrt{t}\\}$.

First, we will show that the mass of $p(t,x)$ in the region
$\\{x>N\sqrt{t}\\}$ is exponentially small in $N$ by an argument that bounds
exponential moments of $p$. Next, using the simple exponential bound (5.19),
we can check that the mass in the region $\\{x<-\mu(t)\\}$ is bounded above by
$\exp\\{-\mu(t)\\}$.

To bound the mass in the middle region, recall that when $\beta=2$, the
traveling wave moving with the minimal speed $c_{*}=2$ is explicit and is
given by (1.14):

$\phi(x)=\frac{1}{1+e^{x}}.$ (5.22)

From this and the steepness comparison in Proposition 1.1, we immediately
deduce the following useful property.

######  Lemma 5.3.

If $u(0,\cdot)$ is steeper than $\phi$ given by (5.22), then, for each
$v\in(0,1)$ and $t>0$, we have

$\displaystyle u(t,x)\geq\
\Big{(}1+\frac{1-v}{v}e^{x-x(t,v)}\Big{)}^{-1},~{}~{}\text{if }x<x(t,v),$
(5.23) $\displaystyle u(t,x)\leq\
\Big{(}1+\frac{1-v}{v}e^{x-x(t,v)}\Big{)}^{-1},~{}~{}\text{if }x>x(t,v).$

Here, $x(t,v)$ is defined implicitly by $v=u(t,x(t,v))$.

By Lemma 5.3, and (5.18), we know that

$p(t,x)=e^{x}\widehat{u}(t,x)\leq\frac{e^{x}}{1+e^{x+\mu(t)}}\leq
e^{-\mu(t)},~{}~{}~{}\hbox{ for $x>-\mu(t)$}.$ (5.24)

It follows that the mass in the region $-\mu(t)\leq x\leq N\sqrt{t}$ is
bounded by $2N\sqrt{t}e^{-\mu(t)}$ (it is easy to show the weak bound
$\mu(t)<N\sqrt{t}$, see (5.28) below). Combining all three bounds and
recalling mass conservation (5.16) of $p$, we deduce that $\mu(t)$ must
satisfy the upper bound in Lemma 5.2. The details are given in Section 5.3.

To prove a lower bound for $\mu(t)$ we use the following lemma.

######  Lemma 5.4.

There exist $C>0$ so that, for all $x\in\mathbb{R}$,

$p(t,x)\leq\frac{C}{\sqrt{t}}.$ (5.25)

We prove Lemma 5.4 in Section 5.4, and its surprisingly delicate proof is
outlined there. When combined with (5.18), Lemma 5.4 implies the lower bound
in Proposition 5.1

$\mu(t)\geq\frac{1}{2}\log t-C,\hbox{ for all $t\geq 1$.}$ (5.26)

Let us make a brief comment that the $t^{-1/2}$ decay rate in (5.25) is
standard for parabolic equations in one dimension, and would be expected for a
solution to (5.13). Nevertheless, the proof of Lemma 5.4 is much less
straightforward than one would naively expect. To illustrate the potential
obstacles, notice that (5.14) can be written as

$p_{t}+(\widehat{u}p)_{x}=p_{xx}.$ (5.27)

One might hope to “forget” the connection between $\widehat{u}$ and $p$, and
prove $t^{-1/2}$ decay for linear divergence form advection-diffusion
equations of the form (5.27) in general, with an advection term
$\widehat{u}(t,x)$ that, say, connects two constants on the left and on the
right. This seems to be a good cartoon for $\widehat{u}(t,x)$. However, such
decay cannot hold in general. Indeed, consider the following explicit example.
Let

$q(x)=\frac{1}{e^{x}+e^{-x}},\quad\text{ and }\quad
v(x)=-\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}},$

so that $q(x)$ is a steady solution to

$q_{t}+(vq)_{x}=q_{xx}.$

The function $q(x)$ is rapidly decaying at infinity but, of course, it does
not decay in time. This shows that the boundedness or existence of the limits
at infinity of $\widehat{u}$ are not sufficient to determine the decay. It is
crucial that $\widehat{u}$ is negligible on $\mathbb{R}_{+}$ and its profile
does not move much – these are, however, exactly the properties that we are
trying to prove. Circumventing these difficulties requires interesting a
priori estimates for our specific problem, which will be discussed in Section
5.4.

To summarize, 5.1 reduces to Lemmas 5.2 and 5.4 and we prove these lemmas in
the rest of this section.

###  5.2 Preliminary weak bounds on $\mu$

Let us first give some very poor bounds on $\mu(t)$ that, at least, ensure
that it does not behave too wildly. These are useful in the sequel.

For a very simple bound, a comparison to the standard KPP equation with
$\beta=0$ that uses monotonicity of $u(t,x)$ in $x$, implies that

$\mu(t)\leq\frac{3}{2}\log t+C,$ (5.28)

with a universal constant $C$.

We now use the conservation of mass for $p$ (5.16) to obtain a lower bound on
$\mu$. Recalling Lemma 5.3 and the definition (5.1) of $\mu$, we find

$p(t,x)=e^{x}\widehat{u}(t,x)\geq\frac{e^{x}}{1+e^{x+\mu(t)}},~{}~{}\hbox{ for
$x<-\mu(t)$.}$

Therefore, we have

$\displaystyle 1$
$\displaystyle=\int_{-\infty}^{\infty}p(0,x)dx=\int_{-\infty}^{\infty}p(t,x)dx\geq\int_{-\infty}^{-\mu(t)}\frac{e^{x}}{1+e^{x+\mu(t)}}dx=e^{-\mu(t)}\int_{-\infty}^{0}\frac{e^{x}}{1+e^{x}}dx\geq\frac{1}{2}e^{-\mu(t)}.$

We conclude that

$\mu(t)\geq-\log 2.$ (5.29)

The bounds (5.28) and (5.29) will be greatly improved below.

###  5.3 An upper bound on the shift: the proof of Lemma 5.2

As we have mentioned, the strategy of the proof is to show that if $\mu(t)$ is
too large then $p(t,x)$ can not have a mass larger than $1/10$ in any of the
three regions

$L=\\{x<-\mu(t)\\},~{}~{}M=\\{-\mu(t)\leq x\leq
N\sqrt{t}\\},~{}~{}R=\\{x>N\sqrt{t}\\},$ (5.30)

provided that $N$ is also chosen sufficiently large. This would contradict
(5.16). Note that (5.29) implies $N\sqrt{t}\geq-\mu(t)$ for $t\geq 1$ and $N$
sufficiently large, so that the regions above are well-defined.

For the left region $L$, we simply apply (5.19) and write

$\int_{L}p(t,x)dx=\int_{-\infty}^{-\mu(t)}p(t,x)dx\leq\int_{-\infty}^{-\mu(t)}e^{x}dx=e^{-\mu(t)}.$
(5.31)

For the middle region $M$, we also have a simple estimate that uses Lemma 5.3:

$\begin{split}\int_{M}p(t,x)dx&=\int_{-\mu(t)}^{N\sqrt{t}}e^{x}\widehat{u}(t,x)dx\leq\int_{-\mu(t)}^{N\sqrt{t}}\frac{e^{x}}{1+e^{x+\mu(t)}}dx\\\
&=e^{-\mu(t)}\int_{0}^{N\sqrt{t}+\mu(t)}\frac{e^{x}}{1+e^{x}}dx\leq
CNe^{-\mu(t)}\sqrt{t},\end{split}$ (5.32)

for all $t\geq 1$. We used (5.28) in the last step above.

Now, we deal with the right region $R$. First we state a bound on an
exponential moment of $p$: fix any $m\in(0,1/2)$ and let

$I_{m}(t)=\int\big{(}e^{mx}+e^{-mx}\big{)}p(t,x)dx.$ (5.33)

We claim that

$I_{m}(t)\leq Ce^{Cm^{2}t}.$ (5.34)

We postpone the proof of (5.34) momentarily and show how to conclude the bound
for the integral over $R$. For $N>1$ large, we estimate this mass by using
(5.33) with $m=1/\sqrt{t}$. This gives

$\displaystyle\int_{R}p(t,x)dx$
$\displaystyle=\int_{N\sqrt{t}}^{\infty}p(t,x)dx\leq\Big{(}\int_{N\sqrt{t}}^{\infty}e^{x/\sqrt{t}}p(t,x)dx\Big{)}^{1/2}\Big{(}\int_{N\sqrt{t}}^{\infty}e^{-x/\sqrt{t}}p(t,x)dx\Big{)}^{1/2}$
(5.35) $\displaystyle\leq
C\Big{(}\int_{N\sqrt{t}}^{\infty}e^{-x/\sqrt{t}}p(t,x)dx\Big{)}\leq
C\Big{(}\int_{N\sqrt{t}}^{\infty}e^{-N}p(t,x)dx\Big{)}^{1/2}\leq Ce^{-N/2}.$

We used the conservation of mass (5.16) in the last step.

We may now put the estimates (5.31), (5.32) and (5.35) together to obtain

$\displaystyle 1$
$\displaystyle=\int_{-\infty}^{\infty}p(0,x)dx=\int_{-\infty}^{\infty}p(t,x)dx=\int_{L}p(t,x)dx+\int_{M}p(t,x)dx+\int_{R}p(t,x)dx$
(5.36) $\displaystyle\leq e^{-\mu(t)}+CN\sqrt{t}e^{-\mu(t)}+Ce^{-N/2}.$

Choosing $N=2\log(2C)$ gives

$\displaystyle\frac{1}{2}$ $\displaystyle\leq
e^{-\mu(t)}+C\sqrt{t}e^{-\mu(t)},$ (5.37)

so that

$e^{\mu(t)}\leq C(1+\sqrt{t}),$ (5.38)

and (5.21) follows, as desired.

All that remains to finish the proof of Lemma 5.2 is to establish (5.34),
which we do now. Recalling $I_{m}$ from (5.33), multiplying (5.14) by
$e^{mx}+e^{-mx}$, and integrating yields

$\displaystyle\frac{d}{dt}I_{m}=m^{2}\int(e^{mx}+e^{-mx})p(t,x)dx+m\int
e^{-x}[e^{mx}-e^{-mx}]p^{2}(t,x)dx.$ (5.39)

We decompose the last integral:

$\displaystyle J_{m}=m\int e^{-x}[e^{mx}-e^{-mx}]p^{2}(t,x)dx=m\int
e^{-x}\tanh(mx)[e^{mx}+e^{-mx}]p^{2}(t,x)dx$ (5.40) $\displaystyle\leq
m\int_{-\infty}^{\log(2)}e^{-x}|\tanh(mx)|[e^{mx}+e^{-mx}]p^{2}(t,x)dx$
$\displaystyle\quad+m\int^{\infty}_{\log(2)}e^{-x}|\tanh(mx)|[e^{mx}+e^{-mx}]p^{2}(t,x)dx=J_{m}^{-}+J_{m}^{+}.$

For $J_{m}^{-}$, we use (5.19), the bound

$|\tanh(mx)|\leq m|x|,$ (5.41)

and the assumption $0<m<1/2$ to obtain

$\displaystyle J_{m}^{-}$ $\displaystyle\leq
m\int_{-\infty}^{\log(2)}|\tanh(mx)|[e^{mx}+e^{-mx}]e^{x}dx\leq
m^{2}\int_{-\infty}^{\log(2)}|x|e^{(1-m)x}dx\leq Cm^{2}.$ (5.42)

On the other hand, as $x>\log(2)>-\mu(t)$, by (5.29), we may apply Lemma 5.3
to obtain

$\displaystyle J_{m}^{+}$
$\displaystyle=m\int_{\log(2)}^{+\infty}|\tanh(mx)|[e^{mx}+e^{-mx}]\widehat{u}(t,x)p(t,x)dx$
$\displaystyle\leq
m\int_{\log(2)}^{+\infty}|\tanh(mx)|[e^{mx}+e^{-mx}]\frac{p(t,x)}{1+e^{x+\mu(t)}}dx$
$\displaystyle\leq
m^{2}\int_{\log(2)}^{+\infty}\frac{|x|}{1+e^{x+\mu(t)}}[e^{mx}+e^{-mx}]p(t,x)dx\leq
m^{2}e^{-\mu(t)}\int_{\log(2)}^{+\infty}[e^{mx}+e^{-mx}]p(t,x)dx.$

In the second inequality, we used (5.41), and in the last inequality, we used
that

$|x|\leq e^{x}\leq e^{-\mu(t)}(1+e^{x+\mu(t)}).$

Recalling that $-\mu(t)\leq\log(2)$ from (5.29), we find

$J_{m}^{+}\leq Cm^{2}I_{m}.$ (5.43)

The combination of (5.42) and (5.43) yields

$\frac{dI_{m}}{dt}\leq C_{0}m^{2}(I_{m}+1),$

for some universal constant $C_{0}$. This implies that

$I_{m}(t)\leq e^{C_{0}m^{2}t}[I_{m}(0)+1]\leq 5e^{C_{0}m^{2}t},$

since, recalling that $m<1/2$,

$I_{m}(0)=\int_{-\infty}^{0}e^{x}(e^{mx}+e^{-mx})dx\leq 4.$

This concludes the proof of (5.34) and, thus, that of Lemma 5.2. $\Box$

###  5.4 A lower bound on the shift: the proof of Lemma 5.4

The proof of Lemma 5.4 is quite a bit more involved than that for Lemma 5.2.
Let us first explain the main steps of the proof. Recall that the standard
$L^{\infty}$-decay for diffusion equations of the self-adjoint form

$z_{t}=\nabla\cdot(a(x)\nabla z),~{}~{}x\in{\mathbb{R}}^{n},$ (5.44)

with a uniformly positive and bounded diffusivity $a(x)$ is obtained as
follows: first, one gets the dissipation inequality

$\frac{1}{2}\frac{d}{dt}\int|z(t,x)|^{2}dx\leq-C\int|\nabla z(t,x)|^{2}dz.$
(5.45)

An application of the Nash inequality leads, after solving an elementary
differential inequality, to the $L^{1}-L^{2}$ decay estimate

$\|z(t,\cdot)\|_{L^{2}}\leq\frac{C}{t^{n/4}}\|z(0,\cdot)\|_{L^{1}}.$ (5.46)

The self-adjoint form of (5.44) and the estimate (5.46) give the dual bound

$\|z(t,\cdot)\|_{L^{\infty}}\leq\frac{C}{t^{n/4}}\|z(0,\cdot)\|_{L^{2}}.$
(5.47)

The last step is to apply the semi-group property and the above estimates to
deduce that

$\|z(t,\cdot)\|_{L^{\infty}}\leq\frac{C}{t^{n/4}}\|z(t/2,\cdot)\|_{L^{2}}\leq\frac{C}{t^{n/2}}\|z(0,\cdot)\|_{L^{1}}.$
(5.48)

It seems not possible to directly obtain a dissipation inequality for the
$L^{2}$-norm of the function $p(t,x)$, starting with (5.14), due to the
spatial inhomogeneity of the nonlinear term. Instead, to get an analogue of
(5.45), we will use a suitably chosen weight $\rho(t,x)$ that weighs
$\mathbb{R}_{+}$ more than $\mathbb{R}_{-}$, and establish an
$L_{w}^{2}$-dissipation inequality for the function

$\varphi=\frac{p}{\rho}.$ (5.49)

Here we use $w$ as a subscript to emphasize that we are working in weighted
Lebesgue spaces. Such weight allows to focus on where advection is negligible
and diffusion dominates the evolution of (5.27).

It turns out that the weight $\rho(t,x)=1-\widehat{u}(t,x)$ can actually be
used to produce a dissipation inequality because, as we will see, the function
$1-\widehat{u}$ is a super-solution for (5.14) satisfied by $p(t,x)$. This
property is not purely algebraic: it will, once again, use the steepness
comparison of $\widehat{u}(t,x)$ to the traveling wave. The dissipation
computation is directly inspired by the relative entropy arguments for linear
advection-diffusion equations in [15, 37]. Here, however, we compute the
entropy relative not to a solution but a super-solution, and the nonlinear
nature of the present situation requires specific cancellations. This is the
subject of Proposition 5.8 below.

We also establish a weighted Nash inequality stated in Proposition 5.9 below.
When adapted to our setting, it yields the appropriate long time decay of the
$L_{w}^{2}$-norm of $\varphi$, up to a (potentially large) boundary layer in
time. An interesting complication is that the weighted Nash inequality holds
for a nontrivial class of weights satisfying certain assumptions, and we need
to control the fact that our weight, coming from the solution to a nonlinear
evolution equation, satisfies these assumptions for all $t>0$ in a uniform
way. This is done in the course of the proof of Lemma 5.7 below.

In contrast to the standard proof for diffusion equations, we cannot directly
pass from $L^{2}_{w}$-decay to $L^{\infty}$-decay. Indeed, due to the weight,
the aforementioned $L_{w}^{2}$-decay estimate is not an $L_{w}^{1}\to
L_{w}^{2}$ estimate as the boundary layer depends on the initial
$L^{2}_{w}$-norm of $\varphi$ and, hence, the usual adjointness trick in
(5.47), used to establish the $L^{\infty}$ decay is not available. This
complication is present even for the linear equation (5.27) when the
relationship between $p$ and $\widehat{u}$ is “forgotten.” In fact, decay in
$L^{\infty}_{w}$ of $\varphi$ is not even expected in the setting in which we
find ourselves. To pass from the $L_{w}^{2}$-bounds for the function
$\varphi(t,x)$ to the $L^{\infty}$ decay for $p(t,x)$, we use time averages to
find a particular intermediate time $T_{g}<T$, at which $\varphi$ satisfies
“good” pointwise bounds in a region of interest. Those bounds, stated in Lemma
5.10, can be transferred to show that $p$ is bounded both by $e^{x}$ and
$C/\sqrt{T}$. We can then “trap” that estimate going forward in time, from the
time $T_{g}$ until the time $T$, by breaking $p$ up into a small mass part,
which necessarily stays small due to conservation of mass and parabolic
regularity theory, and a part that sits under an explicit, small super-
solution. This will conclude the proof.

As the reader will surely have noticed, there is a subtle, but serious, issue
in the above outline. The $L^{2}_{w}$ decay requires that the initial
$L^{2}_{w}$ norm be bounded, which, as can be immediately seen from (5.49), is
not true when $\widehat{u}(0,\cdot)$ is the Heaviside function, the function
$p(0,\cdot)$ is given by (5.15), and $\rho=1-\widehat{u}$. This requires an
extra step where we choose an approximate initial condition

$\widehat{u}_{a}(0,x)=\frac{1}{1+e^{\gamma(x-a)}}\mathbbm{1}(x\leq 0),$ (5.50)

with $\gamma\in(1,2)$, to obtain new solutions $\widehat{u}_{a}$ and $p_{a}$.
Here, $a>0$ is a parameter depending on the final time $T$ at which we wish to
establish the upper bound. A careful analysis shows that, with an appropriate
choice of $a$, the two solutions, $p$ and $p_{a}$, stay $O(1/\sqrt{T})$ away
from each other, due to the error estimate in Lemma 5.6.

Below, we state the upper bound on the modification $p_{a}$ and show how to
bootstrap that bound to the decay of $p$ itself. This is done in Section
5.4.1. Then we give the proof of the upper bound on $p_{a}$ following the
outline above, in Section 5.4.2.

####  5.4.1 The proof of Lemma 5.4

#### A modified initial condition

We first construct the modified solution $p_{a}(t,x)$. Recall that
$\widehat{u}(t,x)$ is the solution to the Burgers-FKPP equation

$\widehat{u}_{t}-2\widehat{u}_{x}+2\widehat{u}\widehat{u}_{x}=\widehat{u}_{xx}+\widehat{u}-\widehat{u}^{2},$
(5.51)

with the initial condition $\widehat{u}_{\rm in}(x)=\mathbbm{1}(x\leq 0)$. For
any $a>0$ and $\gamma\in(1,2)$, let $\widehat{u}_{a}(t,x)$ be the solution to
(5.51) with the initial condition

$\widehat{u}_{a}(0,x)=\frac{1}{1+e^{\gamma(x-a)}}\mathbbm{1}(x\leq 0),$ (5.52)

and set

$p(t,x)=e^{x}\widehat{u}(t,x),~{}~{}p_{a}(t,x)=e^{x}\widehat{u}_{a}(t,x).$
(5.53)

As in (5.14) and (5.27), the function $p_{a}$ satisfies

$\partial_{t}p_{a}+\partial_{x}(p_{a}^{2}e^{-x})=\partial_{x}^{2}p_{a}$ (5.54)

and

$\partial_{t}p_{a}+\partial_{x}(\widehat{u}_{a}p_{a})=\partial_{x}^{2}p_{a}.$
(5.55)

We have made the switch to $\partial_{t,x}$ notation to avoid the awkward
double sub-script. It is easy to observe that $\widehat{u}_{a}\leq 1$, and we
have , by the comparison principle:

$p_{a}(t,x)\leq e^{x}\qquad\text{ for all }t\geq 0,~{}~{}x\in\mathbb{R}.$
(5.56)

Two important quantities of interest for us are

$\begin{split}&M_{a}(t)=\int\left(\frac{p_{a}(t,x)}{1-\widehat{u}_{a}(t,x)}\right)(1-\widehat{u}_{a}(t,x))\,dx=\int
p_{a}(t,x)\,dx=M_{a}(0),\\\
&I_{a}(t)=\int\left(\frac{p_{a}(t,x)}{1-\widehat{u}_{a}(t,x)}\right)^{2}(1-\widehat{u}_{a}(t,x))\,dx=\int\frac{p_{a}(t,x)^{2}}{1-\widehat{u}_{a}(t,x)}\,dx.\end{split}$
(5.57)

The last equality in the first line above follows by conservation of mass.
Note that $M_{a}$ and $I_{a}$ are, respectively, the weighted $L^{1}$ and
$L^{2}$ norms of the function $\varphi=p_{a}/(1-\widehat{u}_{a})$ with weight
$1-\widehat{u}_{a}$. Two easy computations show that $M_{a}(0)$ is uniformly
bounded in $a>0$:

$\log(2)=\int_{-\infty}^{0}\frac{e^{x}}{1+e^{x}}\,dx\leq
M_{a}\leq\int_{-\infty}^{0}e^{x}\,dx=1$ (5.58)

and $I_{a}(0)$ is finite for all $a>0$:

$\displaystyle I_{a}(0)$
$\displaystyle=\int\frac{e^{2x}\widehat{u}_{a}^{2}(0,x)}{1-\widehat{u}_{a}(0,x)}\,dx\leq\int_{-\infty}^{0}\frac{e^{2x}}{1-\widehat{u}_{a}(0,x)}\,dx=\int_{-\infty}^{0}e^{2x}\frac{1+e^{\gamma(x-a)}}{e^{\gamma(x-a)}}\,dx$
(5.59) $\displaystyle\leq 2e^{\gamma
a}\int_{-\infty}^{0}e^{(2-\gamma)x}\,dx=\frac{2}{2-\gamma}e^{\gamma a}$

but is not uniformly bounded as $a\to+\infty$. Note that an analogous estimate
for $I(0)$ with $u_{a}$ and $p_{a}$ replaced by $u$ and $p$, respectively,
does not hold, as

$\frac{p^{2}(0,x)}{1-u(0,x)}=+\infty\cdot\mathbbm{1}(x<0).$ (5.60)

This is what prevents us from establishing the decay of $p$ directly.

As we see in the sequel, another crucial feature of this altered initial
condition is that $\widehat{u}_{a}(0,\cdot)$ is steeper than the traveling
wave $1/(1+e^{x})$. This is why we take $\gamma>1$ in (5.52). The restriction
$\gamma<2$ comes from the upper bound on $I_{a}(0)$ in (5.59).

The main estimate we establish on $p_{a}$ is:

######  Lemma 5.5.

Let $\widehat{u}_{a}(t,x)$ be the solution to (5.51) with the initial
condition (5.52) for some $a>0$ and $1<\gamma<2$, and
$p_{a}(t,x)=e^{x}\widehat{u}_{a}(t,x)$. There exists a universal constant
$C>0$ such that the following holds. Set

$t_{1}(a)=CI_{a}(0),$ (5.61)

with $I_{a}$ defined in (5.57). There exists $K>0$ that does not depend on
$a>1$ so that

$p_{a}(t,x)\leq\frac{K}{\sqrt{t}},~{}~{}\hbox{ for all }t\geq 4t_{1}(a).$
(5.62)

We postpone the proof of Lemma 5.5 until Section 5.4.2. First, we obtain a
closeness estimate on $p$ and $p_{a}$. Its proof is succinct enough to give it
immediately.

######  Lemma 5.6.

Fix $a\in\mathbb{R}$ and $\gamma>0$. Let $\widehat{u}(t,x)$ and
$\widehat{u}_{a}(t,x)$ be the solutions to (5.51) with the respective initial
conditions $\widehat{u}(0,x)=\mathbbm{1}(x\leq 0)$ and (5.52), respectively.
There is a constant $C>0$ that does not depend on $a$ or $\gamma$ such that
$p(t,x)=e^{x}\widehat{u}(t,x)$ and $p_{a}(t,x)=e^{x}\widehat{u}_{a}(t,x)$
satisfy

$0\leq p(t,x)-p_{a}(t,x)\leq Ce^{-\gamma a}\hbox{ for all $x\in{\mathbb{R}}$
and $t\geq 1$.}$ (5.63)

Proof. Let $h(t,x)=p(t,x)-p_{a}(t,x)$ for all $t\geq 0$ and $x\in\mathbb{R}$.
Note that $h(0,\cdot)\geq 0$ and, due to (5.14) and (5.54), that $h$ satisfies
the parabolic equation

$h_{t}-h_{xx}=-(p^{2}e^{-x})_{x}+(p_{a}^{2}e^{-x})_{x}=-(hw)_{x},$ (5.64)

with

$w(t,x)=e^{-x}(p_{a}(t,x)+p(t,x))=\widehat{u}(t,x)+\widehat{u}_{a}(t,x).$

Hence, the comparison principle implies that $h\geq 0$. This finishes the
lower bound in (5.63).

To conclude the upper bound, we use mass conservation for $p(t,x)$ and
$p_{a}(t,x)$. Indeed,

$\begin{split}\int
h(t,x)\,dx&=\int(p(t,x)-p_{a}(t,x))\,dx=\int(p(0,x)-p_{a}(0,x))\,dx=\int_{-\infty}^{0}e^{x}\frac{e^{\gamma(x-a)}}{1+e^{\gamma(x-a)}}\,dx\\\
&=e^{a}\int_{-\infty}^{-a}\frac{e^{(1+\gamma)x}}{1+e^{\gamma
x}}\,dx\leq\frac{1}{1+\gamma}e^{a}e^{-(1+\gamma)a}=\frac{1}{1+\gamma}e^{-\gamma
a}.\end{split}$ (5.65)

Using parabolic regularity theory together with (5.65) and positivity of $h$,
we get immediately that

$\sup_{x}h(t,x)\leq Ce^{-\gamma a}\qquad\hbox{for all $t\geq 1$ and
$x\in{\mathbb{R}}$,}$ (5.66)

finishing the proof. The constant $C$ does not depend on $a$ or $\gamma$.
$\Box$

Proof of Lemma 5.4. We are now in a position to combine Lemma 5.5 with Lemma
5.6, in order to prove Lemma 5.4. One delicate point is that Lemma 5.6
requires us to take $a$ large. On the other hand, $I_{a}(0)$ in (5.59) blows
up as $a\to+\infty$. This will require a careful balancing act.

Given $T>1$ sufficiently large, fix $\gamma=3/2$ and choose $a>0$ such that

$\gamma a=\frac{1}{2}\log(T).$ (5.67)

Recall from (5.59) that

$\displaystyle I_{a}(0)$ $\displaystyle\leq\frac{2}{2-\gamma}e^{\gamma a}.$
(5.68)

By our choice of $\gamma$ and $a$ and (5.61), it follows that, for $T$
sufficiently large,

$4t_{1}(a)\leq C\sqrt{T}\leq T.$ (5.69)

Due to (5.69), we may apply Lemma 5.5 to find

$\sup_{x}\,p_{a}(T,x)\leq\frac{K}{\sqrt{T}}.$ (5.70)

We may also use Lemma 5.6 to see that

$0\leq p(t,x)-p_{a}(t,x)\leq Ce^{-\gamma a},~{}~{}~{}~{}\hbox{ for all
$x\in{\mathbb{R}}$ and $t>1$.}$ (5.71)

Recalling the choice (5.67) of $\gamma$ and $a$ in (5.71) and using (5.70) to
$t=T$, we deduce that

$\sup_{x}\,p(T,x)\leq\sup_{x}\,p_{a}(T,x)+Ce^{-\gamma
a}\leq\frac{K}{\sqrt{T}}+\frac{C}{\sqrt{T}},$

finishing the proof of Lemma 5.4. $\Box$

####  5.4.2 The proof of Lemma 5.5

We now prove Lemma 5.5, following the outline from the beginning of Section
5.4.

#### Step one: decay in a weighted $L^{2}$ space

Our first goal is to obtain an $L^{2}$-decay estimate, an analogue of (5.46)
in the present situation. We will work with norms weighted by

$\rho_{a}=1-\widehat{u}_{a}.$ (5.72)

and make a change of function

$\varphi_{a}=\frac{p_{a}}{1-\widehat{u}_{a}}=\frac{p_{a}}{\rho_{a}}.$ (5.73)

The main goal of this step is the decay rate of $I_{a}$, which we state here.
Recall that $\widehat{u}_{a}$ and $p_{a}$ are defined in Lemma 5.5.

######  Lemma 5.7.

For $a>0$ and $\gamma\in(1,2)$, let $t_{1}(a)$, $M_{a}$, and $I_{a}(0)$ be as
in (5.61) and (5.59). Then

$I_{a}(t_{1})=\int\varphi_{a}^{2}(t_{1},x)\rho_{a}(t_{1},x)\,dx\leq 1,$ (5.74)

and

$I_{a}(t)=\int\varphi_{a}^{2}(t,x)\rho_{a}(t,x)dx\leq\frac{C}{1+\sqrt{t-t_{1}(a)}},~{}~{}\hbox{
for $t\geq t_{1}(a)$.}$ (5.75)

Note that $I_{a}$ may be thought of as
$\|\varphi_{a}\|^{2}_{L^{2}(\rho_{a})}$. To prove Lemma 5.7, we obtain a more
general result for weighted $L^{2}$ decay, identifying assumptions on the
weight under which decay holds. Afterwards, we show that $\rho_{a}$ satisfies
these assumptions and apply the general result to our setting.

The first step is a dissipation inequality, inspired by the relative entropy
arguments for linear equations in [15, 37]. As in [15], given a weight
$\rho(t,x)$ and an advection $v(t,x)$, we consider an operator
$\mathcal{D}_{\rho}$ defined by

$\mathcal{D}_{\rho}q=\rho^{-2}\partial_{x}(\rho^{2}\partial_{x}q)-v\partial_{x}q.$
(5.76)

######  Proposition 5.8.

Let $v(t,x)$ be a smooth bounded function, $q(t,x)$ be a solution to

$q_{t}+(vq)_{x}=q_{xx},$ (5.77)

and $\rho(t,x)$ be a super-solution to (5.77):

$\rho_{t}+(v\rho)_{x}\geq\rho_{xx},$ (5.78)

with initial conditions $p(0,x)\geq 0$ and $\rho(0,x)>0$ such that

$\int\varphi(0,x)^{2}\rho(0,x)dx<+\infty,$ (5.79)

where $\varphi(t,x)=q(t,x)/\rho(t,x)$ for all $t\geq 0$ and $x\in\mathbb{R}$.
Then $\varphi$ satisfies

$\varphi_{t}-\mathcal{D}_{\rho}\varphi=-\varphi^{2}\left(\frac{\rho_{t}-\rho_{xx}+(v\rho)_{x}}{\rho}\right),$
(5.80)

and

$\frac{d}{dt}\int\
\varphi^{2}(t,x)\rho(t,x)dx\leq-2\int\varphi_{x}^{2}(t,x)\rho(t,x)dx.$ (5.81)

The main difference with [15, 37] is that the function $\rho(t,x)$ is not a
solution to the advection-diffusion equation (5.77) but a super-solution.

The second general result is an adaptation of the Nash inequality for weighted
spaces. This allows us to make use of the dissipation inequality in
Proposition 5.8.

######  Proposition 5.9.

Let an increasing function $r\in L^{1}(\mathbb{R}_{-})\cap C(\mathbb{R})$
satisfy the following assumptions:

$\displaystyle(i)~{}~{}~{}\lim_{x\to-\infty}r(x)=0,~{}~{}0<\lim_{x\to+\infty}r(x)=r_{+}<+\infty,$
(5.82) $\displaystyle(ii)~{}~{}~{}\overline{\overline{r}}(x)\leq
C_{1}\max\\{1,\overline{r}(x)^{2}\\}r(x),~{}~{}\hbox{ for all
$x\in{\mathbb{R}}$},$

with

$\bar{r}(x)=\int_{-\infty}^{x}r(y)dy,~{}~{}\overline{\overline{r}}(x)=\int_{-\infty}^{x}\bar{r}(y)dy.$
(5.83)

Then, for any $\theta>0$ and any smooth non-negative function $\varphi(x)$
that is sufficiently rapidly decaying as $x\to+\infty$ and bounded as
$x\to-\infty$ we have

$\int\varphi^{2}(x)r(x)\,dx\leq\frac{2}{\theta}\Big{(}\int\varphi(x)r(x)dx\Big{)}^{2}+8C_{1}\max\\{1,\theta^{2}\\}\int|\varphi_{x}(x)|^{2}r(x)dx.$
(5.84)

Let us point out that the weight

$r(x)=1-\phi_{2}(x)=\frac{e^{x}}{1+e^{x}}$

satisfies assumptions of Proposition 5.9, and this is an example the reader
may want to keep in mind.

Let us also briefly note the connection with the standard Nash inequality. A
key step in the proof of the latter is to establish (5.84) with
$\max\\{1,\theta^{2}\\}$ replaced by $\theta^{2}$. The change here reflects
the fact that the measure induced by $r$ is finite on $\mathbb{R}_{-}$.

The proofs of Proposition 5.8 and Proposition 5.9 are postponed until Section
5.5. We first apply these results to establish the $L^{2}$ decay of $p_{a}$,
that is, Lemma 5.7.

#### Proof of Lemma 5.7

We first assume that the assumptions of the dissipation and Nash inequalities,
that is, Proposition 5.8 and Proposition 5.9, hold in a uniform way for
$\rho_{a}(t,x)$, and show how to conclude. Afterwards, we show how to verify
those assumptions.

Assuming that we can apply Proposition 5.8 and Proposition 5.9, we proceed as
follows. By Proposition 5.8, we have

$\frac{d}{dt}\int\varphi_{a}^{2}(t,x)\rho_{a}(t,x)dx\leq-2\int(\partial_{x}\varphi_{a})^{2}(t,x)\rho_{a}(t,x)dx.$
(5.85)

Moreover, recalling the definition of $M_{a}$ from (5.57) and that it is
uniformly bounded, as in (5.58), and applying Proposition 5.9, we find, for
any $\theta>0$,

$\int\varphi_{a}^{2}(t,x)\rho_{a}(x)\,dx\leq\frac{C}{\theta}+C\max\\{1,\theta^{2}\\}\int|\partial_{x}\varphi_{a}(t,x)|^{2}\rho_{a}(t,x)dx,$
(5.86)

with a constant $C$ that does not depend on $t$, $\gamma$, or $a$.

There are two cases to consider: first, assume that at some $t>0$ we have

$\int|\partial_{x}\varphi_{a}(t,x)|^{2}\rho_{a}(t,x)\,dx\geq 1.$ (5.87)

Then, we get from (5.85):

$\frac{d}{dt}\int\varphi_{a}^{2}(t,x)\rho_{a}(t,x)\,dx\leq-2.$ (5.88)

On the other hand, if (5.87) fails, so that

$\int|\partial_{x}\varphi_{a}(t,x)|^{2}\rho_{a}(t,x)\,dx<1,$

we can use (5.86) with the constant

$\theta=\Big{(}\int|\partial_{x}\varphi_{a}|^{2}\rho_{a}\,dx\Big{)}^{-1/3}\geq
1.$

Keeping in mind the upper bound (5.58) on $M_{a}$, this leads to

$\int\varphi_{a}^{2}(t,x)\rho_{a}(x)\,dx\leq
C\Big{(}\int|\partial_{x}\varphi_{a}(t,x)|^{2}\rho_{a}(t,x)dx\Big{)}^{1/3}.$

Therefore, in the second case we have

$\frac{d}{dt}\int\varphi_{a}^{2}(t,x)\rho_{a}(t,x)\,dx\leq-\frac{1}{C}\Big{(}\int\varphi_{a}^{2}(t,x)\rho_{a}(t,x)\,dx\Big{)}^{3}.$
(5.89)

Putting (5.88) and (5.89) together, we find

$\frac{d}{dt}\int\varphi_{a}^{2}(t,x)\rho_{a}(t,x)\,dx\leq-\frac{1}{C}\min\Big{\\{}1,\Big{(}\int\varphi_{a}^{2}(t,x)\rho_{a}(t,x)\,dx\Big{)}^{3}\Big{\\}}.$
(5.90)

Setting

$t_{1}(a)=CI_{a}(0)\geq C(I_{a}(0)-1)_{+},$ (5.91)

we see from (5.90) that

$\int\varphi_{a}^{2}(t_{1},x)\rho_{a}(t_{1},x)\,dx\leq 1.$ (5.92)

Therefore, for $t\geq t_{1}(a)$, we have

$\frac{d}{dt}\int\varphi_{a}^{2}(t,x)\rho_{a}(t,x)\,dx\leq-C\Big{(}\int\varphi_{a}^{2}(t,x)\rho_{a}(t,x)\,dx\Big{)}^{3},$
(5.93)

and

$\int\varphi_{a}(t,x)^{2}\rho_{a}(t,x)dx\leq\frac{C}{1+\sqrt{t-t_{1}}}.$
(5.94)

Hence, all that remains in the proof of Lemma 5.7 is to verify the assumptions
of Proposition 5.8 and Proposition 5.9.

We consider first the assumptions of Proposition 5.8. In the notation of
Proposition 5.8, we have $q=p_{a}$, $v=\widehat{u}$, and $\rho=\rho_{a}$. In
view of (5.55) and (5.59), the assumptions (5.77) and (5.79) are satisfied.
Hence, the only assumption to check is (5.78), a somewhat miraculous property
that

$\partial_{t}\rho_{a}+\partial_{x}(\widehat{u}_{a}\rho_{a})\geq\partial_{x}^{2}\rho_{a}.$
(5.95)

First, using that $\widehat{u}_{a}$ satisfies (5.51), we find

$\begin{split}\partial_{t}\rho_{a}+\partial_{x}(\widehat{u}_{a}\rho_{a})-\partial_{x}^{2}\rho_{a}&=-\partial_{t}\widehat{u}_{a}+\partial_{x}\widehat{u}_{a}-2\widehat{u}_{a}\partial_{x}\widehat{u}_{a}+\partial_{x}^{2}\widehat{u}_{a}=-\partial_{x}\widehat{u}_{a}-\widehat{u}_{a}(1-\widehat{u}_{a}).\end{split}$
(5.96)

The initial condition for $\widehat{u}_{a}$ in (5.52) was chosen to be steeper
than the traveling wave $\phi$ given by (5.22) – this is why we needed to take
$\gamma>1$ in (5.52). Using Proposition 1.1, we deduce that
$\widehat{u}_{a}(t,x)$ is steeper than $\phi$ for all $t>0$. Applying this
property to (5.96) gives

$\partial_{t}\rho_{a}+\partial_{x}(\widehat{u}_{a}\rho_{a})-\partial_{x}^{2}\rho_{a}=-\partial_{x}\widehat{u}_{a}-\widehat{u}_{a}(1-\widehat{u}_{a})\geq-\phi_{x}(\phi^{-1}(\widehat{u}_{a}))-\phi(\phi^{-1}(\widehat{u}_{a}))(1-\phi(\phi^{-1}(\widehat{u}_{a}))).$
(5.97)

Using the explicit expression (5.22) for $\phi$, it is straightforward to
check that the right side of (5.97) is non-negative everywhere:

$\begin{split}-\phi_{x}-\phi(1-\phi)&=-\left(-\frac{e^{x}}{(1+e^{x})^{2}}\right)-\frac{e^{x}}{1+e^{x}}\frac{1}{1+e^{x}}=0.\end{split}$
(5.98)

This can, of course, be also obtained from (3.24). Hence, (5.95) is
established.

Next, we verify that $\rho_{a}$ satisfies assumptions (i) and (ii) of
Proposition 5.9 uniformly for all $t>0$. Assumption (i) holds automatically
since $\rho_{a}$ is increasing in $x$ (recall that $\widehat{u}_{a}$ is
decreasing in $x$), with $\rho_{a}(t,-\infty)=0$ and $\rho_{a}(t,+\infty)=1$
for all $t>0$. Assumption (ii) in (5.82) requires that

$\displaystyle\overline{\overline{\rho}}_{a}(t,x)\leq
C_{1}\max\\{1,\overline{\rho}_{a}(t,x)^{2}\\}\rho_{a}(t,x),~{}~{}\hbox{ for
all $x\in{\mathbb{R}}$}.$ (5.99)

It is useful to (implicitly) define the analogue $\mu_{a}$ of $\mu$ for
$\widehat{u}_{a}$:

$\widehat{u}_{a}(t,-\mu_{a}(t))=\frac{1}{2}.$ (5.100)

Consider first the case when $x<-\mu_{a}(t)$. As $\widehat{u}_{a}(0,\cdot)$ is
steeper than the traveling wave, so is $\widehat{u}_{a}(t,\cdot)$, by
Proposition 1.1, and we know from Lemma 5.3 that, if $y\leq x$, then

$\rho_{a}(t,y)=1-\widehat{u}_{a}(t,y)\leq\frac{\rho_{a}(t,x)e^{y-x}}{1-\rho_{a}(t,x)+\rho_{a}(t,x)e^{y-x}}.$
(5.101)

This gives

$\displaystyle\bar{\rho}_{a}(t,x)$
$\displaystyle=\int_{-\infty}^{x}\rho_{a}(t,y)\,dy\leq\int_{-\infty}^{x}\frac{\rho_{a}(t,x)e^{y-x}}{1-\rho_{a}(t,x)+\rho_{a}(t,x)e^{y-x}}\,dy$
(5.102)
$\displaystyle\leq\frac{\rho_{a}(t,x)}{1-\rho_{a}(t,x)}\int_{-\infty}^{x}{e^{y-x}}\,dy=\frac{\rho_{a}(t,x)}{1-\rho_{a}(t,x)}\leq
2\rho_{a}(t,x).$

In the last line we used that $x<-\mu_{a}(t)$, so that

$1-\rho_{a}(t,x)=\widehat{u}_{a}(t,x)\geq 1/2.$

This implies (5.99) for $x$ such that $\rho_{a}(t,x)<1/2$:

$\overline{\overline{\rho}}_{a}(t,x)=\int_{-\infty}^{x}\overline{\rho}_{a}(t,y)\,dy\leq
2\int_{-\infty}^{x}\rho_{a}(t,y)\,dy=2\overline{\rho}_{a}(t,x)\leq
4\rho_{a}(t,x).$ (5.103)

On the other hand, when $x\geq-\mu_{a}(t)$, we have $\rho_{a}(t,x)\geq 1/2$,
so that

$\overline{\rho}_{a}(t,x)\leq
2\overline{\rho}_{a}(t,x)\rho_{a}(t,x),~{}~{}\hbox{ for $x>\mu_{a}(t)$.}$
(5.104)

Since $\overline{\rho}_{a}(t,x)$ and $\rho_{a}$ are increasing, we find, using
(5.102) for $y<-\mu_{a}(t)$ and (5.104) for $y>-\mu_{a}(t)$:

$\begin{split}&\overline{\overline{\rho}}_{a}(t,x)=\int_{-\infty}^{x}\overline{\rho}_{a}(t,y)\,dy=\int_{-\infty}^{-\mu_{a}(t)}\overline{\rho}_{a}(t,y)\,dy+\int_{-\mu_{a}(t)}^{x}\overline{\rho}_{a}(t,y)\,dy\\\
&\leq
2\int_{-\infty}^{-\mu_{a}(t)}\rho_{a}(t,y)dy+2\int_{-\infty}^{x}\overline{\rho}_{a}(t,y)\rho_{a}(t,y)\,dy\leq
2\overline{\rho}_{a}(t,-\mu_{a}(t))+2\overline{\rho}_{a}(t,x)\int_{-\infty}^{x}\rho_{a}(t,y)\,dy\\\
&\leq 4\rho_{a}(t,-\mu_{a}(t))+2\overline{\rho}_{a}^{2}(t,x)\leq
4\rho_{a}(t,x)+4\overline{\rho}_{a}(t,x)^{2}\rho_{a}(t,x)\leq
8\max\\{1,\overline{\rho}_{a}(t,x)^{2}\\}\rho_{a}(t,x).\end{split}$ (5.105)

Taking the maximum of (5.103) and (5.105), we arrive at (5.99). We deduce that
the function $\rho_{a}(t,x)$ satisfies the assumptions of the weighted Nash
inequality in Proposition 5.9, for all $t>0$. This concludes the proof of
Lemma 5.7. $\Box$

#### Step two: pointwise bounds at a particular time

We now find a “good time” $T_{g}<T$ when $\varphi_{a}$ satisfies the desired
bounds via a time averaging. The catch is that we do not have control over
$T_{g}$ and, thus, a third step is required afterwards, to control the
solution on the time interval $T_{g}\leq t\leq T$.

######  Lemma 5.10.

For $a>0$ and $\gamma\in(1,2)$, let $t_{1}(a)$ be as in (5.61). Given any
$T>4(t_{1}+1)$, there exists $T_{g}\in[T/2,3T/4]$ such that

$\begin{split}(i)\quad\sup_{x\geq-\mu_{a}(T_{g})}\varphi_{a}(t_{T},x)\leq\frac{C}{\sqrt{T}},\qquad\text{and}\qquad(ii)\quad\mu_{a}(T_{g})\geq\frac{1}{2}\log(T)-C,\end{split}$
(5.106)

where $C$ does not depend on $a$ or $\gamma$.

Proof. We first note that (5.106)(ii) follows immediately from (5.106)(i).
Indeed, (5.100) yields

$\varphi_{a}(t,-\mu_{a}(t))=e^{-\mu_{a}(t)},$ (5.107)

which, in turn, implies, using (5.106)(i):

$\mu_{a}(T_{g})=-\log\varphi_{a}(t,-\mu_{a}(t))\geq\frac{1}{2}\log T-C,$

which is (5.106)(ii). Thus, we focus on proving (5.106)(i).

To this end, we use time averages in order to find the “good time” $T_{g}$
when the (weighted) $L^{2}$ bound of $\partial_{x}\varphi_{a}$ is small. Since
$T/2\geq 2t_{1}(a)$, Lemma 5.7 yields

$\int\varphi_{a}^{2}(t,x)\rho_{a}(t,x)dx\leq\frac{C}{\sqrt{t}},$ (5.108)

for all $t\geq T/2$. Integrating the dissipation inequality (5.81) and using
(5.108), we obtain

$\frac{4}{T}\int_{T/2}^{3T/4}\int|\partial_{x}\varphi_{a}(s,x)|^{2}\rho_{a}(s,x)dxds\leq\frac{C}{T^{3/2}}.$

As a result, there exists $T_{g}\in[T/2,3T/4]$ such that

$\int|\partial_{x}\varphi_{a}(T_{g},x)|^{2}\rho_{a}(T_{g},x)dx\leq\frac{C}{T^{3/2}}.$
(5.109)

Heuristically, we conclude by arguing that, due to (5.109), if $\varphi_{a}$
is “too big” somewhere to the right of $-\mu_{a}(t)$, then it must be “too
big” on a large set. However, in this region, $\rho_{a}$ is bounded above and
below and $p_{a}$ and $\varphi_{a}$ are comparable. As a result, $p_{a}$ will
be “too big” on a large set, violating mass conservation. The key point here
that makes the above reasoning work is that, by the definitions (5.72),
(5.73), and (5.100) of $\rho_{a}$, $\varphi_{a}$, and $\mu_{a}$, we have

$\frac{1}{2}\leq\rho_{a}(t,x)\leq
1,\quad\text{and}\quad\frac{1}{2}\varphi_{a}(t,x)\leq
p_{a}(t,x)\leq\varphi_{a}(t,x),\qquad\text{ for $x>-\mu_{a}(t)$, }$ (5.110)

In particular, working on $x>-\mu_{a}(t)$ is crucial here as $\rho_{a}$ is
potentially small to the left of $-\mu_{a}(t)$.

To make this reasoning rigorous, we take any $x_{0}\geq-\mu_{a}(T_{g})$, and
use the Newton-Leibniz formula for any $y\in[x_{0},x_{0}+\sqrt{T}]$ along with
(5.109)-(5.110), to obtain a lower bound for $\varphi_{a}(T_{g},y)$:

$\begin{split}\varphi_{a}(T_{g},x_{0})&\leq\varphi_{a}(T_{g},y)+\int_{x_{0}}^{y}|\partial_{x}\varphi_{a}|\,dx\leq\varphi_{a}(T_{g},y)+\int_{x_{0}}^{x_{0}+\sqrt{T}}|\partial_{x}\varphi_{a}|\,dx\\\
&\leq\varphi_{a}(T_{g},y)+\sqrt{2}T^{1/4}\Big{(}\int_{x_{0}}^{x_{0}+\sqrt{T}}|\partial_{x}\varphi_{a}|^{2}\rho_{a}\,dx\Big{)}^{1/2}\leq\varphi_{a}(T_{g},y)+\frac{C}{\sqrt{T}}.\end{split}$

Using again (5.110), this yields

$\inf_{x\in[x_{0},x_{0}+\sqrt{T}]}p_{a}(T_{g},x)\geq\frac{1}{2}\Big{(}p_{a}(T_{g},x_{0})-\frac{C}{\sqrt{T}}\Big{)}.$
(5.111)

Recalling mass conservation (5.16) and using (5.111) we arrive at

$\begin{split}1&=\int
p_{a}(T_{g},x)\,dx\geq\int_{x_{0}}^{x_{0}+\sqrt{T}}p_{a}(T_{g},x)\,dx\\\
&\geq\int_{x_{0}}^{x_{0}+\sqrt{T}}\frac{1}{2}\Big{(}p_{a}(T_{g},x_{0})-\frac{C}{\sqrt{T}}\Big{)}\,dx=\frac{\sqrt{T}}{2}p_{a}(T_{g},x_{0})-\frac{C}{2}.\end{split}$

Rearranging the above and using the arbitrariness of $x_{0}>-\mu_{a}(t)$, as
well as (5.110), to translate this into a bound for $\varphi_{a}(t,x_{0})$,
finishes the proof of Lemma 5.10. $\Box$.

#### Step three: preserving the $L^{\infty}$-smallness over $[T_{g},T]$

We are now in a position to combine the results in the first two steps to
finish the proof Lemma 5.5, establishing the upper bound (5.62) on $p_{a}$.

Proof of Lemma 5.5. Combining (5.110), Lemma 5.10, and (5.56), we find $C>1$
such that

$p_{a}(T_{g},x)\leq\min\Big{(}e^{x},\frac{C}{\sqrt{T}}\Big{)}.$ (5.112)

The function

$P(x)=\frac{e^{x}}{1+\frac{\sqrt{T}}{2C}e^{x}}$ (5.113)

is a steady solution to the equation (5.54) that $p_{a}$ satisfies as

$\partial_{x}(P^{2}e^{-x})=\partial_{x}^{2}P.$ (5.114)

Moreover, it obeys a uniform bound

$\sup_{x}\,P(x)\leq\frac{2C}{\sqrt{T}}.$ (5.115)

The next step is to split $p_{a}$ into a portion bounded above by $P$ and a
small error part. Let us write

$p_{a}(t,x)=\psi_{P}(t,x)+\psi_{E}(t,x)\qquad\text{ for $T_{g}\leq t\leq T$.}$
(5.116)

Here, the “$P$-portion” $\psi_{P}\geq 0$ solves

$\partial_{t}\psi_{P}+\partial_{x}(\psi_{P}^{2}e^{-x})=\partial_{x}^{2}\psi_{P},~{}~{}T_{g}\leq
t\leq T,$ (5.117)

with the initial condition

$\psi_{P}(T_{g},x)=\min\\{p_{a}(T_{g},x),P(x)\\},~{}~{}x\in{\mathbb{R}}.$
(5.118)

The “error part” $\psi_{E}\geq 0$ solves

$\partial_{t}\psi_{E}+\partial_{x}((2\psi_{P}+\psi_{E})\psi_{E}e^{-x})=\partial_{x}^{2}\psi_{E},~{}~{}T_{g}\leq
t\leq T,$ (5.119)

with the initial condition

$\psi_{E}(T_{g},x)=p_{a}(T_{g},x)-\min\\{p_{a}(T_{g},x),P(x)\\}.$ (5.120)

We now estimate $\psi_{P}$ and $\psi_{E}$ at time $T$. From (5.114) and
(5.118), it is clear that

$\psi_{P}(T_{g},x)\leq P(x),~{}~{}\hbox{ for all $x\in{\mathbb{R}}$},$

and both functions solve (5.117). Hence, the comparison principle implies that
$\psi_{P}(T,x)\leq P(x)$ for all $x\in{\mathbb{R}}$. Using this and (5.115),
we find

$\sup_{x}\,\psi_{P}(T,x)\leq\sup_{x}\,P(x)\leq\frac{C}{\sqrt{T}},$ (5.121)

as desired.

On the other hand, (5.119) implies that the total mass of $\psi_{E}$ is
conserved:

$\int\psi_{E}(T,x)dx=\int\psi_{E}(T_{g},x)dx.$ (5.122)

We now bound the right hand side. First, by (5.112) and (5.113),

$p_{a}(T_{g},x)\leq\frac{C}{\sqrt{T}}\leq P(x),\quad\text{for all
}x\geq-\frac{1}{2}\log(T)+\log(2C).$

From this and (5.119), it follows that

$\psi_{E}(T_{g},x)=0,\quad\text{for all }x\geq-\frac{1}{2}\log(T)+\log(2C).$

Hence, using (5.120) and (5.56), we obtain

$\displaystyle\int\psi_{E}(T_{g},x)dx$
$\displaystyle=\int_{-\infty}^{-\frac{1}{2}\log(T)+\log(2C)}\psi_{E}(T_{g},x)dx\leq\int_{-\infty}^{-\frac{1}{2}\log(T)+\log(2C)}p_{a}(T_{g},x)dx$
(5.123)
$\displaystyle\leq\int_{-\infty}^{-\frac{1}{2}\log(T)+\log(2C)}e^{x}dx=\frac{2C}{\sqrt{T}}.$

Invoking (5.122), we obtain

$\int\psi_{E}(T,x)dx\leq\frac{2C}{\sqrt{T}}.$

Recall that $T_{g}\leq 3T/4$ and $T$ is sufficiently large. We may, thus,
apply parabolic regularity theory (recall that $\psi_{E}$ solves (5.119)) to
conclude that, up to increasing $C$, we have

$\sup\psi_{E}(T,x)\leq C\int\psi_{E}(T,x)dx\leq\frac{C}{\sqrt{T}}.$ (5.124)

Combining (5.116), (5.121), and (5.124), finishes the proof of Lemma 5.5.
$\Box$