###  5.5 The weighted $L^{2}$ framework: the proof of Propositions 5.8 and 5.9

First, we establish the dissipation inequality in Proposition 5.8.

Proof of Proposition 5.8. Following [15], let us write an equation for
$h=H(\varphi)$, with a given function $H$, and not just for the cases
$H(\varphi)=\varphi$ and $H(\varphi)=\varphi^{2}$ we use here. Setting

$\mathcal{D}_{\rho}h=\frac{1}{\rho^{2}}\partial_{x}(\rho^{2}\partial_{x}h)-v\partial_{x}h,$
(5.125)

we find

$\displaystyle\partial_{t}h-\mathcal{D}_{\rho}h$
$\displaystyle=H^{\prime}(\varphi)\varphi_{t}-\frac{1}{\rho^{2}}(\rho^{2}H^{\prime}(\varphi)\varphi_{x})_{x}+vH^{\prime}(\varphi)\varphi_{x}$
(5.126)
$\displaystyle=H^{\prime}(\varphi)\Big{(}\frac{q_{t}}{\rho}-\frac{q}{\rho}\frac{\rho_{t}}{\rho}\Big{)}-2\frac{\rho_{x}}{\rho}H^{\prime}(\varphi)\left(\frac{q_{x}}{\rho}-\frac{q}{\rho}\frac{\rho_{x}}{\rho}\right)-H^{\prime\prime}(\varphi)\varphi_{x}^{2}$
$\displaystyle\quad-H^{\prime}(\varphi)\left(\frac{q_{xx}}{\rho}-2\frac{q_{x}\rho_{x}}{\rho^{2}}-\frac{q}{\rho}\frac{\rho_{xx}}{\rho}+2q\frac{\rho_{x}^{2}}{\rho^{3}}\right)+vH^{\prime}(\varphi)\left(\frac{q_{x}}{\rho}-\frac{q\rho_{x}}{\rho^{2}}\right).$

We now use (5.77) to obtain

$\displaystyle\partial_{t}h-\mathcal{D}_{\rho}h$
$\displaystyle=-H^{\prime}(\varphi)\frac{q\rho_{t}}{\rho^{2}}-2\frac{\rho_{x}}{\rho}H^{\prime}(\varphi)\Big{(}\frac{q_{x}}{\rho}-\frac{q}{\rho}\frac{\rho_{x}}{\rho}\Big{)}-H^{\prime\prime}(\varphi)\varphi_{x}^{2}$
(5.127)
$\displaystyle\quad+H^{\prime}(\varphi)\Big{(}2\frac{q_{x}\rho_{x}}{\rho^{2}}+\frac{q}{\rho}\frac{\rho_{xx}}{\rho}-2q\frac{\rho_{x}^{2}}{\rho^{3}}\Big{)}+vH^{\prime}(\varphi)\Big{(}-\frac{q\rho_{x}}{\rho^{2}}\Big{)}-v_{x}H^{\prime}(\varphi)\frac{q}{\rho}$
$\displaystyle=-H^{\prime\prime}(\varphi)\varphi_{x}^{2}-H^{\prime}(\varphi)\frac{q}{\rho^{2}}(\rho_{t}-\rho_{xx}+(v\rho)_{x}).$

When $H(\varphi)=\varphi$, this establishes (5.80).

Let us now specialize to the case $h=H(\varphi)=\varphi^{2}$ to prove (5.81).
Multiplying (5.127) by $\rho$ and integrating by parts, we find

$\begin{split}\partial_{t}\int\rho\varphi^{2}&=\partial_{t}\int\rho
h=\int\rho_{t}h+\int\rho\Big{(}\frac{1}{\rho^{2}}(\rho^{2}h_{x})_{x}-vh_{x}-2\varphi_{x}^{2}\Big{)}-2\int
h(\rho_{t}-\rho_{xx}+(u\rho)_{x})\\\
&=\int\rho_{t}h+\int\rho_{x}h_{x}+\int(\rho v)_{x}h-\int
2\rho\varphi_{x}^{2}-2\int h(\rho_{t}-\rho_{xx}+(u\rho)_{x})\\\ &=\int
h(\rho_{t}-\rho_{xx}+(\rho v)_{x})-\int 2\rho\varphi_{x}^{2}-2\int
h(\rho_{t}-\rho_{xx}+(v\rho)_{x})\\\ &=-\int 2\rho\varphi_{x}^{2}-\int
h(\rho_{t}-\rho_{xx}+(v\rho)_{x})\leq-\int 2\rho\varphi_{x}^{2},\end{split}$
(5.128)

finishing the proof. $\Box$

Next, we prove the analogue of Nash’s inequality. Due to the inhomogeneity, we
are not able to use the standard Fourier-based proof. The proof is more
similar to that of Carlen and Loss [13], which is based on the Poincaré
inequality. A complication in our setting is that the decay of $r$ on the left
makes our space more akin to $\mathbb{R}_{+}$ than $\mathbb{R}$; however, we
only have a mild “boundary condition” on the left in that we only know that
$\varphi^{2}r$ is integrable.

Proof of Proposition 5.9. Let us first assume that $\varphi$ is not only non-
negative, bounded, and decays to zero sufficiently fast as $x\to+\infty$ but
also monotonically decreasing. Fix $\theta\in{\mathbb{R}}_{+}$ and notice
that, due to the assumptions on $r$, we may find $L\in{\mathbb{R}}$ such that
$\bar{r}(L)=\theta$. Fix $\varepsilon>0$ to be chosen, and write

$\begin{split}0&\leq\int_{-\infty}^{L}\Big{|}\varphi_{x}\sqrt{\overline{\overline{r}}}+\varepsilon\varphi\frac{\overline{r}}{\sqrt{\overline{\overline{r}}}}\Big{|}^{2}dx=\int_{-\infty}^{L}\Big{[}|\varphi_{x}|^{2}\overline{\overline{r}}+\varepsilon(\varphi^{2})_{x}\overline{r}+\varepsilon^{2}\varphi^{2}\frac{\overline{r}^{2}}{\overline{\overline{r}}}\Big{]}dx\\\
&=\int_{-\infty}^{L}\Big{[}|\varphi_{x}|^{2}\overline{\overline{r}}-\varepsilon\varphi^{2}r+\varepsilon^{2}\varphi^{2}\frac{\overline{r}^{2}}{\overline{\overline{r}}}\Big{]}dx+\varepsilon\overline{r}(L)\varphi^{2}(L).\end{split}$
(5.129)

Note that, since $\varphi(x)$ is decreasing,

$\varphi(L)\overline{r}(L)\leq\int_{-\infty}^{L}\varphi
r\,dx\leq\int_{-\infty}^{\infty}\varphi r\,dx.$ (5.130)

Rearranging (5.129) and using (5.130), we find

$\varepsilon\int_{-\infty}^{\infty}\varphi^{2}r\Big{(}1-\varepsilon\frac{\overline{r}^{2}}{r\overline{\overline{r}}}\Big{)}\,dx\leq\int_{-\infty}^{L}|\varphi_{x}|^{2}\overline{\overline{r}}\,dx+\frac{\varepsilon}{\overline{r}(L)}\Big{(}\int_{-\infty}^{\infty}\varphi
r\,dx\Big{)}^{2}.$ (5.131)

Since $r$ is increasing, we have

$\overline{r}^{2}(x)=2\int_{-\infty}^{x}\bar{r}(y)\bar{r}_{y}(y)\,dy=2\int_{-\infty}^{x}\bar{r}(y)r(y)\,dy\leq
2r(x)\int_{-\infty}^{x}\bar{r}(y)dy=2r(x)\overline{\overline{r}}(x).$ (5.132)

Hence, using (5.132) and choosing $\varepsilon=1/4$, (5.131) becomes

$\displaystyle\int_{-\infty}^{\infty}\varphi^{2}r\,dx$ $\displaystyle\leq
8\int_{-\infty}^{L}|\varphi_{x}|^{2}\overline{\overline{r}}\,dx+\frac{2}{\overline{r}(L)}\Big{(}\int_{-\infty}^{\infty}\varphi
r\,dx\Big{)}^{2}.$ (5.133)

Next, using assumption (ii) in (5.82), together with monotonicity of
$\bar{r}(x)$ gives

$\displaystyle\int_{-\infty}^{\infty}\varphi^{2}r\,dx$ $\displaystyle\leq
8C_{1}\max(1,\bar{r}(L)^{2})\int_{-\infty}^{\infty}|\varphi_{x}|^{2}r\,dx+\frac{2}{\overline{r}(L)}\Big{(}\int_{-\infty}^{\infty}\varphi
r\,dx\Big{)}^{2},$ (5.134)

proving (5.84) for monotonically decreasing functions $\varphi(x)$ as
$\bar{r}(L)=\theta$.

In order to complete the proof for non-monotonically decreasing non-negative
functions $\varphi(x)$, we define an analogue of the decreasing rearrangement
as follows. Let $m_{r}$ be the measure

$m_{r}(A)=\int_{A}r\,dx,$ (5.135)

and for any measurable set $A$ define

$A^{*}=\\{x:\bar{r}(x)<m_{r}(A)\\}.$

Note that $A^{*}$ is a half line of the form $A^{*}=(-\infty,x^{*}(A))$ such
that

$m_{r}(A)=\bar{r}(x^{*}(A))=m_{r}(A^{*}).$ (5.136)

We may then define rearrangements of functions through the “layer cake
decomposition”:

$\varphi^{*}(x)=\int_{0}^{\infty}\mathds{1}_{\\{x:\varphi(x)>t\\}^{*}}(t)dt.$

As with the usual symmetric decreasing rearrangement, we immediately find that
$\varphi^{*}$ is decreasing, the functions $\varphi$ and $\varphi^{*}$ have
super-level sets of equal measure, and, for any $p>0$,

$\int|\varphi^{*}(x)|^{p}r(x)dx=\int|\varphi(x)|^{p}r(x)dx.$ (5.137)

We claim that an analogue of the Pólya-Szegö inequality holds:

$\int|\varphi^{*}_{x}(x)|^{2}r(x)dx\leq\int|\varphi_{x}(x)|^{2}r(x)dx.$
(5.138)

Since we have already established (5.84) for decreasing functions, the fact
that (5.84) holds for a general function $\varphi$ is an immediate consequence
of (5.137) and (5.138).

We now prove (5.138). In principle, it is an immediate consequence of a more
general result in [46]. We present a simpler proof in our present one-
dimensional setting for the convenience of the reader. We may assume without
loss of generality that the function $\varphi(x)$ is positive everywhere,
smooth and takes each value finitely many times. A useful consequence is that
any level set of $\varphi^{*}$ has at most one point.

First, we use the coarea formula to rewrite

$\int\varphi_{x}(x)^{2}r(x)\,dx=\int_{0}^{\infty}\sum_{x\in\varphi^{-1}(t)}|\varphi_{x}(x)|r(x)\,dt.$
(5.139)

Then, we note that

$\sum_{x\in\varphi^{-1}(t)}r(x)\leq\Big{(}\sum_{x\in\varphi^{-1}(t)}r(x)|\varphi_{x}(x)|\Big{)}^{1/2}\Big{(}\sum_{x\in\varphi^{-1}(t)}\frac{r(x)}{|\varphi_{x}(x)|}\Big{)}^{1/2},$
(5.140)

so that (5.139) becomes

$\int\varphi_{x}(x)^{2}r(x)\,dx\geq\int_{0}^{\infty}\frac{\big{(}\sum_{x\in\varphi^{-1}(t)}r(x)\big{)}^{2}}{\sum_{x\in\varphi^{-1}(t)}\frac{r(x)}{|\varphi_{x}(x)|}}dt.$
(5.141)

Next, we show that both the numerator and denominator in the right hand side
of (5.141) may be replaced by their analogues with $\varphi^{*}$ in place of
$\varphi$, up to an inequality.

First, since

$\overline{r}((\varphi^{*})^{-1}(t))=m_{r}(\\{\varphi^{*}>t\\})=m_{r}(\\{\varphi>t\\})\leq\overline{r}(\sup\varphi^{-1}(t)),$

we conclude that $(\varphi^{*})^{-1}(t)\leq\sup\varphi^{-1}(t)$. Using that
$r$ is increasing, we conclude that

$\sum_{x\in\varphi^{-1}(t)}r(x)\geq r(\sup\varphi^{-1}(t))\geq
r((\varphi^{*})^{-1}(t))$ (5.142)

On the other hand, by the coarea formula, we have that, for any $t>0$

$m_{r}(\\{x:\varphi(x)>t\\})=\int_{\\{x:\varphi(x)>t\\}}r(x)dx=\int_{t}^{\infty}\Big{(}\sum_{x\in\varphi^{-1}(s)}\frac{r(x)}{|\varphi_{x}(x)|}\Big{)}ds.$

An immediate consequence is that

$\frac{d}{dt}m_{r}(\\{x:\varphi(x)>t\\})=-\sum_{x\in\varphi^{-1}(t)}\frac{r(x)}{|\varphi_{x}(x)|}.$

The analogous formula for $\varphi^{*}$ holds. Since
$m_{r}(\\{x:\varphi(x)>t\\})=m_{r}(\\{x:\varphi^{*}(x)>t\\})$ for all $t$, it
follows that

$\sum_{x\in\varphi^{-1}(t)}\frac{r(x)}{|\varphi_{x}(x)|}=\sum_{x\in(\varphi^{*})^{-1}(t)}\frac{r(x)}{|\varphi^{*}_{x}(x)|}=\frac{r((\varphi^{*})^{-1}(t))}{\varphi^{*}_{x}((\varphi^{*})^{-1}(t))}.$
(5.143)

The last equality above is due to the fact that $(\varphi^{*})^{-1}(t)$ is a
one point set.

Including (5.142) and (5.143) in (5.141), we conclude that

$\int\varphi_{x}(x)^{2}r(x)\,dx\geq\int_{0}^{\infty}\frac{r((\varphi^{*})^{-1}(t))^{2}}{\frac{r((\varphi^{*})^{-1}(t))}{\varphi^{*}_{x}((\varphi^{*})^{-1}(t))}}dt=\int_{0}^{\infty}\varphi^{*}_{x}((\varphi^{*})^{-1}(t))r((\varphi^{*})^{-1}(t))dt.$

Reapplying the coarea formula to the rightmost quantity above yields

$\int\varphi_{x}(x)^{2}r(x)\,dx\geq\int\varphi^{*}_{x}(x)^{2}r(x)\,dx.$

which concludes the proof of (5.138) and, thus, of Proposition 5.9. $\Box$

###  5.6 A lower bound on $p(t,x)$ in the middle region

We finish this section with a lower bound on $p(t,x)$ in an intermediate
region that will be useful to us later on.

######  Lemma 5.11.

Let $u(t,x)$ be the solution to the Burgers-FKPP equation (1.1) with the
initial condition $u(0,x)=\mathbbm{1}(x\leq 0)$, and $p(t,x)$ be given by
(5.11) and (5.13). There exist $k>0$, $T_{0}>0$ and $c_{0}>0$ so that we have

$p(t,x)\geq\frac{c_{0}}{\sqrt{t}},~{}~{}\hbox{ for all $t\geq T_{0}$ and
$\displaystyle-\frac{1}{2}\log t\leq x\leq k\sqrt{t}$.}$ (5.144)

Proof. Let us take $t>0$ sufficiently large. We argue by contradiction. For
$\varepsilon\in(0,1/10)$ and $k>0$ to be chosen, suppose there exists

$x_{0}\in\big{(}-\frac{1}{2}\log t,k\sqrt{t}\big{)}$ (5.145)

such that

$p(t,x_{0})\leq\frac{\varepsilon}{\sqrt{t}},$ (5.146)

whence

$\widehat{u}(t,x_{0})=e^{-x_{0}}p(t,x_{0})\leq\frac{\varepsilon}{\sqrt{t}}e^{-x_{0}}.$
(5.147)

We will show that this violates mass conservation (5.16):

$\int p(t,x)dx=\int p(0,x)dx,$ (5.148)

if $\varepsilon$ and $k$ are chosen sufficiently small.

The first step is to notice that the mass to the left of $(-1/2)\log t$ is
small. Indeed, due to the upper bound (5.19):

$\int_{-\infty}^{-(1/2)\log t}p(t,x)dx\leq\int_{-\infty}^{-(1/2)\log
t}e^{x}dx=\frac{1}{\sqrt{t}}.$ (5.149)

We also know that the mass to the far right is small. Indeed, recall from
(5.35) that there exist $T_{0}$ and $N_{0}$ so that for all $t>T_{0}$ and
$N>N_{0}$, we have

$\int_{N\sqrt{t}}^{\infty}p(t,x)dx\leq Ce^{-N/2}.$ (5.150)

To estimate the mass of $p$ in the middle region, we split it into two parts.
When $x\in[k\sqrt{t},N\sqrt{t}]$, we use (5.147) together with the steepness
estimate in Lemma 5.3 to find

$\displaystyle p(t,x)$ $\displaystyle=e^{x}\widehat{u}(t,x)\leq
e^{x}\Big{(}1+\Big{(}\frac{1}{\widehat{u}(t,x_{0})}-1\Big{)}e^{x-x_{0}}\Big{)}^{-1}\leq
e^{x}\Big{(}1+\Big{(}\frac{\sqrt{t}}{\varepsilon}e^{x_{0}}-1\Big{)}e^{x-x_{0}}\Big{)}^{-1}$
(5.151) $\displaystyle\leq
e^{x}\Big{(}1+\frac{\sqrt{t}}{2\varepsilon}e^{x_{0}}e^{x-x_{0}}\Big{)}^{-1}\leq\frac{2\varepsilon}{\sqrt{t}},~{}~{}\text{for}~{}~{}x>x_{0}.$

We used (5.145) in the second inequality above. It follows that

$\int_{k\sqrt{t}}^{N\sqrt{t}}p(t,x)dx\leq
N\sqrt{t}\frac{2\varepsilon}{\sqrt{t}}=2\varepsilon N.$ (5.152)

On the other hand, when $x\in[-(1/2)\log t,k\sqrt{t}]$, we use Lemma 5.4 to
find

$\int_{-(1/2)\log t}^{k\sqrt{t}}p(t,x)dx\leq
2k\sqrt{t}\frac{C}{\sqrt{t}}=2kC.$ (5.153)

Putting together (5.149), (5.150), (5.152), and (5.153) yields

$\int p(t,x)dx\leq\frac{1}{\sqrt{t}}+Ce^{-N/2}+2\varepsilon N+2kC.$

Taking $\varepsilon$ and $k$ sufficiently small, and $N$ sufficiently large we
obtain a contradiction to (5.148). The conclusion of Lemma 5.11 follows.
$\Box$

##  6 The proof of Theorem 1.1 for $\beta=2$

We now prove Theorem 1.1 in the critical case $\beta=2$. As in the case
$\beta<2$ considered in Section 4, the proof is based on the analysis in the
self-similar variables, and using the pulled nature of the problem to show
that convergence on the diffusive scales implies convergence to a traveling
wave on scales $x\sim O(1)$. The key difference with the situation for
$\beta<2$ is that, as we have mentioned previously, the Dirichlet boundary
condition at $\eta=0$ for the function $v$, introduced by the weighted Hopf-
Cole transform (3.12), no longer approximately holds in the self-similar
variables. Instead, the function $v$ has a positive but a priori unknown limit
on the left: see Lemma 6.1 below. The bounds in Proposition 5.1 will be a
crucial ingredient in establishing the correct boundary condition in the self-
similar variables. After we pass to the self-similar variables, the non-zero
boundary condition changes the long time behavior in the self-similar
variables. This is an algebraic reason for adjusting the logarithmic shift
$(3/2)\log t$ in the front position to $(1/2)\log t$. With these bounds in
hand, the argument has many similarities with the case $\beta<2$, so we will
omit many of the details, only highlighting the differences. As before, we
will assume without loss of generality that the initial condition is
$u(0,x)=\mathbbm{1}(x\leq 0)$.

#### The weighted Hopf-Cole transform

Motivated by Proposition 5.1, we consider the moving frame: $x\mapsto
x-2t+(1/2)\log(t+1)$, and set

$\tilde{u}(t,x)=u(t,x+2t-\frac{1}{2}\log(t+1)).$ (6.1)

Here, $u(t,x)$ is the solution to the Burgers-FKPP equation (1.1), in the non-
shifted reference frame. We stress the difference between the function
$\widehat{u}(t,x)$ defined in (5.11) and used throughout Section 5, and the
function $\tilde{u}(t,x)$ used in the present section. The former is $u(t,x)$
in the reference frame $x\to x-2t$, while the latter refers to the solution in
the reference frame used in (6.1). In particular, the function $p(t,x)$ used
throughout Section 5 and $\tilde{u}(t,x)$ are related by

$p(t,x)=e^{x}\widehat{u}(t,x)=e^{x}u(t,x+2t)=e^{x}\tilde{u}\Big{(}t,x+\frac{1}{2}\log(t+1)\Big{)}.$
(6.2)

The function $\tilde{u}(t,x)$ satisfies

$\displaystyle\tilde{u}_{t}-2\tilde{u}_{x}+\frac{1}{2(t+1)}\tilde{u}_{x}+2\tilde{u}\tilde{u}_{x}=\tilde{u}_{xx}+\tilde{u}(1-\tilde{u}).$
(6.3)

We will use the weighted Hopf-Cole transform, as in (3.12) with $\beta=2$:

$\displaystyle
v(t,x)=\exp\Big{(}x+\int_{x}^{\infty}\tilde{u}(t,y)dy\Big{)}\tilde{u}(t,x).$
(6.4)

This gives

$\displaystyle
v_{t}-v_{xx}+\frac{1}{2(t+1)}(v_{x}-v)=v\Big{(}\int_{x}^{\infty}\tilde{u}(1-\tilde{u})dy-\tilde{u}\Big{)}.$
(6.5)

The key step in the proof will again be to analyze the long time behavior of
$v$ at scales $O((t+1)^{\gamma})$ for $\gamma\in(0,1/2)$, and this will be
done through the use of self-similar variables. However, the key difference
with the case $\beta<2$ is that $v$ does not decay to zero as $x\to-\infty$
(cf. (4.13)). This is quantified in the following:

######  Lemma 6.1.

There exist $T_{0}$, $k_{0}$, $b_{0}$, $b_{1}>0$ so that

$v(t,x)\leq b_{0}<+\infty,~{}~{}\hbox{ for all $t\geq T_{0}$ and any $x$},$
(6.6)

and

$0<b_{1}\leq v(t,x),~{}~{}\hbox{ for all $x\leq k_{0}\sqrt{t}$ and $t\geq
T_{0}$.}$ (6.7)

Proof. Let us define $\tilde{\mu}(t)$ by

$\tilde{u}(t,\tilde{\mu}(t))=\frac{1}{2}.$ (6.8)

Proposition 5.1 in terms of $\tilde{\mu}(t)$ says that there exists a constant
$K>0$ and $T_{0}>0$ so that

$|\tilde{\mu}(t)|\leq K.$ (6.9)

Lemma 5.3 and (6.9) imply that

$\tilde{u}(t,x)\leq\frac{1}{1+e^{x-K}},~{}~{}\hbox{ for $t\geq T_{0}$ and
$x>K$,}$ (6.10)

and

$\tilde{u}(t,x)\geq\frac{1}{1+e^{x+K}},~{}~{}x<-K,~{}~{}t\geq T_{0}.$ (6.11)

Hence, for $x\leq 0$ we can estimate $v(t,x)$ from above by

$\displaystyle v(t,x)$
$\displaystyle=\tilde{u}(t,x)\exp\Big{(}x+\int_{x}^{\infty}\tilde{u}(t,y)dy\Big{)}\leq\exp\Big{(}\int_{x}^{0}(\tilde{u}(t,y)-1)dy+\int_{0}^{\infty}\tilde{u}(t,y)dy\Big{)}$
(6.12)
$\displaystyle\leq\exp\Big{(}\int_{0}^{\infty}\tilde{u}(t,y)dy\Big{)}\leq\exp\Big{(}K+\int_{K}^{\infty}\frac{dx}{1+e^{x-K}}\Big{)}\leq
C,$

which is (6.6). The proof for $x\geq 0$ is simpler and follows directly from
(6.10).

To obtain (6.7), we first use (6.11) to write for $x\leq-K$:

$\displaystyle v(t,x)$
$\displaystyle=\tilde{u}(t,x)\exp\Big{(}x+\int_{x}^{\infty}\tilde{u}(t,y)dy\Big{)}\geq\frac{1}{2}\exp\Big{(}\int_{x}^{0}(\tilde{u}(t,y)-1)dy+\int_{0}^{\infty}\tilde{u}(t,y)dy\Big{)}$
(6.13)
$\displaystyle\geq\frac{1}{2}\exp\Big{(}-K-\int_{-\infty}^{-K}\Big{(}1-\frac{1}{1+e^{x+K}}\Big{)}dy\Big{)}\geq
C.$

In order to extend this lower bound on the right, we recall the lower bound in
Lemma 5.11. Combined with the change of variables (6.2), it shows that we can
find $c_{0}>0$ and $k_{0}>0$ so that for $0\leq x\leq k_{0}\sqrt{t}$ and
$t\geq T_{0}$ we have

$\tilde{u}(t,x)=\sqrt{t+1}e^{-x}p(t,x-\frac{1}{2}\log(t+1))\geq c_{0}e^{-x}.$
(6.14)

We deduce that

$\displaystyle v(t,x)$
$\displaystyle=\tilde{u}(t,x)\exp\Big{(}x+\int_{x}^{\infty}\tilde{u}(t,y)dy\Big{)}\geq
c_{0},~{}~{}\hbox{ for all $t\geq T_{0}$ and $0\leq x\leq k_{0}\sqrt{t}$.}$
(6.15)

Finally, for $x\in(-K,0)$, we see from the monotonicity of $\tilde{u}(t,x)$
and (6.14) that $\tilde{u}(t,x)\geq c_{0}$, so that

$v(t,x)=\tilde{u}(t,x)\exp\Big{(}x+\int_{x}^{\infty}\tilde{u}(t,y)dy\Big{)}\geq
c_{0}e^{-K},~{}~{}\hbox{ for all $t\geq T_{0}$.}$

This proves (6.7). $\Box$

#### Analysis in the self-similar variables

We now outline the main ingredients required to establish the long-time
behavior of $v$. The thrust of the argument is similar to the case $\beta<2$,
so the outline will be made in reference to the ideas used in Section 4.

First, apply the familiar self-similar change of variables:

$v(t,x)=w\Big{(}\log(t+1),\frac{x}{\sqrt{t+1}}\Big{)}.$ (6.16)

Then, (6.5) leads to the evolution equation

$\displaystyle
w_{\tau}+\mathcal{L}w+\frac{1}{2}e^{-\tau/2}w_{\eta}=e^{\tau}w(\tau,\eta)\Big{(}\int_{\eta
e^{\tau/2}}^{\infty}\tilde{u}(1-\tilde{u})dy-\tilde{u}\Big{)},$ (6.17)

with the operator ${\cal L}$ defined by

$\displaystyle\mathcal{L}w:=-w_{\eta\eta}-\frac{\eta}{2}w_{\eta}-\frac{1}{2}w.$
(6.18)

This is different from the operator $L$ in (4.4) by a multiple of the
identity. Considering $\mathcal{L}$ as an operator on
$H^{1}(e^{\eta^{2}/4}d\eta;\mathbb{R}_{+})$ augmented with Neumann boundary
conditions, its spectrum consists of the eigenvalues $0,1,2,\dots$ with
(unnormalized) principal eigenfunction

$\psi_{0}(\eta)=e^{-\eta^{2}/4}.$

Importantly, Lemma 3.2 shows that the right side of (6.17) is negative:

$\int_{x}^{\infty}\tilde{u}(1-\tilde{u})dy-\tilde{u}\leq 0~{}~{}\hbox{ for all
$x\in\mathbb{R}$.}$ (6.19)

Recall the major ingredients in establishing the long-time dynamics in self-
similar variables when $\beta<2$: (1) a super-solution solving a tractable
equation; (2) a sub-solution solving a (potentially different) tractable
equation; (3) smallness of the right hand side when $\eta\gg e^{-\tau/2}$; (4)
an approximate boundary condition when $\eta\ll-e^{-\tau/2}$. We now check
that analogous ingredients are available in this case.

First, we find a super-solution. This follows directly from Lemma 3.2. Indeed,
as a result, we have that any solution to

$\displaystyle\bar{w}_{\tau}+\mathcal{L}\bar{w}+\frac{1}{2}e^{-\tau/2}\bar{w}_{\eta}=0,~{}~{}\tau>0,~{}~{}\eta>-e^{-(1/2-\gamma)\tau},$
(6.20)

augmented with appropriate boundary conditions on
$\bar{w}_{\eta}(\tau,-e^{-(1/2-\gamma)\tau})$ (see the fourth point, below),
is a super-solution to (6.17).

Second, we see that the solution $\underline{w}$ to

$\underline{w}_{\tau}+\mathcal{L}\underline{w}+\frac{1}{2}e^{-\tau/2}\underline{w}_{\eta}=-e^{\tau}\underline{w}\tilde{u},~{}~{}\tau>0,~{}~{}\eta>e^{-(1/2-\gamma)\tau},$
(6.21)

is a sub-solution to (6.17), up to stating an appropriate boundary condition
that we do now. Due to the fact that $\tilde{u}$ is steeper than $\phi_{2}$,
(3.24), (3.15), and Lemma 3.2, notice that

$w_{\eta}(\tau,\eta)=(\tilde{u}_{x}+\tilde{u}(1-\tilde{u}))\exp\Big{\\{}\tau/2+x+\int_{x}^{\infty}\tilde{u}dy\Big{\\}}\leq
0.$ (6.22)

Hence, we may take Neumann boundary conditions for $\underline{w}_{\eta}$:

$\underline{w}_{\eta}(\tau,e^{-(1/2-\gamma)\tau})=0$ (6.23)

and we are guaranteed that, with ordered initial data, $\underline{w}\leq w$.

Third, we establish the smallness of the right hand side of (6.17) whenever
$\eta\geq e^{-(1/2-\gamma)\tau}$. This follows directly from (6.19), Lemma
6.1, and (6.10):

$\begin{split}0&\geq e^{\tau}\Big{(}\int_{\eta
e^{\tau/2}}^{\infty}\tilde{u}(1-\tilde{u})dy-\tilde{u}(e^{\tau},\eta
e^{\tau/2})\Big{)}w(\tau,\eta)\\\ &\geq-e^{\tau}\tilde{u}(e^{\tau},\eta
e^{\tau/2})w(\tau,\eta)\geq-Ce^{\tau-\eta e^{\tau/2}}\geq-
Ce^{\tau-e^{\gamma\tau}}.\end{split}$ (6.24)

Hence, the right hand side of (6.17) is double exponentially small.

Finally, we address the approximate boundary conditions of $\omega$ at
$\eta=-e^{-(1/2-\gamma)\tau}$. Keeping Lemma 6.1 in mind, an approximate
Dirichlet boundary condition is not possible. This is, of course, part of the
reason for the different shift when $\beta=2$. Instead, we have an approximate
Neumann boundary condition. First notice that, due to (6.11), we have that

$|1-\tilde{u}(e^{\tau},-e^{\gamma\tau})|\leq Ce^{-e^{\gamma\tau}}.$ (6.25)

By parabolic regularity theory, we find

$|\tilde{u}_{x}(e^{\tau},-e^{\gamma\tau})|\leq Ce^{-e^{\gamma\tau}}.$ (6.26)

Using (6.25) and (6.26) in (6.22), we find

$|w_{\eta}(\tau,-e^{-(1/2-\gamma)\tau})|\leq Ce^{\tau-e^{\gamma\tau}}.$ (6.27)

Hence, $w$ satisfies an approximate Neumann boundary condition with double
exponentially small error.

Thus, the main ingredients to an analogous argument as in the case $\beta<2$
are in place, with the only major difference being the change from approximate
Dirichlet boundary conditions to approximate Neumann boundary conditions. The
change in boundary conditions changes the principal eigenfunction of
$\mathcal{L}$ and suggests that the long-time behavior of $w$ should look like
$\alpha_{\infty}e^{-\eta^{2}/4}$ for $\eta\geq 0$. This is confirmed by the
following key estimate, which is the analogue of Lemma 4.2. As the proof
follows from similar arguments as in Lemma 4.2, using the four ingredients
above and our knowledge of the spectrum of $\mathcal{L}$, we omit the details.

######  Lemma 6.2.

Given $w$ solving (6.17) with initial conditions
$w(0,\eta)=\mathds{1}(\eta\leq 0)$, there exists a constant
$\alpha_{\infty}>0$ and a function $R_{\gamma}$ such that

$w(\tau,\eta)=\alpha_{\infty}e^{-\eta^{2}/4}+R_{\gamma}(\tau,\eta)e^{-\eta^{2}/6}\qquad\text{
for }\eta\geq-e^{-(1/2-\gamma)\tau},$ (6.28)

for any $\gamma\in(0,1/2)$ and

$\lim_{\tau\to\infty}|R_{\gamma}(\tau,\eta)|=0\qquad\text{ uniformly for
$\eta\geq-e^{-(1/2-\gamma)\tau}$}.$ (6.29)

Before proceeding, we note that the role of Lemma 6.1 in the proof of Lemma
6.2 is to guarantee the positivity of $\alpha_{\infty}$ and the boundedness of
$w$. From Lemma 6.2, we immediately obtain the long-time behavior of
$\tilde{u}$ at scales between $O(1)$ and $O(\sqrt{t})$.

######  Corollary 6.3.

Let $\gamma\in(0,1/2)$, fix $\varepsilon>0$, and set
$x_{\gamma}=(t+1)^{\gamma}$. There exist $\alpha_{\infty}>0$ and
$T_{\varepsilon,\gamma}$, so that

$\displaystyle|\tilde{u}(t,x_{\gamma})$
$\displaystyle-\alpha_{\infty}e^{-x_{\gamma}}|\leq\varepsilon
e^{-x_{\gamma}},\quad\text{ for all }t>T_{\varepsilon,\gamma}.$ (6.30)

#### Convergence to a single wave: the proof of Theorem 1.1 for $\beta=2$

With Lemma 6.2 and 6.3 in hand, the rest of the proof is quite
straightforward; however, it deviates somewhat from that of the proof for
$\beta<2$ in Section 4. The main reason being that we cannot verify that an
analogously defined $v_{\alpha}$ satisfies a small boundary condition at
$x=-(t+1)^{\gamma}$ (cf. (4.40)).

As in (4.18), we take

$\varphi_{\alpha}(t,x)=\phi_{2}(x+\zeta_{\alpha}(t)),$ (6.31)

and perform the corresponding weighted Hopf-Cole transforms:

$\psi_{\alpha}(t,x)=e^{\Gamma_{\alpha}(t,x)}\varphi_{\alpha}(t,x),\qquad\text{
where }\Gamma_{\alpha}(t,x)=x+\int_{x}^{\infty}\varphi_{\alpha}(t,y)dy.$
(6.32)

Similarly to (4.23), we define a shift of the traveling wave, fixing
$\zeta_{\alpha}(t)$ by the normalization

$\psi_{\alpha}(t,(t+1)^{\gamma})=\alpha.$ (6.33)

Recalling that $\phi_{2}(x)=1/(1+e^{x})$, we can easily compute that

$\int_{x}^{\infty}\varphi_{\alpha}(y)dy=\int_{x+\zeta_{\alpha}}^{\infty}\frac{1}{1+e^{y}}dy=-(x+\zeta_{\alpha})+\log(1+e^{x+\zeta_{\alpha}}),$
(6.34)

so that

$\alpha=\psi_{\alpha}(t,(t+1)^{\gamma})=e^{-\zeta_{\alpha}},\qquad\text{ or,
equivalently}\quad\zeta_{\alpha}(t)=-\log(\alpha).$ (6.35)

Fix $\varepsilon>0$. As in the proof of 1.1 when $\beta<2$, it is enough to
obtain precise upper bounds on
$\varphi_{\alpha_{\infty}-\varepsilon}-\tilde{u}$ and
$\tilde{u}-\varphi_{\alpha_{\infty}+\varepsilon}$ because
$|\varphi_{\alpha_{\infty}}-\varphi_{\alpha_{\infty}\pm\varepsilon}|\leq
C\varepsilon$. We do this now.

First consider $\varphi_{\alpha_{\infty}-\varepsilon}-\tilde{u}$. In exactly
the same manner as we established (4.54), we see that

$\varphi_{\alpha_{\infty}-\varepsilon}-\tilde{u}<0,~{}~{}\text{ for all $t$
sufficiently large and $x<(t+1)^{\gamma}$}.$ (6.36)

Here we used 6.3.

We now turn to $\tilde{u}-\varphi_{\alpha_{\infty}+\varepsilon}$. We claim
that

$\tilde{u}-\varphi_{\alpha_{\infty}+\varepsilon}<C\varepsilon,~{}~{}\text{ for
all $t$ sufficiently large and $|x|<(t+1)^{\gamma}$},$ (6.37)

which is enough to conclude the proof. To this end, we apply Lemma 6.2 to take
$t$ sufficiently large, so that $v\leq\alpha_{\infty}+\varepsilon$ on
$|x|<(t+1)^{\gamma}$. Then, rewriting $\tilde{u}$ and using, (6.34), (6.35),
and (6.36), we find, on $|x|<(t+1)^{\gamma}$,

$\begin{split}\tilde{u}(t,x)&=v(t,x)\exp\Big{\\{}-x-\int_{x}^{\infty}\tilde{u}dy\Big{\\}}\leq(\alpha_{\infty}+\varepsilon)\exp\Big{\\{}-x-\int_{x}^{\infty}\tilde{u}dy\Big{\\}}\\\
&\leq(\alpha_{\infty}+\varepsilon)\exp\Big{\\{}-x-\int_{x}^{\infty}\varphi_{\alpha_{\infty}-\varepsilon}dy+\int_{(t+1)^{\gamma}}^{\infty}\varphi_{\alpha_{\infty}-\varepsilon}dy\Big{\\}}\\\
&=(\alpha_{\infty}+\varepsilon)\frac{e^{\zeta_{\alpha_{\infty}-\varepsilon}}}{1+e^{x+\zeta_{\alpha_{\infty}-\varepsilon}}}e^{Ce^{-(t+1)^{\gamma}}}\\\
&\leq(\alpha_{\infty}+\varepsilon)\frac{e^{\zeta_{\alpha_{\infty}-\varepsilon}}}{1+e^{x+\zeta_{\alpha_{\infty}-\varepsilon}}}+Ce^{-(t+1)^{\gamma}}=(\alpha_{\infty}+\varepsilon)\frac{\frac{1}{\alpha_{\infty}-\varepsilon}}{1+e^{x}\frac{1}{\alpha_{\infty}-\varepsilon}}+Ce^{-(t+1)^{\gamma}}\\\
&=\frac{1}{\frac{\alpha_{\infty}-\varepsilon}{\alpha_{\infty}+\varepsilon}+\frac{1}{\alpha_{\infty}+\varepsilon}e^{x}}+Ce^{-(t+1)^{\gamma}}\leq\frac{1}{\frac{1}{\alpha_{\infty}+\varepsilon}e^{x}+1-\frac{2\varepsilon}{\alpha_{\infty}}}+Ce^{-(t+1)^{\gamma}}\\\
&\leq\frac{1}{\frac{1}{\alpha_{\infty}+\varepsilon}e^{x}+1}+C\varepsilon+Ce^{-(t+1)^{\gamma}}.\end{split}$
(6.38)

Using (6.35), it is easy to see that

$\varphi_{\alpha_{\infty}+\varepsilon}(x)=\frac{1}{1+e^{x+\zeta_{\alpha_{\infty}+\varepsilon}}}=\frac{1}{1+\frac{1}{\alpha_{\infty}+\varepsilon}e^{x}}.$

Using this along with (6.38) yields (6.37), concluding the proof of 1.1 when
$\beta=2$. $\Box$

##  7 Convergence to pushed fronts for $\beta>2$

In this section, we consider convergence to the minimal speed traveling wave
in the case $\beta>2$. As we have mentioned in the introduction, the proof is
quite standard and follows the classical approach of [43, 44, 45] for the
convergence to a traveling wave in the pushed front regime. Let us recall that
for all $\beta>0$ there is a traveling wave solution to the Burgers-FKPP
equation (1.1) of the form

$\phi_{\beta}(x)=\frac{1}{1+e^{{\beta x}/{2}}}$ (7.1)

that moves with the speed

$c_{*}(\beta)=\frac{\beta}{2}+\frac{2}{\beta}\geq 2.$ (7.2)

A key point is that for $\beta\geq 2$, the speed $c_{*}(\beta)$ given by (7.2)
also happens to be the minimal front speed. In other words, for $\beta\geq 2$
the minimal speed wave profile is explicit and given by (7.1). Another
property that will be crucial for the analysis is that for $\beta>2$ we have

$\phi_{\beta}(x)e^{c_{*}x/2}\in L^{2}(\mathbb{R}).$ (7.3)

According to the criterion of [21], this puts the traveling front (7.1) into
the category of pushed fronts, unlike the fronts for $\beta\leq 2$, for which
(7.3) does not hold. Accordingly, the proof of the large time convergence of
the solutions to the initial value problem for (1.1) with $\beta>2$ consists
of three steps that are common in such results for pushed fronts: compactness,
local stability and quasi-convergence.

### Compactness

As the first step, we show that the solution $u(t,x)$ to (1.1) with $\beta>2$
can be trapped between an explicitly constructed super-solution $\bar{u}$ and
a sub-solution $\underline{u}$. Each of them will converge to a separate shift
of the traveling front, exponentially fast in time. Here, we follow the
construction in [43]. We denote by $c_{*}=c_{*}(\beta)$, as given by (7.2),
and drop the subscript $\beta$ in the notation for the traveling wave profile
$\phi_{\beta}(x)$ given by (7.1). Consider the moving frame $z=x-c_{*}t$,
setting

$\widehat{u}(t,z)=u(t,z+c_{*}t).$ (7.4)

This function satisfies

$\displaystyle{\cal
M}[\widehat{u}]:=\widehat{u}_{t}-c_{*}\widehat{u}_{x}+\beta\widehat{u}\widehat{u}_{x}-\widehat{u}_{xx}-\widehat{u}(1-\widehat{u})=0,$
(7.5)

with the initial condition $\widehat{u}(0,z)=u_{\rm in}(z)$. The sub- and
super-solutions in the moving frame are described by the following lemma.

######  Lemma 7.1.

Fix $\lambda\in(2/\beta,\beta/2)$ and $z_{0}^{1,2}\in\mathbb{R}$. For
$q_{0}^{(1,2)},\mu>0$, let

$\begin{split}&\underline{v}(t,z):=\phi(z+\xi_{1}(t))-q_{1}(t,z+z_{0}^{(1)})\qquad\text{and}\\\
&\overline{v}(t,z):=\phi(z-\xi_{2}(t))+q_{2}(t,z-z_{0}^{(2)})\qquad\text{ for
all }t>0,z\in\mathbb{R}\end{split}$ (7.6)

with

$q_{1,2}(t,z)=q_{0}^{(1,2)}e^{-\mu t}\min\\{\exp\big{(}-\lambda z\big{)},1\\}$

and, for $K>0$,

$\xi_{1,2}^{\prime}(t)=Kq_{0}e^{-\mu t}.$ (7.7)

Suppose, for some $D_{0}>0$,

$\frac{Kq_{0}^{(1,2)}}{\mu},~{}~{}|\xi_{1,2}(0)+z_{0}^{(1,2)}|\leq D_{0}.$
(7.8)

Then, if $q_{0}^{(1,2)}$ and $\mu$ are sufficiently small and $K$ is
sufficiently large, all depending only on $\beta$, $\lambda$, and $D_{0}$,
then $\underline{v}$ and $\overline{v}$ are sub- and super-solutions of (7.5),
respectively.

Proof. We only prove the claim for $\underline{v}$. We drop the subscripts of
$q_{1,2}$ and $\xi_{1,2}$ and superscripts of $q_{0}^{(1,2)}$, and
$z_{0}^{(1,2)}$ in order to simplify the notation. Let us insert the ansatz
for $\underline{v}(t,z)$ given by the left side of (7.6) into the desired
inequality

${\cal M}[\underline{v}]\leq 0,$ (7.9)

that needs to hold for $\underline{v}(t,x)$ to be a sub-solution to (1.1).
This gives

$\displaystyle{\cal M}[\underline{v}]$
$\displaystyle=\xi^{\prime}(t)\phi^{\prime}-q_{t}+\beta(\phi-q)(\phi^{\prime}-q_{z})-c_{*}(\phi^{\prime}-q_{z})-\phi^{\prime\prime}+q_{zz}-F(\phi-q)$
(7.10) $\displaystyle=\xi^{\prime}(t)\phi^{\prime}-q_{t}+c_{*}q_{z}-\beta\phi
q_{z}-\beta q\phi^{\prime}+\beta qq_{z}+q_{zz}+F(\phi)-F(\phi-q).$

Here, we have set

$F(u)=u(1-u).$

We fix a sufficiently small $\delta>0$, and consider (7.10) in three regions
of $z\in{\mathbb{R}}$ separately.

The far right region:

$R=\\{z:~{}\phi(z-\xi(t))\leq\delta\\}.$ (7.11)

In other words $z\geq\xi(t)+\phi^{-1}(\delta)$. Thus,

$z+z_{0}\geq\phi^{-1}(\delta)+\xi(t)+z_{0}.$ (7.12)

By (7.7)-(7.8),

$|\xi(t)+z_{0}|\leq 2D_{0}.$ (7.13)

Hence, by choosing $\delta>0$ sufficiently small depending only on $D_{0}$, we
have

$z+z_{0}\geq 0,$ (7.14)

so that

$q(t,z+z_{0})=q_{0}\exp\\{-\mu t-\lambda(z-z_{0})\\},~{}~{}\hbox{ for all
$z\in R$}.$ (7.15)

It follows that

$q_{t}=-\mu q,~{}~{}q_{z}=-\lambda q,~{}~{}q_{zz}=\lambda^{2}q,~{}~{}\hbox{
for all $z\in R$}.$

As $\xi^{\prime}(t)>0$, $\phi^{\prime}<0$, and $\phi(t,x)<\delta$ for all
$x\in R$, we deduce that

$\displaystyle{\cal M}[\underline{v}]$
$\displaystyle=\xi^{\prime}(t)\phi^{\prime}+(\mu-
c_{*}\lambda+\lambda^{2}+\beta\lambda\phi-\beta\phi^{\prime})q+F(\phi)-F(\phi-q)-\lambda\beta
q^{2}$ (7.16) $\displaystyle\leq(\mu-
c_{*}\lambda+\lambda^{2}+\beta\delta\lambda-\beta\phi^{\prime})q+F(\phi)-F(\phi-q).$

Notice that the traveling wave satisfies

$\displaystyle
c_{*}\phi-\frac{\beta}{2}\phi^{2}=-\phi^{\prime}+\int_{x}^{\infty}\phi(1-\phi)dy,$
(7.17)

we see that

$\phi^{\prime}=-c_{*}\phi+\frac{\beta}{2}\phi^{2}+\int_{x}^{\infty}\phi(1-\phi)dy\geq-
c_{*}\phi~{}~{}~{}\hbox{ for all $x\in\mathbb{R}$}.$ (7.18)

In addition, we have

$F(\phi)-F(\phi-q)=\phi-\phi^{2}-\phi+q+(\phi-q)^{2}=q-2\phi
q+q^{2}\leq(1+q_{0})q.$ (7.19)

Using these inequalities, together with (7.11), in (7.16), gives

$\displaystyle{\cal M}[\underline{v}]$
$\displaystyle\leq(\lambda^{2}-c_{*}\lambda+\beta\delta\lambda+\mu+1+q_{0}+c_{*}\beta\delta)q.$
(7.20)

Due to the assumption on $\lambda$, we have $\lambda^{2}-c_{*}\lambda+1<0$.
Then take $\delta>0$, $\mu>0$ and $q_{0}>0$ all sufficiently small, depending
on $\lambda$, so that

${\cal M}[\underline{v}]\leq 0.$ (7.21)

We stress that here we use the assumption that $\beta>2$, which makes
$c_{*}>2$.

The middle region:

$M=\\{z:~{}\delta\leq\phi(z-\xi(t))\leq 1-\delta\\}.$ (7.22)

There exists $\alpha_{\delta}>0$, which depends on $\delta>0$, so that for all
$z$ in the middle region we have

$\phi^{\prime}(z-\xi(t))\leq-\alpha_{\delta}.$ (7.23)

We need to consider separately the points where $z\geq z_{0}$, or $z<z_{0}$,
as this changes the definition of $q(t,z)$. If $z>z_{0}$, then $q$ satisfies
(7.15), and we use (7.23) to obtain, using positivity of $\xi^{\prime}(t)$
once again,

$\displaystyle{\cal M}[\underline{v}]$
$\displaystyle=\xi^{\prime}(t)\phi^{\prime}-q_{t}+c_{*}q_{z}-\beta\phi
q_{z}-\beta q\phi^{\prime}+\beta qq_{z}+q_{zz}+F(\phi)-F(\phi-q)$ (7.24)
$\displaystyle\leq-\alpha_{\delta}\xi^{\prime}(t)+(\mu+\lambda^{2}-c_{*}\lambda+\beta\phi\lambda+c_{*}\beta\phi)q+(1+q)q-\lambda\beta
q^{2}\leq-\alpha\xi^{\prime}(t)+Cq_{0}e^{-\mu t}.$

Above we used that $F(\phi)-F(\phi-q)\leq q(1+q)$. Due to the assumption on
$\lambda$, $\lambda\beta>1$, which implies that $q^{2}-\lambda\beta q^{2}\leq
0$. It follows that $C$ does not depend on $q_{0}$.

On the other hand, if $z\leq z_{0}$, then

$q_{t}=-\mu q,~{}~{}q_{z}=0,$ (7.25)

and, as long as we choose $q_{0}<\delta$, we get, after another appplication
of (7.18):

$\begin{split}{\cal
M}[\underline{v}]\leq\xi^{\prime}(t)\phi^{\prime}+(\mu+c_{*}\beta\phi)q+F(\phi)-F(\phi-q)&\leq-\alpha_{\delta}\xi^{\prime}(t)+(\mu+c_{*}\beta+1)q\\\
&\leq-\alpha_{\delta}\xi^{\prime}(t)+Cq_{0}e^{-\mu t}.\end{split}$ (7.26)

It follows that $M[\underline{v}]\leq 0$ both in (7.24) and (7.26) since, by
(7.7)

$\displaystyle\xi^{\prime}(t)=Kq_{0}e^{-\mu
t}\geq\frac{C}{\alpha_{\delta}}q_{0}e^{-\mu t},$ (7.27)

after possibly increasing $K$ depending only on $\delta$, which, in turn,
depends on $D_{0}$ and $\lambda$.

The far left region:

$L=\\{z:~{}\phi(z-\xi(t))\geq 1-\delta\\}.$ (7.28)

Arguing as in the far right region setting, we see that, up to further
decreasing $\delta$ depending on $D_{0}$, we have $z+z_{0}\leq 0$. Therefore,
$q(t)$ satisfies (7.25), and we have

$\displaystyle{\cal M}[\underline{v}]$
$\displaystyle=\xi^{\prime}(t)\phi^{\prime}-q_{t}+c_{*}q_{z}-\beta\phi
q_{z}-\beta q\phi^{\prime}+\beta qq_{z}+q_{zz}+F(\phi)-F(\phi-q)$ (7.29)
$\displaystyle=\xi^{\prime}(t)\phi^{\prime}+(\mu-\beta\phi^{\prime})q+F(\phi)-F(\phi-q).$

The explicit form (7.1) of the profile $\phi(x)$ as $x\to-\infty$ implies that
in the far left region we have

$-\frac{\beta}{2}\delta\leq\phi^{\prime}<0.$ (7.30)

As we also have $F^{\prime}(1)=-1$, we can assure that

$\displaystyle
M[\underline{v}]\leq\Big{(}\mu+\frac{\beta^{2}\delta}{2}\Big{)}q+F(\phi)-F(\phi-q)\leq
0,$ (7.31)

as long as we choose $\mu>0$ and $\delta>0$ sufficiently small.

Thus, in all cases, we have show that $M[\underline{v}]\leq 0$, which finishes
the proof of Lemma 7.1. $\Box$

### Local stability

Lemma 7.1 implies a certain compactness for the solution $v(t,z)$ to (7.5)
that eventually will lead to the nonlinear stability result. To formulate it,
we need to introduce the weighted Banach space

$B_{\lambda}=\\{u\in C(\mathbb{R},\mathbb{R}):\|u\|_{\lambda}<+\infty\\},$

with the norm

$\displaystyle\|v\|_{\lambda}=\max_{z\in\mathbb{R}}\frac{|v(z)|}{\eta(z)}.$
(7.32)

We also introduce the norm

$\|v\|_{\lambda,1}=\|v\|_{\lambda}+\|v_{z}\|_{\lambda}$ (7.33)

for a later usage. Here, we have set

$\displaystyle\eta(z)=\min\\{1,\exp(-\lambda z)\\}.$

Let us define the $\omega$-limit set of $u_{\rm in}\in B_{\lambda}$ with
respect to the evolution (7.5) as

$\displaystyle\omega(u_{\rm in}):=\\{\psi\in B_{\lambda}\cap
C^{2}(\mathbb{R}):~{}~{}\exists~{}t_{n}\to+\infty~{}\text{ s.t
}~{}v(t_{n},\cdot)\to\psi(\cdot)\text{ in }C_{loc}^{2}({\mathbb{R}})\\}.$

By the standard parabolic regularity theory, we know that the “orbit”
$\\{v(t,\cdot):t\geq 1\\}$ is relatively compact in
$C^{2}_{loc}({\mathbb{R}})$. Thus, the set $\omega(u_{\rm in})$ is nonempty
for a given initial condition $u_{\rm in}\in B_{\lambda}$. Proposition 2.2
implies that each element of $\omega(u_{\rm in})$ is either a constant $0$ or
$1$, or a shift of the traveling wave $\phi(x)$. Lemma 7.1 rules out the first
possibility. Our goal is to prove that it consists of exactly one element, and
that the solution converges exponentially fast to that particular shift of a
traveling wave.

Lemma 7.1 implies the following local stability result.

######  Lemma 7.2.

Fix $\lambda\in(2/\beta,\beta/2)$. For every $\varepsilon>0$, there exists
$\gamma>0$ such that whenever

$\|v(0,\cdot)-\phi(\cdot-s_{0})\|_{\lambda}<\gamma,$ (7.34)

we have

$\|v(t,\cdot)-\phi(\cdot-s_{0})\|_{\lambda}<\varepsilon~{}~{}\hbox{ for all
$t\geq 0$.}$ (7.35)

Proof. Let us assume that $s_{0}=0$. We see from (7.34) that

$\phi(z)-\gamma\eta(z)<v(0,z)<\phi(z)+\gamma\eta(z),~{}~{}z\in\mathbb{R}.$
(7.36)

This allows us to apply Lemma 7.1 with $\xi(t)$ satisfying (7.27) and
$\xi(0)=0$:

$\xi(t)=\frac{K\gamma}{\mu}\Big{(}1-e^{-\mu t}\Big{)},$

leading to

$\displaystyle\phi(z-\xi(t))-\gamma\eta(z)<v(t,z)<\phi(z+\xi(t))+\gamma\eta(z),~{}~{}t\geq
0,~{}z\in\mathbb{R}.$

Because

$\phi(z)\sim e^{-{\beta z}/{2}}~{}~{}~{}\hbox{ as $x\to+\infty$,}$

and $\lambda<{\beta}/{2}$, there exists a constant $K_{1}$ such that

$\displaystyle\|\phi(\cdot)-\phi(\cdot\pm\xi(t))\|_{\lambda}\leq
K_{1}\xi(t)\leq K_{1}\gamma,~{}~{}t\geq 0,$

finishing the proof. $\Box$

### Quasi-convergence and exponential stability

Now, convergence in shape in Proposition 2.2 and the steepness comparison in
1.1, together with Lemma 7.2 imply the following quasi-convergence property.

######  Corollary 7.3.

Fix $\lambda\in(2/\beta,\beta/2)$. Let $v(t,x)$ be the solution to (7.5) with
the initial condition $v_{0}(x)=\mathbbm{1}(x\leq 0)$. Then there exists a
sequence $t_{n}\to+\infty$ and $s_{0}\in{\mathbb{R}}$ so that

$\lim_{n\to+\infty}\|v(t_{n},\cdot)-\phi(\cdot-s_{0})\|_{\lambda}=0.$ (7.37)

To improve the result of Corollary 7.3 to the exponential convergence, we will
use the method of [44, 45], which consider convergence to traveling waves for
equations of the form

$\displaystyle u_{t}=u_{xx}+f(u,u_{x}).$

In the Burgers-KPP case, we have

$f(u,u_{x})=-\beta uu_{x}+u(1-u).$

If one looks for solutions that are perturbations of a traveling wave, that is

$\displaystyle u(t,x)=\phi(x-c_{*}t)+w(t,x),$

then in the moving coordinate $z=x-c_{*}t$, the equation of $w$ can be
decomposed as

$\displaystyle\frac{\partial w}{\partial t}={\cal L}w+R(w).$

Here, the linearized operator is

$\displaystyle{\cal L}w=w_{zz}+c_{*}w_{z}+\frac{\partial
f}{\partial\phi^{\prime}}w_{z}+\frac{\partial
f}{\partial\phi}w=w_{zz}+(c_{*}-\beta\phi)w_{z}+(-\beta\phi^{\prime}+1-2\phi)w.$
(7.38)

The remainder $R$ is a nonlinear operator whose Fréchet derivative vanishes at
$w=0$.

The main result of [45] is that a local stability result, as we have in
Corollary 7.3, implies exponential convergence to a traveling wave, as long as
the operator ${\cal L}$ satisfies certain spectral assumptions that we will
now recall and specify to the Burgers-FKPP equation. Let us put the linearized
operator in the from

$\displaystyle{\cal
L}w=w^{\prime\prime}-2bw^{\prime}+qw,~{}~{}b=\frac{1}{2}(\beta\phi-
c_{*}),~{}~{}q=-\beta\phi^{\prime}+1-2\phi,$ (7.39)

and set

$B(z)=\int_{0}^{z}b(s)ds.$

To remove the drift term in the operator ${\cal L}$ in (7.39), Sattinger
introduces the operator

$\widetilde{\cal M}=e^{-B}{\cal L}e^{B},$

so that

$\displaystyle\widetilde{\cal
M}w=w^{\prime\prime}+pw,~{}~{}p=b^{\prime}-b^{2}+q.$ (7.40)

In our case, with $\phi(x)$ given explicitly by (7.1), and $c_{*}$ by (7.2),
we have the left and right limits

$\displaystyle
b_{+}=\lim_{x\to+\infty}b(x)=-\frac{c_{*}}{2}=-\frac{\beta}{4}-\frac{1}{\beta},\qquad
b_{-}=\lim_{x\to-\infty}b(x)=\frac{1}{2}(\beta-
c_{*})=\frac{1}{2}\Big{(}\frac{\beta}{2}-\frac{2}{\beta}\Big{)},$
$\displaystyle q_{\pm}=\lim_{x\to\pm\infty}q(x)=\pm 1.$

We see that

$p_{\pm}=\lim_{z\to\pm\infty}p(z)$

are given by

$p_{+}=-b_{+}^{2}+q_{+}=-\Big{(}\frac{\beta}{4}+\frac{1}{\beta}\Big{)}^{2}+1=-\Big{(}\frac{\beta}{4}-\frac{1}{\beta}\Big{)}^{2}<0,$

as $\beta>2$, and

$p_{-}=-b_{-}^{2}+q_{-}=-\Big{(}\frac{1}{\beta}+\frac{\beta}{4}\Big{)}^{2}-1<0.$

As both $p_{+}<0$ and $p_{-}<0$, the operator $\widetilde{\cal M}$ is stable
both as $x\to+\infty$ and $x\to-\infty$, and Theorems 1 and 2 in [45] imply
that convergence in Corollary 7.3 is actually exponential in time.

######  Corollary 7.4.

Let $v(t,x)$ be the solution to (7.5) with the initial condition
$v_{0}(x)=\mathbbm{1}(x\leq 0)$. There exist $C>0$ and $\omega_{0}>0$, as well
as $s_{0}\in{\mathbb{R}}$ so that

$\|v(t,\cdot)-\phi(\cdot-s_{0})\|_{\lambda}\leq Ce^{-\omega_{0}t}.$ (7.41)

This completes the proof of Theorem 1.1 for $\beta>2$.

##  8 An informal derivation of the higher corrections

We explain in this section how the non-rigorous but extremely interesting
methodology of [7] can be used to predict the higher order corrections to the
logarithmic shift in the front position. This strategy was applied in [7] for
the classical Fisher-KPP equation, and the first two extra terms in the
expansion were rigorously confirmed in [24, 41], leading to the long-time
front position asymptotics

$\displaystyle
m(t)=2t-\frac{3}{2}\log(t+1)+x_{\infty}-\frac{3\sqrt{\pi}}{\sqrt{t+1}}+\frac{9}{8}(5-6\log
2)\frac{\log(t+1)}{t+1}+o\Big{(}\frac{\log t}{t}\Big{)},$ (8.1)

a significant refinement of Theorem 1.1. The terms that appear in (8.1),
except for $x_{\infty}$, do not depend on the initial conditions. Moreover,
they are expected to be universal for a large class of Fisher-KPP type
problems. That is, this expansion has been shown to hold, with exactly the
same coefficients, for all equations of the form

$u_{t}=u_{xx}+f(u),$ (8.2)

with a nonlinearity $f(u)$ of the Fisher-KPP type, normalized so that
$f^{\prime}(0)=1$ as long as $f\in C^{1,\delta}$ near $0$ (see [8] for the
treatment of the less regular case).

An analysis similar to what we do in this section for $\beta=2$ would show
that (8.1) holds with exactly the same coefficients for the Burgers-FKPP
equation (1.1) for all $\beta<2$, confirming the universality prediction of
[7]. We omit the details. The goal of this section is to show that when
$\beta=2$ this expansion changes the coefficients but not the form of the
individual terms to

$\displaystyle
m(t)=2t-\frac{1}{2}\log(t+1)-x_{\infty}-\frac{\sqrt{\pi}}{2\sqrt{t+1}}+\frac{(1-\log
2)}{4}\frac{\log(t+1)}{t+1}++o\Big{(}\frac{\log t}{t}\Big{)}.$ (8.3)

It is tempting to conjecture that the expansion (8.3) is also universal for
problems that combine the pulled nature with traveling waves that decay as
$e^{-x}$ rather than the Fisher-KPP asymptotics $xe^{-x}$. This regime is
referred to as Case (II) in Chapter 2 of [34] for reaction-diffusion equations
of the form (8.2).

We begin with the following observation, inspired by [7], that we state for
all $\beta\leq 2$. Let $u(t,x)$ be the solution to (1.1) with an initial
condition $u_{\rm in}(x)$ and set

$\displaystyle\varphi(t,r)=\int_{\mathbb{R}}u^{2}(t,z)e^{rz}dz,~{}~{}\Phi_{1}(r)=(1-r\beta/2)\int_{0}^{\infty}\varphi(t,r)e^{-(r^{2}+1)t}dt,$
(8.4)

and

$\displaystyle\Phi(r)=\int_{\mathbb{R}}u_{\rm in}(x)e^{rx}dx.$ (8.5)

Note that the function $\Phi(r)$ is smooth for all $r>0$, as long as the
initial condition $u_{\rm in}(x)$ is compactly supported on the right. On the
other hand, as we will see below, the function $\Phi_{1}(r)$ may potentially
blow up as $r\to 1^{-}$. This possibility is removed by the following
identity.

######  Proposition 8.1.

Assume that the initial condition $u_{\rm in}(x)$ for (1.1) satisfies

$u_{\rm in}(x)\leq\frac{1}{1+e^{x-L_{0}}},~{}~{}\hbox{ for all
$x\in{\mathbb{R}}$},$ (8.6)

with some $L_{0}>0$. Then, we have

$\displaystyle\Phi(r)=\Phi_{1}(r),~{}~{}\hbox{for all~{}$\beta\leq 2$ and
$r\in(0,1)$}.$ (8.7)

Proof. The proof is a modification of the argument in [7]. Without loss of
generality we assume that $L_{0}=0$. First, we recall that, as we have shown
in (4.9)-(4.10), for all $\beta\leq 2$, we have the upper bound

$u(t,x)\leq\bar{u}(t,x):=\frac{1}{1+e^{x-2t}},~{}~{}\hbox{ for all
$x\in{\mathbb{R}}$}.$ (8.8)

It follows that $\varphi(t,r)$ is defined and differentiable in $r$ for all
$r<2$. This also shows that for all $r\in(0,1)$ we have

$g(t,r):=\int_{\mathbb{R}}u(t,x)e^{rx}dx\leq\int_{\mathbb{R}}\frac{e^{rx}}{1+e^{x-2t}}dx=I_{0}(r)e^{2rt},$
(8.9)

with

$I_{0}(r)=\int_{\mathbb{R}}\frac{e^{rx}}{1+e^{x}}dx<\infty,$

since $r\in(0,1)$. Next, differentiating $g(t,r)$ in $t$, we obtain

$\displaystyle g_{t}(t,r)$
$\displaystyle=\int_{\mathbb{R}}u_{t}(t,x)e^{rx}dx=\int_{\mathbb{R}}\Big{(}-\frac{\beta}{2}(u^{2})_{x}+u_{xx}+u-u^{2}\Big{)}e^{rx}dx$
(8.10) $\displaystyle=(1+r^{2})g(t,r)-(1-r\beta/2)\varphi(t,r).$

Integrating this identity from $0$ to $t$ and using the definition of
$\Phi(r)$, we see that

$\displaystyle
g(t,r)e^{-(1+r^{2})t}=\Phi(r)-(1-r\beta/2)\int_{0}^{t}\varphi(s,r)e^{-(1+r^{2})s}ds.$

As $2r<1+r^{2}$, we may use (8.9) to pass to the limit $t\to+\infty$ and
obtain (8.7). $\Box$

An immediate consequence of Proposition 8.1 is that the function $\Phi_{1}(r)$
remains regular and even infinitely differentiable, with bounded derivatives
as $r\to 1^{-}$. A surprising discovery of [7] is that for the classical
Fisher-KPP equation one can perform a careful analysis of that limit in terms
of the front location $m(t)$ and this regularity alone can be used to obtain
the asymptotics (8.1) of $m(t)$ as $t\to+\infty$. As noted above, this
approach can be applied nearly verbatim for $\beta<2$, so we only consider
$\beta=2$ below.

#### Assumptions on the rate of convergence

Let us now formalize the assumptions that go into the derivation of (8.1) and
(8.3). We know from Proposition 2.2 that there is a reference frame $m(t)$
such that

$u(t,x+m(t))\to\phi_{2}(x),$ (8.11)

and from Theorem 1.1 that, when $\beta=2$:

$m(t)=2t-\gamma(t),~{}~{}\gamma(t)=a\log(t+1)-\alpha(t),~{}~{}\alpha(t)=x_{\infty}+o(1)~{}~{}\hbox{as
$t\to+\infty$}.$ (8.12)

Of course, we already know that $a=1/2$ when $\beta=2$ but we leave this
coefficient undetermined for now, to show how this value can be discovered by
the arguments below. One assumption of [7], later proved in [24], both in the
case of the standard Fisher-KPP equation, is that the analogue of (8.11) with
$\beta=0$ holds at the rate $O(1/t)$. For the Burgers-FKPP equation this
translates into the assumption that

$u(t,x+m(t))=\phi_{2}(x)+\frac{1}{t}\eta(t,x),~{}~{}\hbox{ as $t\to+\infty$,}$
(8.13)

with a rapidly decaying in space (but not necessarily in time) function
$\eta(t,x)$. A result of [24] for $\beta=0$ is that $\eta(t,x)$ has a positive
limit as $t\to+\infty$, and the rate $O(t^{-1})$ can not be improved. We
stress that (8.13) is an assumption and not a rigorous claim, even though we
believe that it holds, as it does for the classical Fisher-KPP equation.

Let us now re-write the function $\Phi_{1}(r)$ in terms of $m(t)$. We
introduce

$\displaystyle\varphi_{m}(t,r)=\int_{\mathbb{R}}u(t,z+m(t))^{2}e^{rz}dz=e^{-rm(t)}\int_{\mathbb{R}}u(t,z)^{2}e^{rz}dz=e^{-rm(t)}\varphi(t,r),$
(8.14)

and write

$\Phi_{1}(r)=(1-r)\int_{0}^{\infty}\varphi_{m}(t,r)e^{rm(t)-(r^{2}+1)t}dt.$
(8.15)

Given the exponential decay of $u(t,x)$ and $\phi_{2}(x)$, and assumption
(8.13), we deduce that there exists a function $\tilde{\eta}(t,r)$, so that

$\varphi_{m}(t,r)=\int_{\mathbb{R}}u(t,z+m(t))^{2}e^{rz}dz=\tilde{\varphi}(r)+\frac{1}{t+1}\tilde{\eta}(t,r),$
(8.16)

with

$\tilde{\varphi}(r):=\int_{\mathbb{R}}\phi_{2}(z)^{2}e^{rz}dz,$ (8.17)

and

$|\tilde{\eta}(t,r)|\leq K,~{}~{}\hbox{ for all $t>0$ and $r\in(1/2,3/2)$. }$
(8.18)

Inserting (8.16) into (8.15) gives

$\Phi_{1}(r)=(1-r)\int_{0}^{\infty}\Big{(}\tilde{\varphi}(r)+\frac{1}{t+1}\tilde{\eta}(t,r)\Big{)}e^{rm(t)-(r^{2}+1)t}dt.$
(8.19)

#### Higher order corrections from the limit $r\to 1^{-}$

We now pass to the limit $r\to 1^{-}$ in (8.19) and use the regularity of
$\Phi_{1}(r)$ in this limit, implied by Proposition 8.1, to get extra terms in
the asymptotic expansion for $m(t)$ as $t\to+\infty$. We take
$r=1-\varepsilon$ in (8.19), with $\varepsilon\in(0,1)$ and use expression
(8.12) for $m(t)$:

$\displaystyle\Phi_{1}(1-\varepsilon)$
$\displaystyle=\varepsilon\int_{0}^{\infty}\Big{(}\tilde{\varphi}(1-\varepsilon)+\frac{1}{t+1}\tilde{\eta}(t,1-\varepsilon)\Big{)}e^{(1-\varepsilon)(2t-\gamma(t))-((1-\varepsilon)^{2}+1)t}dt$
(8.20)
$\displaystyle=\tilde{\varphi}(1-\varepsilon)I(\varepsilon)+E(\varepsilon).$

Here, the main term is

$I(\varepsilon)=\varepsilon\int_{0}^{\infty}e^{-(1-\varepsilon)\gamma(t)-\varepsilon^{2}t}dt,$
(8.21)

and

$E(\varepsilon)=\varepsilon\int_{0}^{\infty}\frac{1}{t+1}\tilde{\eta}(t,1-\varepsilon)e^{-(1-\varepsilon)\gamma(t)-\varepsilon^{2}t}dt$
(8.22)

is the error term.

In order to avoid additional technicalities in an argument that is not
rigorous (because assumption (8.13) has not been justified), we use (8.13) to
replace the rigorous claim that $\Phi_{1}(1-\varepsilon)$ remains regular as
$\varepsilon\to 0$ by the assumption that

$I(\varepsilon)\hbox{ remains strictly positive and regular as $\varepsilon\to
0$,}$ (8.23)

neglecting the error term $E(\varepsilon)$. Note that the function
$\tilde{\varphi}(r)$ is regular near $r=1$, as can be seen immediately from
its definition (8.17). The positivity of $I(\varepsilon)$ holds because we
know from (8.7) that $\Phi_{1}(1-\varepsilon)$ is finite and not small for any
$\varepsilon>0$, and so is $\tilde{\varphi}(1)$.

The surprising fact is that (8.23) by itself leads to the asymptotic expansion
(8.3) for $m(t)$. We first explain how we can find from (8.23) that the
coefficient $a$ that appears in (8.12) equals to $1/2$, as expected. Using the
expression for $\gamma(t)$ in (8.12), we write

$\displaystyle I(\varepsilon)$
$\displaystyle=\varepsilon\int_{0}^{\infty}\frac{e^{-\varepsilon^{2}t+(1-\varepsilon)\alpha(t)}}{(t+1)^{a(1-\varepsilon)}}dt=\varepsilon\int_{0}^{\infty}\frac{e^{-s+(1-\varepsilon)\alpha(s/\varepsilon^{2})}}{(s/\varepsilon^{2}+1)^{a(1-\varepsilon)}}\frac{ds}{\varepsilon^{2}}$
(8.24)
$\displaystyle=\varepsilon\big{(}1+o(1)\big{)}e^{(1-\varepsilon)x_{\infty}}\int_{0}^{\infty}\frac{\varepsilon^{2a(1-\varepsilon)-2}}{(s+\varepsilon^{2})^{a(1-\varepsilon)}}e^{-s}ds,~{}~{}\hbox{
as $\varepsilon\to 0$. }$

We replaced $\alpha(s/\varepsilon^{2})$ by its limit $x_{\infty}$ in the last
step above. The positivity and finiteness of the limit of $I(\varepsilon)$ as
$\varepsilon\to 0$ imply that the integral in the right side of (8.24) should
blow up as $\sim\varepsilon^{-1}$ as $\varepsilon\to 0$. Thus, the only
possible choice for $a$ is $1/2$, as in Theorem 1.1.

Next, we show how the remaining terms in (8.3) come about. Taking $a=1/2$ in
(8.24) and writing

$\alpha(t)=x_{\infty}+p(t),~{}~{}p(t)\to 0\hbox{ as $t\to+\infty$},$ (8.25)

leads to

$\displaystyle I(\varepsilon)$ $\displaystyle=\varepsilon
e^{(1-\varepsilon)x_{\infty}}\int_{0}^{\infty}\frac{1}{\sqrt{t+1}}e^{(\varepsilon/2)\log(t+1)+(1-\varepsilon)p(t)-\varepsilon^{2}t}dt.$
(8.26)

Let us assume without loss of generality that $x_{\infty}=0$, and note that
$I(\varepsilon)$ has the limit

$\displaystyle
I(\varepsilon)\to\int_{0}^{\infty}\frac{e^{-t}}{\sqrt{t}}dt=\sqrt{\pi},~{}~{}\hbox{
as $\varepsilon\to 0$},$ (8.27)

so the positivity requirement in (8.23) holds automatically. Expanding further
gives

$\displaystyle I(\varepsilon)$
$\displaystyle=\sqrt{\pi}+\varepsilon\int_{0}^{\infty}\frac{1}{\sqrt{t+1}}\Big{(}\frac{\varepsilon}{2}\log(t+1)+p(t)\Big{)}e^{-\varepsilon^{2}t}dt+o(\varepsilon\log\varepsilon).$
(8.28)

Note that the first term in the integral above has the asymptotics

$\displaystyle\frac{\varepsilon^{2}}{2}$
$\displaystyle\int_{0}^{\infty}\frac{\log(t+1)}{\sqrt{t+1}}e^{-\varepsilon^{2}t}dt=\frac{\varepsilon^{2}}{2}\int_{1}^{\infty}\frac{\log
t}{\sqrt{t}}e^{-\varepsilon^{2}t}dt+o(\varepsilon\log\varepsilon)$ (8.29)
$\displaystyle=\frac{\varepsilon}{2}\int_{\varepsilon^{2}}^{\infty}\frac{\log(t/\varepsilon^{2})}{\sqrt{t}}e^{-t}dt+o(\varepsilon\log\varepsilon)=-\sqrt{\pi}\varepsilon\log\varepsilon+o(\varepsilon\log\varepsilon).$

We now make an ansatz

$p(t)=\frac{b}{\sqrt{t}}+o\big{(}t^{-1/2}\big{)},~{}~{}\hbox{ as
$t\to+\infty$},$ (8.30)

with the constant $b$ to be determined, and insert it into the second integral
in (8.28):

$\displaystyle\varepsilon\int_{0}^{\infty}\frac{p(t)}{\sqrt{t+1}}e^{-\varepsilon^{2}t}dt=\varepsilon
b\int_{1}^{\infty}\frac{e^{-\varepsilon^{2}t}}{t}dt+o(\varepsilon\log\varepsilon)=\varepsilon
b\log(\varepsilon^{-2})+o(\varepsilon\log\varepsilon).$ (8.31)

Taking into account the regularity of $I(\varepsilon)$ together with
(8.28)-(8.29) and (8.31), we conclude that, since the overall coefficient in
front of $\varepsilon\log\varepsilon$ has to cancel, we must have

$b=\frac{\sqrt{\pi}}{2}.$ (8.32)

We deduce that when $\beta=2$ the front location has the asymptotics

$m(t)=2t-\frac{1}{2}\log
t-x_{\infty}-\frac{\sqrt{\pi}}{2\sqrt{t}}+o(t^{-1/2}),~{}~{}\hbox{ as
$t\to+\infty$,}$ (8.33)

recovering all but the last term in the right side of (8.3).

A similar computation, expanding $I(\varepsilon)$ further in $\varepsilon$,
and observing that the terms of the order $\varepsilon^{2}\log\varepsilon$
must cancel, improves (8.33) to

$\displaystyle m(t)=2t-\frac{1}{2}\log
t-x_{\infty}-\frac{\sqrt{\pi}}{2\sqrt{t}}+\frac{(1-\log 2)}{4}\frac{\log
t}{t}+....$ (8.34)

which is (8.3). We leave the computational details to an interested reader.

##  Appendix A Traveling waves for the Burgers-FKPP equation

In this appendix, we recall some basic facts on the Burgers-FKPP traveling
waves. Most of them can be found in Section 13.4 of [38], at least on a formal
level, and also in [49] for $\beta<0$.

We will use the notation

$f(u)=u(1-u)$ (A.1)

for the reaction term, and write the traveling wave Burgers-FKPP equation as

$-cU^{\prime}+\beta
UU^{\prime}=U^{\prime\prime}+f(U),~{}~{}~{}U(-\infty)=1,~{}U(+\infty)=0.$
(A.2)

### Existence of the traveling waves

We first state the result on the existence of the traveling waves.

######  Proposition A.1.

A traveling wave solution to (A.2) exists for all $c\geq c_{*}(\beta)$, with

$c_{*}(\beta)=\left\\{\begin{matrix}2\sqrt{f^{\prime}(0)}=2,~{}~{}~{}&\text{if
}\beta<2,\cr\dfrac{\beta}{2}+\dfrac{2}{\beta},&\text{if }\beta\geq
2.\cr\end{matrix}\right.$ (A.3)

Proof. First, we note that the boundary conditions in (A.2) allow us to use
the sliding method to deduce that any traveling wave solution $U(x)$ to (A.2)
is monotonically decreasing. Introducing $V=-U^{\prime}>0$, we write (A.2) as
a system

$\frac{dU}{dx}=-V,~{}~{}\frac{dV}{dx}=(\beta U-c)V+f(U).$ (A.4)

This leads to

$\displaystyle\frac{dV}{dU}=c-\beta U-\frac{f(U)}{V}.$ (A.5)

Consider the plane formed by the horizontal $U$-axis and the vertical
$V$-axis. The explicit form of $f(u)$ in (A.1) shows that the system (A.4) has
two equilibrium points

$E_{1}=(0,0)$ and $E_{2}=(1,0)$.

A traveling wave is a heteroclinic orbit of (A.4) that goes from $E_{2}$ to
$E_{1}$.

The linearization of (A.4) around $E_{1}$ is

$\displaystyle\frac{d}{dx}\begin{pmatrix}\tilde{U}\\\
\tilde{V}\end{pmatrix}=\begin{pmatrix}0&-1\\\
f^{\prime}(0)&-c\end{pmatrix}\begin{pmatrix}\tilde{U}\\\
\tilde{V}\end{pmatrix}.$

The eigenvalues of this matrix are

$\lambda_{1,2}=\frac{-c\pm\sqrt{c^{2}-4}}{2},$ (A.6)

and are both real and negative if $c\geq 2$. Thus, the point $E_{1}$ is a
stable equilibrium of $c\geq 2$.

On the other hand, the linearization of (A.2) around $E_{2}$ is

$\displaystyle\frac{d}{dx}\begin{pmatrix}\tilde{U}\\\
\tilde{V}\end{pmatrix}=\begin{pmatrix}0&-1\\\
f^{\prime}(1)&\beta-c\end{pmatrix}\begin{pmatrix}\tilde{U}\\\
\tilde{V}\end{pmatrix}.$

The eigenvalues of this matrix are

$\lambda_{\pm}=\frac{\beta-c\pm\sqrt{(\beta-c)^{2}+4}}{2}.$ (A.7)

They are real and have opposing signs, so $E_{2}$ is a saddle point.

#### Existence of traveling waves for $c\geq c_{*}(\beta)$

We prove existence of a heteroclinic orbit by constructing an invariant region
in the $(U,V)$-plane that contains the unstable manifold of the point $(1,0)$.
The regions are different for $\beta\geq 2$ and $\beta<2$, so we consider them
separately.

When $\beta\geq 2$, we consider the region $D_{1}$ formed by the interval
$\ell_{1}=[0,1]\times\\{0\\}$ along the $U$-axis, and the curve
$\ell_{2}=\\{V=(\beta/{2})f(U)\\}$ that connects the equilibrium points
$E_{1}$ and $E_{2}$ and lies in the upper half-plane $\\{V>0\\}$. Note that
the slope of $\ell_{2}$ at the point $E_{2}$ is $(-\beta/2)$, while the slope
$e_{u}$ of the unstable orbit at $E_{2}$ is $(-\lambda_{+})$. Hence, the
unstable orbit starting at $E_{2}$ enters the region $D_{1}$ at the point
$E_{2}$ if $\lambda_{+}<\beta/2$. This condition is satisfied if

$\lambda_{+}=\frac{\beta-c+\sqrt{(\beta-c)^{2}+4}}{2}<\frac{\beta}{2},$ (A.8)

or

$c>c_{*}(\beta)=\frac{\beta}{2}+\frac{2}{\beta}.$ (A.9)

Now, we check that the region $D_{1}$ is invariant. Along the interval
$\ell_{1}$, we have, from (A.4):

$\frac{dU}{dx}=0,~{}~{}\frac{dV}{dx}=f(U)>0,$ (A.10)

so the trajectories point upward, into $D_{1}$. Along the curve $\ell_{2}$, we
have, from (A.5):

$\frac{dV}{dU}=c-\beta U-\frac{2}{\beta}.$ (A.11)

The slope of the curve $\ell_{2}$ itself is

$\frac{\beta}{2}f^{\prime}(U)=\beta\Big{(}\frac{1}{2}-U\Big{)}.$ (A.12)

Thus, the trajectories along $\ell_{2}$ point into $D_{1}$ if

$\displaystyle c-\beta
U-\frac{2}{\beta}>\beta\Big{(}\frac{1}{2}-U\Big{)},~{}~{}\hbox{ for all
$U\in[0,1]$,}$ (A.13)

or, equivalently,

$c>c_{*}(\beta)=\frac{\beta}{2}+\frac{2}{\beta}.$ (A.14)

This shows that traveling waves exist for all $c>c_{*}(\beta)$ when $\beta\geq
2$. It is easy to see that the curve $\ell_{2}$ itself forms a heteroclinic
connection between $E_{1}$ and $E_{2}$ when $c=c_{*}(\beta)$.

When $0<\beta<2$, we consider a different region $D_{2}$ formed by the same
interval $\ell_{1}=[0,1]\times\\{0\\}$ and the curve $\ell_{3}=\\{V=f(U)\\}$,
also connecting the points $E_{1}$ and $E_{2}$ through the upper half-plane.
The slope of $\ell_{3}$ at $E_{2}$ is $(-1)$, hence the condition that the
unstable direction $e_{u}$ points into the region $D_{1}$ at the point $E_{2}$
is $\lambda_{+}<1$, which is

$\lambda_{+}=\frac{\beta-c+\sqrt{(\beta-c)^{2}+4}}{2}<1.$ (A.15)

This condition holds as soon as $c>\beta$.

To check that the region $D_{2}$ is invariant, we first note that, as for
$\beta\geq 2$, the trajectories point upward and inside $D_{2}$ along the
interval $\ell_{1}$. The slope of the curve $\ell_{3}$ is $1-2U$, and we have,
from (A.5):

$\displaystyle\frac{dU}{dx}<0,~{}~{}\frac{dV}{dU}=c-\beta U-1\geq
1-2U,~{}~{}\hbox{for all $U\in[0,1]$},$ (A.16)

as long as $c\geq c_{*}=2$. We use here the fact that $\beta<2$. Thus, the
region $D_{2}$ is invariant. As the unstable direction $e_{u}$ enters $D_{2}$,
existence of the heteroclinic orbit connecting $E_{2}$ and $E_{1}$ follows.

#### Non-existence of traveling waves for $c<c_{*}(\beta)$

Consider first the case $\beta\leq 2$. Suppose that $c<c_{*}(\beta)=2$. Given
a wave $U(x)$ that satisfies (A.2), define the normalized translates

$U_{n}(x)=\frac{U(x+n)}{U(n)},$

that satisfy

$-cU_{n}^{\prime}+\beta
U(n)U_{n}U_{n}^{\prime}=U_{n}^{\prime\prime}+U_{n}-U(n)U_{n}^{2},~{}~{}U_{n}(0)=1.$
(A.17)

As $U(n)\to 0$ as $n\to+\infty$, the Harnack inequality and normalization
$U_{n}(0)=1$ imply that the sequence $U_{n}(x)$ converges locally uniformly to
a limit $\bar{U}(x)>0$ that satisfies

$-c\bar{U}^{\prime}=\bar{U}^{\prime\prime}+\bar{U},~{}~{}\bar{U}(0)=1.$ (A.18)

Since $c<2$, solutions to (A.18) are exponentials of the form

$\bar{U}(x)=A_{1}e^{\lambda_{1}x}+A_{2}e^{\lambda_{2}x},$

with $\lambda_{1,2}\in\mathbb{C}\setminus\mathbb{R}$ given by (A.6). On the
other hand, $\bar{U}$ is real and positive. This is only possible if
$A_{1}=A_{2}=0$, which violates the normalization $\bar{U}(0)=1$. This is a
contradiction, and, thus, $c<2$ is not possible when $\beta\leq 2$.

In the case $\beta\geq 2$ we argue as follows to show that no wave can exist
for $c<c_{*}(\beta)$. First, the same argument as for $\beta<2$ implies that
there exist no waves for $c<2$. Let us assume that there there exists a
heteroclinic orbit connecting $E_{2}$ to $E_{1}$ for some $c<c_{*}(\beta)$.
Then, (A.6) implies that its slope at the point $E_{1}=(0,0)$ is

$\frac{dV}{dU}\Big{|}_{(0,0)}=\frac{c-\sqrt{c^{2}-4}}{2}.$ (A.19)

On the other hand, its slope at the point $E_{2}$ is still given by
$(-\lambda_{+})$, and we see from (A.8) that it goes out of the region $D_{1}$
as it leaves $E_{2}$. Moreover, looking at (A.11) and (A.12) we see that if
$c<c_{*}(\beta)$, then the heteroclinic orbit can not re-enter the domain
$D_{1}$ along the curve $\ell_{2}$ at a point with $0<U<1$. Therefore, if it
arrives to $E_{1}$, it has to do that above the curve $\ell_{2}$, and its
slope at this point has to satisfy

$\frac{dV}{dU}\Big{|}_{(0,0)}>\frac{\beta}{2}.$ (A.20)

Combining (A.19) and (A.20) gives

$\frac{c-\sqrt{c^{2}-4}}{2}>\frac{\beta}{2}.$ (A.21)

It follows that

$\beta<\frac{4}{c+\sqrt{c^{2}-4}}\leq\frac{4}{c}\leq 2,$ (A.22)

as we already know that $c\geq 2$. This is a contradiction to the assumption
that $\beta\geq 2$, finishing the proof of Proposition A.1. $\Box$

##### Remark.

When $c=2$ and $\beta\in(0,2)$, the above proof shows that for
$\beta\in(0,2)$, all trajectories are trapped in the region bounded by the
curves $\ell_{2}=\\{V=({\beta}/{2})f(U)\\}$ and $\ell_{3}=\\{V=f(U)\\}$, that
is $D_{2}\setminus D_{1}$. The reasons for this are that the inequality in
(A.13) is reversed and (A.16) holds, so this region is invariant, and
$\beta/2<\lambda_{+}<1$, so trajectories enter $D_{2}\setminus D_{1}$ from
$E_{2}$. This implies that for any $0<z<1$ we have

$\frac{z(1-z)}{|\bar{E}(z)|}\leq\frac{2}{\beta},$ (A.23)

with $\bar{E}(z)$ defined in (2.6).

#### The asymptotic profile of the traveling waves

We briefly summarize asymptotic profiles of traveling wave solutions. The
details are essentially identical to the Fisher-KPP equation, see, for
instance, [34] for a detailed analysis.

When $\beta<2$, the minimal speed traveling wave has the following asymptotics
on the right:

$\displaystyle U(z)=(Az+B)e^{-z}+O(e^{-(1+\delta)z}),~{}~{}\text{as
}z\to+\infty.$ (A.24)

with $A>0$ and $\delta>0$. When $\beta\geq 2$, the critical front has an
explicit form

$\displaystyle U(z)=\frac{1}{1+e^{\beta z/2}}.$ (A.25)

#### Steepness comparison of the waves

Finally, we present a steepness comparison for the traveling waves that is
used to show convergence of the solution in shape to the minimal speed
traveling wave.

######  Lemma A.2.

Fix any $\beta\in{\mathbb{R}}$. Let $U_{1}$ and $U_{2}$ be two traveling waves
with speeds $c_{1}$ and $c_{2}$, respectively. If $c_{1}\leq c_{2}$, then
$U_{1}$ is steeper than $U_{2}$.

Proof. The proof follows an approach from [17]. If we define

$\bar{E}_{i}(z)=-U_{i}^{\prime}(U_{i}^{-1}(z)),$ (A.26)

it is enough to show that

$\bar{E}_{1}(z)\leq\bar{E}_{2}(z),~{}~{}\hbox{ for all $z\in(0,1)$.}$ (A.27)

Using (A.5), we obtain

$\displaystyle\bar{E}_{i}^{\prime}(z)+\beta
z+\frac{f(z)}{\bar{E}_{i}(z)}=c_{i},$ (A.28)

so that

$\displaystyle\bar{E}_{1}^{\prime}(z)-\bar{E}_{2}^{\prime}(z)-\frac{f(z)}{\bar{E}_{1}(z)\bar{E}_{2}(z)}(\bar{E}_{1}(z)-\bar{E}_{2}(z))=c_{1}-c_{2}.$
(A.29)

It follows that the function

$\displaystyle
F(z)=(\bar{E}_{1}(z)-\bar{E}_{2}(z))\exp\Big{(}-\int_{1/2}^{z}\frac{f(z^{\prime})}{\bar{E}_{1}(z^{\prime})\bar{E}_{2}(z^{\prime})}dz^{\prime}\Big{)},$

satisfies

$\displaystyle
F^{\prime}(z)=(c_{1}-c_{2})\exp\Big{(}-\int_{1/2}^{z}\frac{f(z^{\prime})}{\bar{E}_{1}(z^{\prime})\bar{E}_{2}(z^{\prime})}dz^{\prime}\Big{)}.$
(A.30)

Observe that $\bar{E}_{1}-\bar{E}_{2}\to 0$ as $z\to 1$. It follows that
$F(z)\to 0$, as well. On the other hand, since $c_{1}\leq c_{2}$, we see from
(A.30) that $F(z)$ is decreasing. It follows that $F(z)\leq 0$ for all $z$,
which implies (A.27) and concludes the proof. $\Box$

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