# Blowup solutions for a nonlinear heat equation involving a critical power
nonlinear gradient term

###### Abstract.

We consider the following exponential reaction-diffusion equation involving a
nonlinear gradient term:

$\partial_{t}U=\Delta U+\alpha|\nabla
U|^{2}+e^{U},\quad(x,t)\in\mathbb{R}^{N}\times[0,T),\quad\alpha>-1.$

We construct for this equation a solution which blows up in finite time $T>0$
and satisfies some prescribed asymptotic behavior. We also show that the
constructed solution and its gradient blow up in finite time $T$
simultaneously at the origin, and find precisely a description of its final
blowup profile. It happens that the quadratic gradient term is critical in
some senses, resulting in the change of the final blowup profile in comparison
with the case $\alpha=0$. The proof of the construction inspired by the method
of Merle and Zaag in 1997, relies on the reduction of the problem to a finite
dimensional one, and uses the index theory to conclude. One of the major
difficulties arising in the proof is that outside the blowup region, the
spectrum of the linearized operator around the profile can never be made
negative. Truly new ideas are needed to achieve the control of the outer part
of the solution. Thanks to a geometrical interpretation of the parameters of
the finite dimensional problem in terms of the blowup time and the blowup
point, we obtain the stability of the constructed solution with respect to
perturbations of the initial data.

###### Key words and phrases:

Finite-time blowup, Blowup profile, Stability, Semilinear heat equations.

###### 1991 Mathematics Subject Classification:

Primary: 35K58, 35K55; Secondary: 35B40, 35B44.

H. Zaag is supported by the ERC Advanced Grant no. 291214, BLOWDISOL and by
the ANR project ANAÉ ref. ANR-13-BS01-0010-03.
—————–

Tej-Eddine Ghoul†, Van Tien Nguyen† and Hatem Zaag∗

†New York University in Abu Dhabi, P.O. Box 129188, Abu Dhabi, United Arab
Emirates.

∗Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), F-93430,
Villetaneuse, France.

##  1\. Introduction.

We are interested in the following nonlinear heat equation:

$\left\\{\begin{array}[]{rcl}\partial_{t}U&=&\Delta U+\alpha|\nabla
U|^{r}+e^{U},\\\ U(0)&=&U_{0},\end{array}\right.$ (1.1)

where $U(t):x\in\mathbb{R}^{N}\to U(x,t)\in\mathbb{R}$, $\Delta$ and $\nabla$
stand for the Laplacian and the gradient in $\mathbb{R}^{N}$ with $N\geq 1$,

$r=2\quad\text{and}\quad\alpha>-1.$

Equation (1.1) can be viewed as the limiting case of the following critical
splitting as $p\to+\infty$, which was introduced by Chipot and Weissler
[CW89]:

$\partial_{t}U=\Delta U+\alpha|\nabla
U|^{r}+|U|^{p-1}U,\quad\text{with}\;\;p>1\;\text{and}\;r=\frac{2p}{p+1}.$
(1.2)

The Cauchy problem for (1.1) can be solved in several functional spaces
$\mathcal{F}$, for example $\mathcal{F}=W^{1,\infty}(\mathbb{R}^{N})$ or in a
special affine space $\mathcal{F}=\mathcal{H}_{a}$ for some positive constant
$a$ with

$\mathcal{H}_{a}=\\{u\in\psi+W^{1,\infty}(\mathbb{R}^{N})\;\text{with}\;\psi(x)=-\ln(1+a|x|^{2})\\}.$
(1.3)

In particular, the problem (1.1) has a unique classical solution
$U(t)\in\mathcal{F}$ defined on $[0,T)$ with $T\leq+\infty$ (see Remark 1.3
for more details). In the case $T=+\infty$, we have the global existence for
$U(t)$; on the contrary, i.e $T<+\infty$, we say that the solution $U(t)$
blows up in finite time $T$, namely

$\displaystyle\lim_{t\to T}\|U(t)\|_{W^{1,\infty}(\mathbb{R}^{N})}=+\infty,$
or $\displaystyle\lim_{t\to
T}\|U(t)-\psi\|_{W^{1,\infty}(\mathbb{R}^{N})}=+\infty\;\;\text{in the second
case.}$

As for the case of equation (1.1), the value $r=2$ is a critical exponent for
different reasons ($r<2$ and $r>2$ correspond to the sub-critical and super-
critical cases). One reason is that, when $r=2$, equation (1.1) is invariant
under the following transformation:

$\forall\lambda>0,\quad U_{\lambda}(x,t)=2\ln\lambda+U(\lambda
x,\lambda^{2}t),$ (1.4)

as for the equation without the gradient term, i.e $\alpha=0$. Recalling that
equation (1.2) is invariant under the transformation

$\forall\lambda>0,\quad U_{\lambda}(x,t)=\lambda^{\frac{2}{p-1}}U(\lambda
x,\lambda^{2}t).$ (1.5)

Let us now sketch the main results for the case of the equation

$\partial_{t}U=\Delta U+e^{U}.$ (1.6)

One of the first result for the finite-time blow up problem (1.6) is due to
Friedman and McLoed [FM85] who proved the following upper and lower bounds for
the blowup rate of $U$ under some conditions on the initial data,

$-c\leq U(0,t)+\ln(T-t)\leq C,$ (1.7)

for some $c,C$ positive (see also Berbenes and Eberly [BE89] for this
estimate).

The study of the blowup behavior for solution (1.6) is done through the
introduction of similarity variables

$W_{a}(y,s)=U(x,t)+\ln(T-t),\quad y=\frac{x-a}{\sqrt{T-t}},\quad s=-\ln(T-t),$
(1.8)

where $a$ may or not be a blowup point for $U$. From (1.6), we see that
$W_{a}$ satisfies the following equation: for all
$(y,s)\in\mathbb{R}^{N}\times[-\ln T,+\infty)$,

$\partial_{s}W_{a}=\Delta W_{a}-\frac{y}{2}\cdot\nabla W_{a}+e^{W_{a}}-1.$

Since $s\to+\infty$ as $t\to T$, the change of variables (1.8) converts any
question about the blowup of $U$ into one about the large time asymptotic of
$W_{a}$.

According to Berbenes and Eberly [BE89], we know that if $U(x,t)$ is a
solution of (1.6) which blows up at $x=a$ and $t=T$, then

$\lim_{t\to
T}\big{[}U(a+y\sqrt{T-t},t)+\ln(T-t)\big{]}=\lim_{s\to+\infty}W_{a}(y,s)=0,$
(1.9)

uniformly on compact sets $|y|\leq R$.

This estimate has been refined until the higher order by Bebernes and Briche
[BB92], Herrero and Velázquez [HV93] in one dimensional case. More precisely,
they classified the behavior of $W_{a}$ for $|y|$ bounded, and showed that one
of the following cases occurs:

$\sup_{|y|\leq
R}\left|W_{a}(y,s)+\frac{1}{4s}(y^{2}-2)\right|=o\left(\frac{1}{s}\right),$
(1.10)

or

$\sup_{|y|\leq R}|W_{a}(y,s)|=\mathcal{O}(e^{-\mu s})\quad\text{for some
$\mu>0$,}$

(the exponential convergence has been refined up to order $1$ in [HV93]). It
is remarkable that a similar result can be extended to higher dimensional
cases by using the technique of [Vel92] for the equation

$\partial_{t}U=\Delta U+|U|^{p-1}U.$ (1.11)

When (1.10) occurs, the authors of [BB92] established the following blowup
profile in the variable $z=\frac{y}{\sqrt{s}}$ (which is the intermediate
scale that separates the regular and singular parts)

$\lim_{t\to
T}\big{[}U(a+z\sqrt{(T-t)|\ln(T-t)|},t)+\ln(T-t)\big{]}=\lim_{s\to+\infty}W_{a}(z\sqrt{s},s)=\Phi_{0}(z),$
(1.12)

uniformly on compacts in $z$, where

$\Phi_{0}(z)=-\ln\left(1+\frac{|z|^{2}}{4}\right).$ (1.13)

Note that (1.12) was formally obtained in [Dol89], [Dol85] and [VGH91] by
means of the method of matched asymptotic expansions. The asymptotic behavior
(1.12) leads to the limiting profile in the $U(x,t)$ variable, in the sense
that $U(x,t)\to U^{*}_{0}(x)$ when $t\to T$ if $x\neq a$ and $x$ is in the
neighborhood of $a$ with

$U^{*}_{0}(x)\sim-2\ln|x-a|+\ln|\ln|x-a||+\ln 8.$ (1.14)

In [Bre90] and [Bre92], Bressan proved the existence of solutions to (1.6)
which blow up in finite time and verify the behavior (1.12). He also obtained
the stability of such blowup behavior with respect to small perturbations of
the the initial data. For more results related to equation (1.6), see Fila and
Pulkkinen [FP08], [Pul11] and the references therein.

By considering $\alpha\neq 0$ and $r=2$ in (1.1), we want to ask whether the
nonlinear gradient term appearing in the equation affects the blowup profile
of the case $\alpha=0$, i.e equation (1.6). Note that the case $r<2$ can be
considered as a perturbation of equation (1.6) because the nonlinear gradient
term has a sub-critical size in the sense that in the similarity variables
setting (1.8), equation (1.1) yields

$\partial_{s}W_{a}=\Delta W_{a}-\frac{y}{2}\cdot\nabla W_{a}+\alpha e^{-\delta
s}|\nabla W_{a}|^{r}+e^{W_{a}}-1,\quad\text{with}\;\;\delta=1-\frac{r}{2}.$
(1.15)

This gives another explanation to the fact that problem (1.1) is critical when
$r=2$. In fact, our problem is motivated by the work of Tayachi and Zaag
[TZ16] treated for equation (1.2). In that paper, the authors construct for
(1.2) a solution which blows up in finite time $T$ only at the origin and give
a sharp description of its blowup profile in the case where $\alpha>0$ and
$p>3$. The originality of [TZ16] lays in the fact that the constructed
solution does not exist in the case of the standard nonlinear heat equation
(1.11). In particular, their solution has a profile depending on the variable

$z=\frac{x}{|\ln(T-t)|^{\mu}\sqrt{T-t}}\quad\text{with}\quad\mu=\frac{p+1}{2(p-1)}>\frac{1}{2},$
(1.16)

which is different from the results for equation (1.11) by Herrero and
Velázquez [HV93], [Vel92], [Vel93], Bricmont and Kupianen [BK94], where the
blowup profiles depend on the reduced variables

$z=\frac{x}{|\ln(T-t)|^{\frac{1}{2}}\sqrt{T-t}}\quad\text{or}\quad
z=\frac{x}{(T-t)^{\frac{1}{2m}}}\;\;\text{where $m\geq 2$ is an integer.}$

This evidently shows the effect of the forcing gradient term in equation (1.2)
with $\alpha>0$ in the equation. It is worth to mention the work by Ebde and
Zaag [EZ11] where the authors considered equation (1.2) in the case where $r$
is sub-critical, i.e $r<\frac{2p}{p+1}$. They showed that the involved
nonlinear gradient term in (1.2) does not affect to the final blowup profile,
leading to the same result (construction and stability of a solution whose
blowup profile depends on the variable
$z=\frac{x}{|\ln(T-t)|^{\frac{1}{2}}\sqrt{T-t}}$) for the standard nonlinear
heat equation (1.11) obtained in [BK94] and [MZ97b]. For more results related
to (1.2), see Galaktionov and Vázquez [GV93], [GV96], Snoussi, Tayachi and
Weissler [ST07], Chipot and Weissler [CW89], Fila [Fil91], Souplet, Tayachi
and Weissler [STW96], [Sou01], [ST01], Chlebík, Fila and Quittner [CFQ03].

In this paper, we aim at constructing a solution for equation (1.1) and giving
precisely the description of its blowup profile. Our main result is the
following:

######  Theorem 1.1 (Existence of a blowup solution for equation (1.1) with
the description of its profile).

Let $\alpha>-1$ and $r=2$, there exists $T_{0}\in(0,1)$ such that for each
$T\in(0,T_{0}]$, equation (1.1) has a solution $U(x,t)$ such that $U$ and
$\nabla U$ blow up in finite time $T$ at the origin simultaneously. Moreover,

  * (i)

For all $t\in[0,T)$,

$\left\|(T-t)e^{U(x,t)}-e^{\Phi_{\alpha}\left(z\right)}\right\|_{W^{1,\infty}(\mathbb{R}^{N})}\leq\frac{C}{\sqrt{|\ln(T-t)|}},$
(1.17)

where $C$ is some positive constant and

$\Phi_{\alpha}(z)=-\ln\left(1+\frac{|z|^{2}}{4+4\alpha}\right),\quad
z=\frac{x}{\sqrt{(T-t)|\ln(T-t)|}}.$ (1.18)

  * (ii)

The functions $e^{U}$ and $\nabla U$ blow up at the origin and only there.

  * (iii)

There exists a function $U^{*}_{\alpha}(x)$ defined on
$\mathbb{R}^{N}\setminus\\{0\\}$ such that $U(x,t)\to U^{*}_{\alpha}(x)$ and
$\nabla U(x,t)\to\nabla U^{*}_{\alpha}(x)$ as $t\to T$, uniformly on compact
sets of $\mathbb{R}^{N}\setminus\\{0\\}$, where

$U^{*}_{\alpha}(x)\sim\ln\left(\frac{(8+8\alpha)|\ln|x||}{|x|^{2}}\right)\quad\text{as
$|x|\to 0$,}$ (1.19)

and

$|\nabla U^{*}_{\alpha}(x)|\leq\frac{C}{|x|}\quad\text{as $|x|\to 0$.}$ (1.20)

######  Remark 1.2.

Although our problem can be considered as the limiting case of (1.2) treated
in [TZ16], the construction is far from a simple adaptation of the method in
[TZ16]. Compared to the paper [TZ16], our work has the following major
difficulty: the spectrum of the linearized operator around the profile outside
the blowup region can never be made negative. This requires new ideas as far
as the control of the outer component is concerned, and this is one of the
main novelties of our approach. Note that the linearized operator in the power
case (equation (1.2)) behaves as one with fully negative spectrum, which
greatly simplifies the analysis in the outer region.

######  Remark 1.3.

The initial data for which equation (1.1) has a solution blowing up in finite
time $T$ at the origin and verifying the behavior (1.17) and (1.19) is given
by formulation (3.12) below. Note that the initial datum $U_{0}(x)$ defined in
(3.12) does not belong to $\in L^{\infty}(\mathbb{R}^{N})$. However, the local
in time Cauchy problem for equation (1.1) is well-posed in $\mathcal{H}_{a}$,
where $\mathcal{H}_{a}$ is defined by (1.3). Indeed, if we define
$\tilde{U}(x,t)$ by $U(x,t)=\tilde{U}(x,t)+\psi(x)$, equation (1.1) is
equivalent to

$\partial_{t}\tilde{U}=\Delta\tilde{U}+2\alpha\nabla\psi\cdot\nabla\tilde{U}+\alpha|\nabla\tilde{U}|^{2}+e^{\psi}e^{\tilde{U}}+\alpha|\nabla\psi|^{2}+\Delta\psi.$
(1.21)

Since $\|e^{\psi}\|_{L^{\infty}(\mathbb{R}^{N})}$,
$\|\nabla\psi\|_{L^{\infty}(\mathbb{R}^{N})}$ and
$\|\Delta\psi\|_{L^{\infty}(\mathbb{R}^{N})}$ are bounded, we see by classical
arguments that equation (1.21) is well-posed in
$W^{1,\infty}(\mathbb{R}^{N})$. Thus, there exists a unique maximum solution
on $[0,T)$ for equation (1.1) with $T\leq+\infty$ corresponding to the initial
datum $U_{0}$ given by (3.12).

######  Remark 1.4.

The derivation of the profile $\Phi_{\alpha}$ can be understood through a
formal analysis given in Section 2.1 below. We impose the condition
$\alpha>-1$ in order to have a bounded profile.

######  Remark 1.5.

From part $(iii)$ of Theorem 1.1, we see that the point $x=0$ is an isolated
blowup point. In order to prove this result, we need to establish a new ”no
blowup under some threshold” of a parabolic inequality with a nonlinear
gradient term, see Proposition 3.6 below.

######  Remark 1.6.

The results stated in Theorem 1.1 also hold for more general equations than
(1.1), that is

$\partial_{t}U=\Delta U+\alpha|\nabla U|^{2}+e^{U}+F(U,\nabla U),$

where $F:\mathbb{R}\times\mathbb{R}^{N}\to\mathbb{R}$ is Lipschitz and
satisfies

$|F(u,v)|\leq C(1+e^{r_{1}u}+|v|^{r_{2}}),\quad
r_{1}\in[0,1),\;\;r_{2}\in[0,2).$

Indeed, the nonlinear term $F(U,\nabla U)$ can be considered as subcritical
(in size) in comparison with the main nonlinear terms in the equation,
therefore, a simple perturbation of our proof can be extended to this case
without difficulties.

######  Remark 1.7.

We see from part $(i)$ of Theorem 1.1 that the constructed non self-similar
solution has the profile depending on the variable

$z=\frac{x}{\sqrt{(T-t)|\ln(T-t)|}},$

which is the same for the case of equations (1.1) and (1.2) without the
nonlinear gradient term ($\alpha=0$). Unlike the work done by Tayachi and Zaag
in [TZ16], where the authors constructed for equation (1.2) with $\alpha>0$
and $p>3$ a stable blowup solution having a blowup profile depending on the
variable (1.16) and satisfying

$U(x,t)\sim\left(p-1+b|z|^{2}\right)^{-\frac{1}{p-1}},$

where $b=b(\alpha,p,N)$ is a positive constant which goes to $+\infty$ as
$\alpha\to 0$. Thus, their result does not recover the result obtained in
[BK94] and [MZ97b] for the case $\alpha=0$ where $\mu=\frac{1}{2}$ and $b>0$;
on the contrary, our result obtained for equation (1.1) recovers the case
$\alpha=0$.

######  Remark 1.8.

By the space translation invariance of equation (1.1), the result of Theorem
1.1 can be stated for any arbitrary considered point $x_{0}\in\mathbb{R}^{N}$
as a blowup point of the solution, in particular, we have for all $t\in[0,T)$,

$\left\|(T-t)e^{U(x,t)}+\left(1+\frac{|x-x_{0}|^{2}}{(4+4\alpha)(T-t)|\ln(T-t)|}\right)^{-1}\right\|_{W^{1,\infty}(\mathbb{R}^{N})}\leq\frac{C}{\sqrt{|\ln(T-t)|}},$

and the final blowup profile satisfies

$U_{\alpha}^{*}(x)\sim\ln\left(\frac{(8+8\alpha)|\ln|x-x_{0}||}{|x-x_{0}|^{2}}\right)\quad\text{as}\;\;|x-x_{0}|\to
0.$

######  Remark 1.9.

We conjecture that the identity (1.19) also holds after differentiating,
namely that

$|\nabla U^{*}_{\alpha}(x)|\sim\frac{c^{*}}{|x|},\quad\text{as}\quad|x|\to 0,$

where $c^{*}$ is some positive constant. We strongly believe that with some
better refined estimate of (1.17), we could obtain such a final profile for
the gradient solution. Unfortunately, we are only able to derive in this work
a weaker result given in (1.20).

######  Remark 1.10.

In comparison with the work in [GV96], where the authors deal with the study
of the behavior of the profile near blowup of the following quasilinear
parabolic equation

$\partial_{t}U=\Delta U+|\nabla U|^{2}+U^{\beta},\quad\beta>1,$ (1.22)

our problem can be considered as the limit case of equation (1.22) as
$\beta\to+\infty$, and our result obtained $(ii)$ of Theorem 1.1 is quite
different from that of [GV96]. More precisely, they showed that in the case of
the single point blowup in the equation (1.22) and under the radially
symmetric, i.e $U(x,t)=U(r,t)$ with $r=|x|$, and nonincreasing assumption (see
Theorem 8, page 36 in [GV96]), the profile near blowup is given by

$\forall\beta>2,\quad U(r,T)\sim
C_{*}r^{-\frac{2}{\beta-2}}\quad\text{as}\;\;r\to 0,$

where $C_{*}=C_{*}(\beta)$ is a positive appreciated constant.

As in [MZ97b] and [TZ16] (see also [Zaa02], [MZ08]), it is possible to make
the interpretation of the finite-dimensional variable in terms of the blowup
time and the blowup point. This allows us to derive the stability of the
profile $\Phi_{\alpha}$ in Theorem 1.1 with respect to perturbations of the
initial data. More precisely, we have the following:

######  Theorem 1.11 (Stability of the blowup profile (1.17)).

Let us denote by $\hat{U}(x,t)$ the solution constructed in Theorem 1.1 and by
$\hat{T}$ its blowup time. Then, there exists a neighborhood $\mathcal{V}_{0}$
of $\hat{U}(x,0)$ in $\mathcal{H}_{a}$ defined in (1.3) such that for any
$U_{0}\in\mathcal{V}_{0}$, equation (1.1) has a unique solution $U(x,t)$ with
initial data $U_{0}$, and $U(x,t)$ blows up in finite time $T(U_{0})$ at point
$a(U_{0})$. Moreover, estimate (1.17) is satisfied by $U(x-a,t)$ and

$T(U_{0})\to\hat{T},\quad a(U_{0})\to 0\quad\text{as $U_{0}\to\hat{U}_{0}$ in
$\mathcal{H}_{a}$}.$

######  Remark 1.12.

As in [Bre92], we conjecture that the blowup pattern just described in part
$(i)$ of Theorem 1.1 is generic, i.e there exists an open, everywhere dense
set $\mathcal{V}_{0}$ of initial data whose corresponding solutions either
converge to a steady state, or blowup in finite time at a single point,
according to the estimate (1.17). Up to our knowledge, the only proof for the
genericity is given by Herrero and Velázquez [HV92] for equation (1.2) without
the nonlinear gradient term ($\alpha=0$) in the one dimensional case.

######  Remark 1.13.

We will not give the proof of Theorem 1.11 because the stability result
follows from the reduction to a finite dimensional case as in [MZ97b] (see
Theorem 2 and its proof in Section 4) and [TZ16] (see Theorem 9 and its proof
in Section 6) with the same argument. Hence, we only prove the exsitence
result (Theorem 1.1) and kindly refer the reader to [MZ97b] and [TZ16] for the
proof of the stability.

Let us now give the main ideas of the proof of Theorem 1.1. Note that our
proof is quite different from that of [Bre92] (see also [Bre90]) treated for
equation (1.6). Here, we follow the method developed by Bricmont and Kupiainen
[BK94], and modified by Merle and Zaag [MZ97b] for the construction of a
stable blowup solution to equation (1.11). Note that the method of [MZ97b] has
been proved to be successful for various situations including parabolic and
hyperbolic equations. For the parabolic equations, we would like to mention
the works by Masmoudi and Zaag [MZ08] (see also the earlier work by Zaag
[Zaa98]) for the complex Ginzburg-Landau equation with no gradient structure,
by Nguyen and Zaag [NZ16a], [NZ16b] for a logarithmically perturbed nonlinear
heat equation and for a refined blowup profile for equation (1.11), or by
Nouaili and Zaag [NZ15] for a non-variational complex-valued semilinear heat
equation, by Ghoul, Nguyen and Zaag [GNZ16] for a non-variational parabolic
system. There are also the cases for the construction of multi-solitons for
the semilinear wave equation in one space dimension by Côte and Zaag [CZ13].

Our goal is to construct for equation (1.1) a solution $U(x,t)$ which blows up
in finite time $T$ and verifies the behaviors (1.17). The proof is performed
in the framework of the similarity variables defined in (1.8). By the space
translation invariance of equation (1.1), we may assume $a=0$ in (1.8) and
write $W=W_{a}$ for simplicity of the notation. We recall from (1.15) that
when $r=2$, $W$ solves the equation

$\partial_{s}W=\Delta W-\frac{y}{2}\cdot\nabla W+\alpha|\nabla
W|^{2}+e^{W}-1.$ (1.23)

If we introduce

$Z=e^{W},$

then $Z$ solves

$\partial_{s}Z=\Delta Z-\frac{y}{2}\cdot\nabla Z+(\alpha-1)\frac{|\nabla
Z|^{2}}{Z}+Z^{2}-Z.$ (1.24)

Constructing a solution for (1.1) satisfying (1.17) reduces to the
construction of a solution for (1.24) such that

$\begin{array}[]{ll}Q(y,s)&=e^{W(y,s)}-e^{\Phi_{\alpha}\left(\frac{y}{\sqrt{s}}\right)}\equiv
Z(y,s)-\Gamma_{\alpha}(y,s)\to 0,\end{array}\quad\text{as}\;s\to+\infty,$
(1.25)

where $Q$ satisfies the equation

$\partial_{s}Q=\big{(}\mathscr{L}+V(y,s)\big{)}Q+Q^{2}+(\alpha-1)\frac{|\nabla
Q+\nabla\Gamma_{\alpha}|^{2}}{Q+\Gamma_{\alpha}}+R(y,s),$ (1.26)

and

$\mathscr{L}=\Delta-\frac{y}{2}\cdot\nabla+1,\quad
V=2(\Gamma_{\alpha}-1),\quad\|R(y,s)\|_{L^{\infty}}\leq\frac{C}{s}.$

Satisfying such a property is guaranteed by a condition that $Q(s)$ belongs to
some set $\mathcal{V}_{A,K_{0}}(s)$ which shrinks to $0$ as $s\to+\infty$ (see
item $(i)$ in Definition 3.1 below for an example). Let us insist on the fact
that we do not work with equation (1.23), but with equation (1.24) (see
explaination below). Since the linearization of equation (1.24) around the
profile $e^{\Phi_{\alpha}}\equiv\Gamma_{\alpha}$ gives $N+1$ positive modes,
$\frac{N(N+1)}{2}$ zero modes, and an infinite dimensional negative part, we
can use the method of [BK94] and [MZ97b] which relies on two arguments:

  * -

The use of the bounding effect of the heat kernel to reduce the problem of the
control of $Q(s)$ in $\mathcal{V}_{A,K_{0}}(s)$ to the control of its positive
modes (in fact, we control a modified version of $Q$ in
$\mathcal{V}_{A,K_{0}}$; see Proposition 3.5). Let us insist on the fact that
if we linearize equation (1.23) around $\Phi_{\alpha}$, say that
$q=W-\Phi_{\alpha}$, we could not able to control $q$ outside the blowup
region because in this region the linear part of the equation satisfied by $q$
does not have a fully negative spectrum. In the contrary, the spectrum of the
linear operator of the equation satisfied by $Q$ is negative outside the
blowup region, which makes the control of $Q$ in that region easily.

  * -

The control of the $(N+1)$ positive modes thanks to a topological argument
based on index theory (see the arguments at page 3.5).

(Note that the topological argument is also used in [Bre92] to solve the
finite dimensional problem, however, the approach of the reduction of the
problem to a finite dimensional one in our proof is totally different from
that in [Bre92]).

Since the gradient term in the equation (1.1) in the critical case contributes
to the change of the blowup profile and given the fact that the blowup profile
$\Phi_{\alpha}$ is different from the one considered in [BK94] and [MZ97b],
our proof truly requires crucial modifications of the methods of [BK94] and
[MZ97b] and special arguments in order to handle the nonlinear gradient term.
These modifications lay in the following places:

  * (i)

We no longer work with the linearization of equation (1.23) around
$\Phi_{\alpha}$, but instead the equation (1.26) satisfied by $Q(y,s)$. Note
that linearizing (1.23) around the profile $\Phi_{\alpha}$ generates the
linear operator $\mathscr{L}+\tilde{V}$ which behaves similarly as in the
power case (equation (1.2)) inside the blowup region, but there is a quite
difference outside the blowup region. Indeed, Bricmont and Kupiainen’s
approach for equation (1.2) gives that the spectrum of $\mathscr{L}+\tilde{V}$
in the outer region is controlled by $-\frac{1}{p-1}+\epsilon$ which can be
made negative by taking $\epsilon$ small enough. Whereas, the same estimate
for equation (1.1) gives that the spectrum of $\mathscr{L}+\tilde{V}$ is
controlled by $0+\epsilon$, which can never be made negative. In order to
overcome this difficulty, we no longer work with the linearization of equation
(1.23) around $\Phi_{\alpha}$, but instead the equation (1.26) satisfied by
$Q$. In fact, the spectrum of the linear part of equation (1.26) is fully
negative in the outer region, which make it easy to control. However, this
manner gives additionally a term of the form $\frac{|\nabla
Q+\nabla\Gamma_{\alpha}|^{2}}{Q+\Gamma_{\alpha}}$ (see (1.26)) which is needed
new ideas to achieve the control.

  * (ii)

Defining the shrinking set $\mathcal{S}^{*}$ (see Definition 3.1 below) to
trap the solution. Note that our definition of $\mathcal{S}^{*}$ is defferent
from the one in [MZ97b] designed for the standard nonlinear heat equation
(1.2) with $\alpha=0$. Note also that equation (1.26) is almost the same as in
[MZ97b], except for the nonlinear gradient term which causes serious
difficulties in the analysis. In [MZ97b], the authors introduced estimates of
$Q$ in the blowup region $|y|\leq K_{0}\sqrt{s}$, and in the regular region
$|y|\geq K_{0}\sqrt{s}$. However, the estimates in the region $|y|\geq
K_{0}\sqrt{s}$ imply smallness of $Q$ only, and do not allow any control of
the nonlinear gradient term in this region. In other words, the analysis based
on the method of [MZ97b], that is to estimate the solution in the
$z=\frac{y}{\sqrt{s}}$ variable is not sufficient and must be improved. In
particular, we introduce estimates of $Q$ in three regions in a different
variable scale, which follows the approach of [MZ97a] using for a finite time
quenching problem of vortex reconnection with the boundary. More precisely, in
the blowup region, i.e. $|x|\leq K_{0}\sqrt{(T-t)|\ln(T-t)|}$, we do an
asymptotic analysis around the profile $\Gamma_{\alpha}$ through equation
(1.26). In the intermediate region, i.e.
$\frac{K_{0}}{4}\sqrt{(T-t)|\ln(T-t)|}\leq|x|\leq\epsilon_{0}$, we control the
solution by using classical parabolic estimates through introducing a rescaled
function of $U$ (see (3.4) below). In the regular region, i.e.
$|x|\geq\frac{\epsilon_{0}}{4}$, we estimate directly $U$. The estimates in
the intermediate and regular regions allow us to control the nonlinear
gradient term appearing in equation (1.26).

The derivation of the final profile $U_{\alpha}^{*}(x)$ stated in part $(ii)$
of Theorem 1.1 uses the method of [Zaa98] and [Mer92]. In particular, we need
to establish a new no blowup under some threshold criterion for a parabolic
inequality with a nonlinear gradient term, whose proof follows ideas given in
[GK89] (see Proposition 3.6).

The organization of the rest of this paper is as follows:
\- In Subsection 2.1, we first explain formally how we obtain the profile
$\Phi_{\alpha}$ and give a suggestion for an appreciated profile to be
linearized around. In Subsection 2.2, we give a formulation of the problem in
order to justify the formal argument.
\- In Section 3, we give all the arguments of the proof of part $(i)$ of
Theorem 1.1 assuming technical results, which are left to the next section.
\- In Section 4, we give the proof of the technical results used in the
existence’s proof, that is the proof of Proposition 3.5. We divide its proof
in two subsections. In Subsection 4.1, we derive a priori estimates of $U(t)$
in the set $\mathcal{S}^{*}(t)$. In Subsection 4.2, we show that all the
estimates given in Definition 3.1 of $\mathcal{S}^{*}$ can be improved, except
for the positive modes, which concludes the proof of the reduction to a finite
dimensional one.
\- In Appendix A, we give properties of the shrinking set $\mathcal{S}^{*}(t)$
to trap the solution as well as properties of the initial data corresponding
to the blowup solution described in Theorem 1.1. In Appendix B, we give the
proof of a no blowup under some threshold criterion for a parabolic inequality
with a nonlinear gradient term, which is an ingredient in the proof of the
existence of the final blowup profile.

##  2\. Formulation of the problem.

###  2.1. A formal analysis.

In this subsection, we use matching asymptotics to formally derive the blowup
behavior of the solution to (1.1). More precisely, we will explain how to
deduce the following behavior

$U(x,t)+\ln(T-t)\sim\Phi_{\alpha}\left(\frac{x}{\sqrt{(T-t)|\ln(T-t)|}}\right)\quad\text{as}\;\;t\to
T,$ (2.1)

where $\Phi_{\alpha}$ is defined by (1.18).

In the similarity variables setting (1.8), justifying (2.1) is equivalent to
showing that

$W(y,s)\sim\Phi_{\alpha}\left(\frac{y}{\sqrt{s}}\right)\quad\text{as}\;\;s\to+\infty,$

where $W$ satisfies (1.23). Let us rewrite the equation on $W$ as follows:

$\partial_{s}W=\mathscr{L}W+\alpha|\nabla W|^{2}+Q(W),$ (2.2)

where

$\mathscr{L}=\Delta-\frac{y}{2}\cdot\nabla+1,$ (2.3)

and

$Q(W)=e^{W}-1-W.$

Note that we have

$\big{|}Q(W)-\frac{1}{2}W^{2}\big{|}\leq C|W|^{3}\quad\text{as}\;\;W\to 0.$

The linear operator $\mathscr{L}$ is self-adjoint in
$L^{2}_{\rho}(\mathbb{R}^{N})$, where $L^{2}_{\rho}(\mathbb{R}^{N})$ is the
weighted $L^{2}$ space associated with the weight $\rho$ defined by

$\rho(y)=\frac{1}{(4\pi)^{N/2}}e^{-\frac{|y|^{2}}{4}}.$

It can be shown that the spectrum of $\mathscr{L}$ is explicitly given by (see
[FK92] for instance)

$\text{spec}(\mathscr{L})=\left\\{1-\frac{n}{2},n\in\mathbb{N}\right\\}.$

For $N=1$, all the eigenvalues are simple and the eigenfunctions are dilations
of Hermite polynomials: the eigenvalue $1-\frac{n}{2}$ corresponds to the
following eigenfunction:

$h_{n}(y)=\sum_{i=0}^{\big{[}n/2\big{]}}\frac{n!}{i!(n-2i)!}(-1)^{i}y^{n-2i}.$
(2.4)

The first three eigenfunctions are

$h_{0}(y)=1,\quad h_{1}(y)=y,\quad h_{2}(y)=y^{2}-2.$

Notice that $h_{n}$ satisfies

$\int_{\mathbb{R}}h_{n}(\xi)h_{m}(\xi)\rho(\xi)d\xi=2^{n}n!\delta_{n,m},\quad\mathscr{L}h_{n}=\big{(}1-\frac{n}{2}\big{)}h_{n}.$
(2.5)

We also introduce $k_{n}(y)=\|h_{n}\|^{-2}_{L^{2}_{\rho}}h_{n}(y)$. In higher
dimensions, the eigenfuntions are formed by taking products of the polynomials
$\\{h_{k}\\}_{k=0}^{\infty}$, namely that for
$n=(n_{1},\cdots,n_{N})\in\mathbb{N}^{N}$, the eigenfunction corresponding to
$1-\frac{|n|}{2}$ ($|n|=n_{1}+\cdots+n_{N}$) is

$h_{n}(y)=h_{n_{1}}(y_{1})\cdots h_{n_{N}}(y_{N}).$ (2.6)

Since the eigenfunctions of $\mathscr{L}$ span the whole space
$L^{2}_{\rho}(\mathbb{R}^{N})$, we can expand the solution $W$ as follows:

$W(y,s)=\sum_{m\in\mathbb{N}^{N}}^{\infty}W_{m}(s)h_{m}(y).$

For simplicity, let us assume that the solution is radially symmetric in $y$.
Since $h_{m}$ with $m\geq 3$ correspond to negative eigenvalues of
$\mathscr{L}$, we may consider that

$W(y,s)=W_{0}(s)+W_{2}(s)(|y|^{2}-2N),$ (2.7)

with $W_{0}(s),W_{2}(s)\to 0$ as $s\to+\infty$ (for simplicity in the
notation, we write $W_{2}(s)$ instead of $W_{(2,0,\cdots,0)}(s)$).

Writing $\alpha|\nabla W|^{2}=4\alpha(h_{2}+2Nh_{0})W_{2}^{2}$, multiplying
equation (2.2) by $k_{0}\rho$ and $k_{2}\rho$ respectively and integrating
over $\mathbb{R}^{N}$, we derive the following ODE system for $W_{0}$ and
$W_{2}$:

$\displaystyle W_{0}^{\prime}$
$\displaystyle=W_{0}+\frac{1}{2}W_{0}^{2}+N(4+8\alpha)W_{2}^{2}+\mathcal{O}(|W_{0}|^{3}+|W_{2}|^{3})$
(2.8) $\displaystyle W_{2}^{\prime}$
$\displaystyle=0+W_{0}W_{2}+(4+4\alpha)W_{2}^{2}+\mathcal{O}(|W_{0}|^{3}+|W_{2}|^{3}).$
(2.9)

Assuming that

$|W_{0}(s)|=o(|W_{2}(s)|)\quad\text{as}\quad s\to+\infty,$ (2.10)

we then rewrite equation (2.9) as follows:

$W_{2}^{\prime}=(4+4\alpha)W_{2}^{2}+o(|W_{2}|^{2})\quad\text{as}\quad
s\to+\infty,$

which yields

$W_{2}(s)=-\frac{1}{(4+4\alpha)s}+o\left(\frac{1}{s}\right)\quad\text{as}\quad
s\to+\infty.$

From equation (2.8) and the above formula of $W_{2}$, we have
$W_{0}^{\prime}=W_{0}+\mathcal{O}\left(\frac{1}{s^{2}}\right)$, which gives

$|W_{0}(s)|=\mathcal{O}\left(\frac{1}{s^{2}}\right)\quad\text{as}\quad
s\to+\infty.$

From the above formulas of $W_{0}$ and $W_{2}$, we have by equation (2.9),

$W_{2}^{\prime}=(4+4\alpha)W_{2}^{2}+\mathcal{O}\left(\frac{1}{s^{3}}\right)\quad\text{as}\quad
s\to+\infty,$

hence,

$W_{2}(s)=-\frac{1}{(4+4\alpha)s}+\mathcal{O}\left(\frac{\ln
s}{s^{2}}\right)\quad\text{as}\quad s\to+\infty.$

Such $W_{0}$ and $W_{2}$ are compatible with the assumption (2.10). Therefore,
we write by (2.7),

$W(y,s)=-\frac{1}{4+4\alpha}\frac{y^{2}}{s}+\frac{N}{(2+2\alpha)s}+\mathcal{O}\left(\frac{\ln
s}{s^{2}}\right)\quad\text{as}\quad s\to+\infty,$ (2.11)

in $L^{2}_{\rho}$, and also uniformly in compact sets by standard parabolic
regularity.

Since (2.11) provides a relevant variable for blowup, namely
$z=\frac{y}{\sqrt{s}}$, we then try to search formally solutions of (1.23) of
the form

$\tilde{W}(y,s)=\sum_{j=0}^{\infty}\frac{1}{s^{j}}w_{j}(z),\quad
z=\frac{y}{\sqrt{s}},$ (2.12)

and compare elements of order $\frac{1}{s^{j}}$. For $j=0$ and $j=1$, we find
that

$-\frac{z}{2}\cdot\nabla w_{0}(z)+e^{w_{0}(z)}-1=0,$ (2.13)
$F(z):=\frac{z}{2}\cdot\nabla
w_{1}(z)-e^{w_{0}(z)}w_{1}(z)-\frac{z}{2}\cdot\nabla w_{0}(z)-\Delta
w_{0}(z)-\alpha|\nabla w_{0}(z)|^{2}=0.$ (2.14)

Recalling that $\tilde{W}\to 0$ as $s\to+\infty$ on every compact set, we
naturally impose the condition

$w_{0}(0)=0.$

Solving equation (2.13) with this condition, we then obtain by radial symmetry

$w_{0}(z)=-\ln(1+c_{0}|z|^{2}),$

for an integration constant $c_{0}$. Because we want a bounded solution, then
it requires $c_{0}>0$. From the Taylor expansion for $|y|$ bounded, we write

$\tilde{W}(y,s)\sim-\ln\left(1+c_{0}\frac{|y|^{2}}{s}\right)\sim-
c_{0}\frac{|y|^{2}}{s}-\frac{c_{0}^{2}}{2}\frac{|y|^{4}}{s^{2}}+\cdots,$

from which and (2.11), we find that the coefficient
$c_{0}=\frac{1}{4+4\alpha}$. Thus,

$w_{0}(z)=-\ln\left(1+\frac{|z|^{2}}{4+4\alpha}\right).$

Substituting this formula into (2.14) and evaluating $F$ at $z=0$, we obtain

$w_{1}(0)=2Nc_{0}=\frac{N}{2+2\alpha},$

which matches with the expansion (2.11).

In conclusion, the first term in the expansion (2.12) of $\tilde{W}$ is
precisely the profile function $\Phi_{\alpha}$ as expected in (2.1).

###  2.2. Transformation of the problem.

In this subsection, we set up the problem of constructing a solution $U$ for
equation (1.1) which blows up in finite time $T$ only at the origin and
satisfies (1.17). In particular, we want to prove for suitable initial data
$U_{0}$ of (1.1) that

$\lim_{t\to
T}\left\|(T-t)e^{U(z\sqrt{(T-t)|\ln(T-t)|},t)}-e^{\Phi_{\alpha}(z)}\right\|_{W^{1,\infty}(\mathbb{R}^{N})}=0,$
(2.15)

where $\Phi_{\alpha}$ is defined by (1.18). Or, in the self-similar setting
(1.8), our aim is to construct initial data $W(s_{0})$ such that the equation
(1.23) has a solution $W(y,s)$ defined for all
$(y,s)\in\mathbb{R}^{N}\times[s_{0},+\infty)$ and satisfies

$\lim_{s\to+\infty}\left\|e^{W(y,s)}-e^{\Phi_{\alpha}\left(\frac{y}{\sqrt{s}}\right)}\right\|_{W^{1,\infty}(\mathbb{R}^{N})}=0.$
(2.16)

In the previous subsection, we formally obtain for $W(y,s)$ an expansion of
the form $\sum_{j=0}^{+\infty}\frac{1}{s^{j}}w_{j}(\frac{y}{\sqrt{s}})$ with
$w_{0}=\Phi_{\alpha}$ and $w_{1}$ satisfying (2.14). Hence, we will not
linearize $W$ around $\Phi_{\alpha}$, but we will study the difference
$W(y,s)-w_{0}(\frac{y}{\sqrt{s}})-\frac{1}{s}w_{1}(\frac{y}{\sqrt{s}})$. Since
the expression of $w_{1}$ is too complicated, we study instead
$W(y,s)-w_{0}(\frac{y}{\sqrt{s}})-\frac{1}{s}w_{1}(0)$. Let us introduce

$Q(y,s)=e^{W(y,s)}-\psi_{\alpha}(y,s)\equiv
Z(y,s)-\psi_{\alpha}(y,s)\;\;\text{where}\;\;\psi_{\alpha}(y,s)=e^{\frac{N}{(2+2\alpha)s}}e^{\Phi_{\alpha}\left(\frac{y}{\sqrt{s}}\right)},$
(2.17)

where $\Phi_{\alpha}$ is defined in (1.18). Then, we see from (1.24) that $Q$
satisfies

$\partial_{s}Q=\big{(}\mathscr{L}+V(y,s)\big{)}Q+Q^{2}+G(Q)+R(y,s),$ (2.18)

where $\mathscr{L}$ is defined by (2.3) and

$\displaystyle V(y,s)$ $\displaystyle=2(\psi_{\alpha}(y,s)-1),$ (2.19)
$\displaystyle G(Q)$ $\displaystyle=(\alpha-1)\left[\frac{|\nabla
Q+\nabla\psi_{\alpha}|^{2}}{Q+\psi_{\alpha}}-\frac{|\nabla\psi_{\alpha}|^{2}}{\psi_{\alpha}}\right],$
(2.20) $\displaystyle R(y,s)$
$\displaystyle=-\partial_{s}\psi_{\alpha}+\Delta\psi_{\alpha}-\frac{y}{2}\cdot\nabla\psi_{\alpha}+\psi_{\alpha}^{2}-\psi_{\alpha}+(\alpha-1)\frac{|\nabla\psi_{\alpha}|^{2}}{\psi_{\alpha}}.$
(2.21)

(Note that the equation satisfied by $Q$ is almost the same as in [MZ97b],
except the term $G(Q)$).

Satisfying (2.16) reduces to the construction of initial data $Q(y,s_{0})$
such that the equation (2.18) has a solution $Q(y,s)$ defined for all
$(y,s)\in\mathbb{R}^{N}\times[s_{0},+\infty)$ such that

$\lim_{s\to+\infty}\|Q(s)\|_{W^{1,\infty}(\mathbb{R}^{N})}=0.$ (2.22)

Our analysis uses the Duhamel formulation of equation (2.18): for each
$s\geq\sigma\geq s_{0}$, we have

$Q(s)=\mathcal{K}(s,\sigma)Q(\sigma)+\int_{\sigma}^{s}\mathcal{K}(s,\tau)\left[Q^{2}(\tau)+G(Q(\tau))+R(\tau)\right]d\tau,$
(2.23)

where $\mathcal{K}$ is the fundamental solution of the linear operator
$\mathscr{L}+V$ defined for each $\sigma>0$ and $s\geq\sigma$ by

$\partial_{s}\mathcal{K}(s,\sigma)=(\mathscr{L}+V)\mathcal{K}(s,\sigma),\quad\mathcal{K}(\sigma,\sigma)=\text{Identity}.$
(2.24)

Since we want to construct for (2.18) a solution $Q$ satisfying (2.22) and the
fact that $\|R(s)\|_{L^{\infty}}\leq\frac{C}{s}$ for $s$ large, it is then
reasonable to think that the dynamics of equation (2.18) are influenced by the
linear part, namely $\mathscr{L}+V$. The properties of the self-adjoint
operator $\mathscr{L}$ are given in the previous subsection. In particular,
$\mathscr{L}$ is predominant on all the modes, except on the null modes where
the terms $VQ$ and $G(Q)$ will play a crucial role. As for potential $V$, it
has two fundamental properties which will strongly influence our strategy:

  * (i)

$\|V(s)\|_{L^{2}_{\rho}(\mathbb{R}^{N})}\to 0$ as $s\to+\infty$. In practice,
the effect of $V$ in the blowup region $\\{|y|\leq K_{0}\sqrt{s}\\}$ is
regarded as a perturbation of the effect of $\mathscr{L}$ (except for the null
mode).

  * (ii)

outside of the blowup region, we have the following property: for all
$\epsilon>0$, there exist $K_{\epsilon}>0$ and $s_{\epsilon}$ such that

$|V(y,s)-(-1)|\leq\epsilon-1,\quad\forall s\geq s_{\epsilon},\;\forall|y|\geq
K_{\epsilon}\sqrt{s},$

which $1$ is the largest eigenvalue of the operator $\mathscr{L}$. Thus, the
spectrum of the linear operator $\mathscr{L}+V$ is fully negative, hence, the
control of $Q$ in $L^{\infty}$ ouside the blowup region will be done without
difficulties. Note that linearizing equation (1.23) around $\Phi_{\alpha}$,
say $q=W-\Phi_{\alpha}$, generates the linear operator $\mathscr{L}+\tilde{V}$
whose spectrum in the outer region is fully positive. This is the major reason
why we do not work with $q$, but with $Q=Z-\psi_{\alpha}$.

Since the behavior of the potential $V(y,s)$ inside and outside of the blowup
region is different, let us decompose $Q$ as follows:

$Q(y,s)=\chi(y,s)Q(y,s)+(1-\chi(y,s))Q(y,s)\equiv Q_{b}(y,s)+Q_{e}(y,s),$
(2.25)

where

$\chi(y,s)=\chi_{0}\left(\frac{|y|}{K_{0}\sqrt{s}}\right),$ (2.26)

and $\chi_{0}\in\mathcal{C}_{0}^{\infty}([0,+\infty))$ with
$\text{supp}(\chi_{0})\subset[0,2]$ and $\chi_{0}\equiv 1$ on $[0,1]$, and
$K_{0}>0$ is to be fixed large enough. Note that
$supp(Q_{b}(s))\subset\mathbf{B}(0,2K_{0}\sqrt{s})$ and
$supp(Q_{e}(s))\subset\mathbb{R}^{N}\setminus\mathbf{B}(0,K_{0}\sqrt{s})$.

Since the eigenfunctions of $\mathscr{L}$ span the whole space
$L^{2}_{\rho}(\mathbb{R}^{N})$, let us write

$\displaystyle Q_{b}(y,s)$ $\displaystyle=Q_{0}(s)+Q_{1}(s)\cdot
y+y^{T}Q_{2}(s)y-2tr(Q_{2}(s))+Q_{-}(y,s)$
$\displaystyle=Q_{0}(s)+Q_{1}(s)\cdot y+Q_{\bot}(y,s),$ (2.27)

where

$Q_{0}(s)=P_{0}(Q_{b})(y,s),\quad Q_{1}(s)\cdot y=P_{1}(Q_{b})(y,s),$
$Q_{-}(y,s)=P_{-}(Q_{b})(y,s)=\sum_{m\geq 3}P_{m}(Q_{b})(y,s),$
$Q_{\bot}(y,s)=P_{\bot}(Q_{b})(y,s)=\sum_{m\geq 2}P_{m}(Q_{b})(y,s),$

and $P_{m}$ is the projector on the eigenspace corresponding to the eigenvalue
$1-\frac{m}{2}$ defined by

$P_{m}(Q_{b})(y,s)=\sum_{\beta\in\mathbb{N}^{N},|\beta|=m}\frac{h_{\beta}(y)}{\|h_{\beta}\|^{2}_{L^{2}_{\rho}}}\int_{\mathbb{R}^{N}}h_{\beta}(y)Q_{b}(y,s)\rho(y)dy,$
(2.28)

where $h_{\beta}$ is defined by (2.6), and $Q_{2}(s)$ is a symmetric $(N\times
N)$ matrix defined by

$Q_{2}(s)=\int_{\mathbb{R}^{N}}Q_{b}(y,s)\mathcal{M}(y)\rho(y)dy,$ (2.29)

and

$\mathcal{M}(y)=\left\\{\frac{1}{8}y_{i}y_{j}-\frac{1}{4}\delta_{ij}\right\\}_{1\leq
i,j\leq N}.$ (2.30)

According to (2.25) and (2.27), we have

$Q(y,s)=Q_{0}(s)+Q_{1}(s)\cdot
y+y^{T}Q_{2}(s)y-2tr(Q_{2}(s))+Q_{-}(y,s)+Q_{e}(y,s),$ (2.31)

and

$Q(y,s)=Q_{0}(s)+Q_{1}(s)\cdot y+Q_{\bot}(y,s)+Q_{e}(y,s).$ (2.32)

The reader should keep in mind that $Q_{m},\,m=0,1,2$, $Q_{-}$ and $Q_{\bot}$
are the coordinates of $Q_{b}$ and not those of $Q$.

##  3\. Proof of the existence without technical details.

This section is devoted to the proof of Theorem 1.1, that is the existence of
the solution $U(t)$ of equation (1.1) satisfying

$\lim_{t\to
T}\left\|(T-t)e^{U(x,t)}-e^{\Phi_{\alpha}\left(\frac{x}{\sqrt{(T-t)|\ln(T-t)|}}\right)}\right\|_{W^{1,\infty}(\mathbb{R}^{N})}=0,$
(3.1)

and

$\lim_{t\to T}U(x,t)=U^{*}_{\alpha}(x),$ (3.2)

where $\Phi_{\alpha}$ is defined by (1.18) and $U^{*}_{\alpha}$ behaves as
(1.19) as $x\to 0$. According to the transformation (2.17), we see that the
proof of (3.1) is equivalent to proving the existence of the solution $Q(s)$
of equation (2.18) such that

$\lim_{s\to+\infty}\|Q(s)\|_{W^{1,\infty}(\mathbb{R}^{N})}=0.$ (3.3)

We shall give all the arguments of the proof for (3.3) and (3.2) assuming
technical results which are left to the following sections.

In order to prove (3.3), we use ideas given in Merle and Zaag [MZ97a] where
the authors suggested a modification of the argument of [MZ97b] for the
standard nonlinear heat equation (1.2) with $\alpha=0$. In particular, we
shall control the solution in three different regions covering
$\mathbb{R}^{N}$, defined as follows: for $K_{0}>0$, $\epsilon_{0}>0$ and
$t\in[0,T)$, we set

$\displaystyle\mathcal{D}_{1}(t)$ $\displaystyle=\left\\{x\;\Big{|}\;|x|\leq
K_{0}\sqrt{|\ln(T-t)|(T-t)}\right\\}$
$\displaystyle\quad\equiv\left\\{x\;\big{|}\;|y|\leq
K_{0}\sqrt{s}\right\\}\equiv\left\\{x\;\Big{|}\;|z|\leq K_{0}\right\\},$
$\displaystyle\mathcal{D}_{2}(t)$
$\displaystyle=\left\\{x\;\Big{|}\;\frac{K_{0}}{4}\sqrt{|\ln(T-t)|(T-t)}\leq|x|\leq\epsilon_{0}\right\\}$
$\displaystyle\quad\equiv\left\\{x\;\Big{|}\;\frac{K_{0}}{4}\sqrt{s}\leq|y|\leq\epsilon_{0}e^{\frac{s}{2}}\right\\}\equiv\left\\{x\;\Big{|}\;\frac{K_{0}}{4}\leq|z|\leq\frac{\epsilon_{0}}{\sqrt{s}}e^{\frac{s}{2}}\right\\},$
$\displaystyle\mathcal{D}_{3}(t)$
$\displaystyle=\left\\{x\;\Big{|}\;|x|\geq\frac{\epsilon_{0}}{4}\right\\}\equiv\left\\{x\;\big{|}\;|y|\geq\frac{\epsilon_{0}}{4}e^{\frac{s}{2}}\right\\}\equiv\left\\{x\;\Big{|}\;|z|\geq\frac{\epsilon_{0}}{4\sqrt{s}}e^{\frac{s}{2}}\right\\}.$

\- In $\mathcal{D}_{1}$, the blowup region of $U$, we make the change of
variables (2.17), resulting in equation (2.18), to do an asymptotic analysis
around the profile $e^{\Phi_{\alpha}(y/\sqrt{s})}$ according to the
decomposition (2.31) and (2.32).

\- In the intermediate region $\mathcal{D}_{2}$, we control $U$ by using
classical parabolic estimates on $\mathcal{U}$, a rescaled function of $U$
defined for $x\neq 0$ by

$\mathcal{U}(x,\xi,\tau)=\ln\big{(}T-t(x)\big{)}+U\big{(}x+\xi\sqrt{T-t(x)},t(x)+\tau(T-t(x))\big{)},$
(3.4)

where $t(x)$ is uniquely defined for $|x|$ sufficiently small by

$|x|=\frac{K_{0}}{4}\sqrt{(T-t(x))|\ln(T-t(x))|}\equiv\frac{K_{0}}{4}\sqrt{\theta(x)|\ln\theta(x)|},$
(3.5)

with

$\theta(x)=T-t(x).$ (3.6)

From (1.1), we see that $\mathcal{U}$ satisfies the same equation as $U$: for
all $\xi\in\mathbb{R}^{N}$ and $\tau\in\left[-\frac{t(x)}{T-t(x)},1\right)$,

$\partial_{\tau}\mathcal{U}=\Delta_{\xi}\mathcal{U}+\alpha|\nabla_{\xi}\mathcal{U}|^{2}+e^{\mathcal{U}}.$
(3.7)

We will in fact prove that $U$ behaves for

$|\xi|\leq\alpha_{0}\sqrt{|\ln(T-t(x))|}\quad\text{and}\quad\tau\in\left[\frac{t_{0}-t(x)}{T-t(x)},1\right)$

for some $t_{0}<T$ and $\alpha_{0}>0$, like the solution of

$\partial_{\tau}\hat{\mathcal{U}}=e^{\hat{\mathcal{U}}},$ (3.8)

subject to the initial data

$\hat{\mathcal{U}}(0)=\Phi_{\alpha}\left(\frac{K_{0}}{4}\right)=-\ln\left(1+\frac{K_{0}^{2}/16}{4+4\alpha}\right),$

namely that

$\hat{\mathcal{U}}(\tau)=-\ln\left((1-\tau)+\frac{K_{0}^{2}/16}{4+4\alpha}\right).$
(3.9)

As we will see that the analysis in $\mathcal{D}_{2}$ will imply the
conclusion of (3.2).

\- In $\mathcal{D}_{3}$, we estimate directly $U$ by using the local in time
well-posedness of the Cauchy problem for equation (1.1).

As described above, satisfying (3.3) and (3.2) is guaranteed if we can show
that

$U(t)\in\mathcal{S}^{*}(t),\quad\forall t\in[t_{0},T),$ (3.10)

where $\mathcal{S}^{*}(t)$ is precisely defined as follows:

######  Definition 3.1 (Definition of shrinking set to trap solutions).

For all $t_{0}<T$, $K_{0}>0$, $\epsilon_{0}>0$, $\alpha_{0}>0$, $A>0$,
$\delta_{0}>0$, $\eta_{0}>0$, $C_{0}>0$ and $C_{0}^{\prime}>0$, for all
$t\in[t_{0},T)$, we define
$\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},\eta_{0},C_{0},C_{0}^{\prime},t)$
being the set of all functions $U\in\mathcal{H}_{a}$ (see Definition (1.3))
satisfying:

$(i)$ Estimates in $\mathcal{D}_{1}$: $Q(s)\in\mathcal{V}_{A,K_{0}}(s)$ where
$s=-\ln(T-t)$, $Q(s)$ is defined as in (2.17) and $\mathcal{V}_{A,K_{0}}(s)$
is the set of all functions $Q$ in $W^{1,\infty}(\mathbb{R}^{N})$ such that

$|Q_{0}(s)|\leq\frac{A}{s^{2}},\quad|Q_{1,i}(s)|\leq\frac{A}{s^{2}},\quad|Q_{2,ij}(s)|\leq\frac{A^{2}\ln
s}{s^{2}},\quad\forall i,j\in\\{1,\cdots,N\\},$
$|Q_{-}(y,s)|\leq\frac{A}{s^{2}}(|y|^{3}+1),\quad|(\nabla_{y}Q)_{\bot}(y,s)|\leq\frac{A}{s^{2}}(|y|^{3}+1),\quad\forall
y\in\mathbb{R}^{N},$
$\|Q_{e}(s)\|_{L^{\infty}(\mathbb{R}^{N})}\leq\frac{A^{2}}{\sqrt{s}},$

where $Q_{m}$ ($m=0,1,2$), $Q_{-}$, $Q_{\bot}$ and $Q_{e}$ are defined as in
(2.31) and (2.32).

$(ii)$ Estimates in $\mathcal{D}_{2}$: For all
$|x|\in\left[\frac{K_{0}}{4}\sqrt{|\ln(T-t)|(T-t)},\epsilon_{0}\right]$,
$\tau=\tau(x,t)=\frac{t-t(x)}{\theta(x)}$ and
$|\xi|\leq\alpha_{0}\sqrt{\ln\theta(x)}$,

$\left|\mathcal{U}(x,\xi,\tau)-\hat{\mathcal{U}}(\tau)\right|\leq\delta_{0},\quad|\nabla_{\xi}\mathcal{U}(x,\xi,\tau)|\leq\frac{C_{0}}{\sqrt{|\ln\theta(x)|}},\quad|\nabla^{2}_{\xi}\mathcal{U}(x,\xi,\tau)|\leq
C_{0}^{\prime},$

where $\mathcal{U}$, $\hat{\mathcal{U}}$, $t(x)$ and $\theta(x)$ are defined
in (3.4), (3.9), (3.5) and (3.6) respectively.

$(iii)$ Estimates in $\mathcal{D}_{3}$: For all
$|x|\geq\frac{\epsilon_{0}}{4}$,

$|U(x,t)-U(x,t_{0})|\leq\eta_{0},\quad|\nabla_{x}U(x,t)-\nabla_{x}U(x,t_{0})|\leq\eta_{0}.$

For all $t_{0}<T$, we define
$\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},\eta_{0},C_{0},C_{0}^{\prime})$
being the set of all functions $U\in\mathcal{C}([t_{0},T),\mathcal{H}_{a})$
such that

$U(t)\in\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},\eta_{0},C_{0},C_{0}^{\prime},t),\quad\forall
t\in[t_{0},T).$

######  Remark 3.2.

The estimates on $\nabla_{y}Q$ in $\mathcal{D}_{1}$, $\nabla_{\xi}\mathcal{U}$
in $\mathcal{D}_{2}$ and $\nabla_{x}U$ in $\mathcal{D}_{3}$ allow us to
control the nonlinear gradient term $G(Q)$ appearing in equation (2.18). Note
that in the case when $G(Q)$ does not appear, the only estimates on
$\mathcal{D}_{1}$ are enough to fully control the solution (see [MZ97b]).
Therefore, this part makes the originality of the paper.

We will show that the proof of Theorem 1.1 reduces to find suitable parameters
$t_{0}<T$, $K_{0}$, $\epsilon_{0}$, $\alpha_{0}$, $A$, $\delta_{0}$,
$\eta_{0}$, $C_{0}$ and $U_{0}\in\mathcal{H}_{a}$ so that the solution $U$ of
equation (1.1) with data $U(t_{0})=U_{0}$ belongs to
$\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},\eta_{0},C_{0},C_{0}^{\prime},t_{0})$.
As a matter of fact, through a priori estimate, we will show that the control
of $U(t)$ in
$\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},\eta_{0},C_{0},C_{0}^{\prime},t)$
for $t\in[t_{0},T)$ reduces to the control of $(Q_{0},Q_{1})(s)$ in

$\hat{\mathcal{V}}_{A}(s)=\left[-\frac{A}{s^{2}},\frac{A}{s^{2}}\right]^{N+1},$
(3.11)

for $s\geq-\ln(T-t_{0})$ (recall that $Q_{0}$ and $Q_{1}$ correspond to
expanding eigenvalues in the $Q$ variable (2.17)). Hence, we will consider
initial data $U_{0}$ depending on $(N+1)$ parameters
$(d_{0},d_{1})\in\mathbb{R}\times\mathbb{R}^{N}$ of the form:

$\displaystyle U_{d_{0},d_{1}}(x,t_{0})=\hat{U}^{*}(x)(1-\chi_{1}(x,t_{0}))$
(3.12)
$\displaystyle+\left\\{s_{0}+\ln\left[\frac{A}{s_{0}^{2}}\left(d_{0}+d_{1}\cdot
xe^{\frac{s_{0}}{2}}\right)\chi\left(16xe^{\frac{s_{0}}{2}},s_{0}\right)+\psi_{\alpha}\left(xe^{\frac{s_{0}}{2}},s_{0}\right)\right]\right\\}\chi_{1}(x,t_{0}),$

where $s_{0}=-\ln(T-t_{0})$, $\psi_{\alpha}$ and $\chi$ are defined in (2.17)
and (2.26),

$\chi_{1}(x,t_{0})=\chi_{0}\left(\frac{|x|}{|\ln(T-t_{0})|\sqrt{T-t_{0}}}\right),$
(3.13)

with $\chi_{0}$ being defined right before (2.26), and
$\hat{U}^{*}\in\mathcal{C}^{\infty}(\mathbb{R}^{N}\backslash\\{0\\})$ is
defined by

$\hat{U}^{*}(x)=\left\\{\begin{array}[]{ll}\ln\left(\frac{(8+8\alpha)|\ln|x||}{|x|^{2}}\right)&\quad\text{for}\quad|x|\leq
C(a,\alpha),\\\ -\ln\left(1+a|x|^{2}\right)&\quad\text{for}\quad|x|\geq
1.\end{array}\right.$ (3.14)

Note that
$\chi\left(16xe^{\frac{s_{0}}{2}},s_{0}\right)(1-\chi_{1}(x,t_{0}))=0$ for
$s_{0}$ large, hence, the initial data (3.12) has an equivalence in the $Q$
variable (2.17),

$\bar{Q}_{d_{0},d_{1}}(y)=Q_{d_{0},d_{1}}(y,s_{0})=\frac{A}{s_{0}^{2}}(d_{0}+d_{1}\cdot
y)\chi(16y,s_{0})e^{\chi_{1}(x,t_{0})}.$ (3.15)

In what follows, the solution of equation (1.1) with initial data (3.12) will
be denote by $U_{d_{0},d_{1}}(x,t)$ or $U(x,t)$ when there is no ambiguity. We
also write $\mathcal{S}^{*}(t)$ and $Q(y,s)$ instead of
$\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},\eta_{0},C_{0},C_{0}^{\prime},t)$
and $Q_{d_{0},d_{1}}(y,s)$ (the solution of equation (2.18) with initial data
(3.15)) for simplicity.

We aim at proving the following central proposition which implies Theorem 1.1:

######  Proposition 3.3 (Existence of a solution of equation (1.1) trapped in
$\mathcal{S}^{*}(t)$).

We can choose parameters $t_{0}<T$, $K_{0}$, $\epsilon_{0}$, $\alpha_{0}$,
$A$, $\delta_{0}$, $\eta_{0}$ and $C_{0}$ such that the following holds: there
exists $(d_{0},d_{1})\in\mathbb{R}\times\mathbb{R}^{N}$ such that if $U(x,t)$
is the solution of (1.1) with initial data at $t=t_{0}$ given by (3.12), then
$U(x,t)$ exists for all $(x,t)\in\mathbb{R}^{N}\times[t_{0},T)$ and satisfies

$U(t)\in\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},\eta_{0},C_{0},C_{0}^{\prime},t),\quad\forall
t\in[t_{0},T).$

Before going to the proof of Proposition 3.3, let us first make sure that the
initial data (3.12) starts in $\mathcal{S}^{*}(t_{0})$ by selecting the good
parameters $(d_{0},d_{1})$. More precisely, we have the following:

######  Proposition 3.4 (Properties of the initial data (3.12)).

There exists $K_{0,1}>0$ such that for each $K_{0}\geq K_{0,1}$ and
$\delta_{0,1}>0$, there exist $\alpha_{0,1}(K_{0},\delta_{0,1})>0$ and
$C_{0,1}(K_{0})>0$ such that for all $\alpha_{0}\in(0,\alpha_{0,1}]$, there
exists $\epsilon_{0,1}(K_{0},\delta_{0,1},\alpha_{0})>0$ such that for all
$\epsilon_{0}\in(0,\epsilon_{0,1}]$ and $A\geq 1$, there exists
$t_{0,1}(K_{0},\delta_{0,1},\epsilon_{0},A,C_{0,1})<T$ such that for all
$t_{0}\in[t_{0,1},T)$,

$(I)$ There exists a subset

$\mathcal{D}_{t_{0},A}\subset[-2,2]\times[-2,2]^{N}$

such that the mapping

$\displaystyle\Lambda:\mathbb{R}\times\mathbb{R}^{N}\;\;$
$\displaystyle\to\;\;\mathbb{R}\times\mathbb{R}^{N}$
$\displaystyle(d_{0},d_{1})\;\;$
$\displaystyle\mapsto\;\;(\bar{Q}_{0},\bar{Q}_{1})$

(where $\bar{Q}$ stands for $Q_{d_{0},d_{1}}(y,s_{0})$ given by (3.15) and
$s_{0}=-\log(T-t_{0})$) is linear, one to one from $\mathcal{D}_{t_{0},A}$
onto $\hat{\mathcal{V}}_{A}(s_{0})$ defined by (3.11) and maps
$\partial\mathcal{D}_{t_{0},A}$ into $\partial\hat{\mathcal{V}}_{A}(s_{0})$.

$(II)$ For all $(d_{0},d_{1})\in\mathcal{D}_{t_{0},A}$, we have
$U_{d_{0},d_{1}}(x,t_{0})$ defined by (3.12) belongs in
$\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0,1},\eta_{0}=0,C_{0,1},t_{0})$.
More precisely,

$(i)$ Estimates in $\mathcal{D}_{1}$:
$\bar{Q}(s_{0})\in\mathcal{V}_{A,K_{0}}(s_{0})$ with strict inequalities,
except for $(\bar{Q}_{0},\bar{Q}_{1})(s_{0})$ in the sense that

$|\bar{Q}_{0}(s_{0})|\leq\frac{A}{s_{0}^{2}},\quad|\bar{Q}_{1,i}(s_{0})|\leq\frac{A}{s_{0}^{2}},\quad|\bar{Q}_{2,ij}(s_{0})|\leq\frac{\ln
s_{0}}{s_{0}^{2}},\quad\forall i,j\in\\{1,\cdots,N\\},$
$|\bar{Q}_{-}(y,s_{0})|\leq\frac{1}{s_{0}^{2}}(|y|^{3}+1),\quad|(\nabla_{y}\bar{Q})_{\bot}(y,s_{0})|\leq\frac{1}{s_{0}^{2}}(|y|^{3}+1),\quad\forall
y\in\mathbb{R}^{N},$
$\|\bar{Q}_{e}(s_{0})\|_{L^{\infty}(\mathbb{R}^{N})}\leq\frac{1}{\sqrt{s}_{0}},$

where $\bar{Q}_{m}$ ($m=0,1,2$), $\bar{Q}_{-}$, $\bar{Q}_{\bot}$ and
$\bar{Q}_{e}$ are defined as in (2.31) and (2.32).

$(ii)$ Estimates in $\mathcal{D}_{2}$: For all
$|x|\in\left[\frac{K_{0}}{4}\sqrt{|\ln(T-t_{0})|(T-t_{0})},\epsilon_{0}\right]$,
$\tau_{0}=\tau_{0}(x,t_{0})=\frac{t_{0}-t(x)}{\theta(x)}$ and
$|\xi|\leq\alpha_{0}\sqrt{\ln\theta(x)}$,

$\left|\mathcal{U}(x,\xi,\tau_{0})-\hat{\mathcal{U}}(\tau_{0})\right|\leq\delta_{0,1},\;\;|\nabla_{\xi}\mathcal{U}(x,\xi,\tau_{0})|\leq\frac{C_{0,1}}{\sqrt{|\ln\theta(x)|}},\;\;|\nabla^{2}_{\xi}\mathcal{U}(x,\xi,\tau_{0})|\leq
C_{0,1},$

where $\mathcal{U}$, $\hat{\mathcal{U}}$, $t(x)$ and $\theta(x)$ are defined
in (3.4), (3.9), (3.5) and (3.6) respectively.

The proof of Proposition 3.4 is left to Appendix A. Let us assume that
Proposition 3.4 holds and continue the proof of Proposition 3.3. The proof of
Theorem 1.1 will follow from Proposition 3.3 afterward.

###### Proof of Proposition 3.3.

The proof of Proposition 3.3 follows from the general ideas developed in
[MZ97b]. We proceed into two steps:
\- In the first step, we reduce the problem of controlling $U(t)$ in
$\mathcal{S}^{*}(t)$ to the control of $(Q_{0},Q_{1})(s)$ in
$\hat{\mathcal{V}}_{A}(s)$, where $Q_{0}$ and $Q_{1}$ are the components of
$Q(s)$ corresponding to the positive modes given in the decomposition (2.31)
and $\hat{\mathcal{V}}_{A}(s)$ defined by (3.11). This means that we reduce an
infinite dimensional problem to a finite dimensional one.
\- In the second step, we argue by contradiction to solve the finite
dimensional problem thanks to the dynamics of $(Q_{0},Q_{1})(s)$ and a
topological argument based on the variation of the finite dimensional
parameters $(d_{0},d_{1})$ appearing in the definition of initial data (3.15).

Step 1: Reduction to a finite dimensional problem.

In this step, we show through a priori estimate that the control of $U(t)$ in
$\mathcal{S}^{*}(t)$ reduces to the control of $(Q_{0},Q_{1})(s)$ in
$\hat{\mathcal{V}}_{A}(s)$ defined by (3.11). This result crucially follows
from a good understanding of the properties of the linear operator
$\mathscr{L}+V$ of equation (2.18) in the blowup region $\mathcal{D}_{1}$
together with advanced parabolic techniques applied to equation (1.1)
involving a nonlinear gradient term for analysis in the intermediate and
regular regions $\mathcal{D}_{2}$ and $\mathcal{D}_{3}$. In particular, we
claim the following which is the heart of our contribution:

######  Proposition 3.5 (Control of $U(t)$ by $(Q_{0},Q_{1})(s)$ in
$\hat{\mathcal{V}}_{A}(s)$).

We can choose parameters $t_{0}<T$, $K_{0}$, $\epsilon_{0}$, $\alpha_{0}$,
$A$, $\delta_{0}$, $\eta_{0}$ and $C_{0}$ such that the following properties
hold. Assume that $U(x,t_{0})$ is given by (3.12) with
$(d_{0},d_{1})\in\mathcal{D}_{t_{0},A}$. Assume in addition that for some
$t^{*}\in[t_{0},T)$,

$U(t)\in\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},\eta_{0},C_{0},C_{0}^{\prime},t),\quad\forall
t\in[t_{0},t^{*}],$

and

$U(t^{*})\in\partial\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},\eta_{0},C_{0},C_{0}^{\prime},t^{*}).$

Then, we have
$(i)$ (Finite dimensional reduction)
$(Q_{0},Q_{1})(s^{*})\in\partial\hat{\mathcal{V}}_{A}(s^{*})$ with
$s^{*}=-\log(T-t^{*})$.
$(ii)$ (Transversality) There exists $\mu_{0}>0$ such that for all
$\mu\in(0,\mu_{0})$,

$(Q_{0},Q_{1})(s^{*}+\mu)\not\in\hat{\mathcal{V}}_{A}(s^{*}+\mu),$

hence,

$U(t^{*}+\mu^{\prime})\not\in\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},\eta_{0},C_{0},C_{0}^{\prime},t^{*}+\mu^{\prime}),\quad\mu^{\prime}=\mu^{\prime}(t^{*},\mu)>0.$

###### Proof.

The proof uses ideas of [MZ97a] where the authors adapted the technique of a
priori estimates developed in [BK94] and [MZ97b] treated for equation (1.2)
with $\alpha=0$. Let us emphasize that the techniques introduced in [BK94] and
[MZ97b] are not enough to handle the nonlinear gradient term appearing in
equation (2.18). Truly new ideas are needed to achieve the control of this
term and this is one of the main novelties in this paper. The essential
feature of the proof is that the given bootstrap bounds in Definition 3.1 can
be improved, except for the bounds on $(Q_{0},Q_{1})$. More precisely, the
improvement of the bounds in the blowup region $\mathcal{D}_{1}$ (except for
$Q_{0},Q_{1}$) is done through projecting equation (2.18) on the different
components of $Q$ introduced in (2.31) and (2.32). One can see that the
components $Q_{2}$, $Q_{-}$, $Q_{\bot}$ and $Q_{e}$ corresponding to
decreasing directions of the flow are small at $s=s_{0}$ and they remain small
up to $s=s^{*}$, hence, they can not touch their boundary. In
$\mathcal{D}_{2}$ and $\mathcal{D}_{3}$, we use advanced parabolic techniques
applied to equation (1.1) involving a nonlinear gradient term in order to
achieve the improvement. Therefore, only $Q_{0}$ and $Q_{1}$ may touch their
boundary at $s=s^{*}$ and the conclusion follows. Since we would like to keep
the proof of Proposition 3.3 short, we leave the proof of Proposition 3.5 to
Section 4 below. ∎

Step 2: Topological argument for the finite dimensional problem.

From Proposition 3.5, we claim that there exist
$(d_{0},d_{1})\in\mathcal{D}_{t_{0},A}$ such that the equation (1.1) with
initial data (3.12) has a solution

$U_{d_{0},d_{1}}(t)\in\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},\eta_{0},C_{0},C_{0}^{\prime},t)\quad\text{for
all $t\in[t_{0},T)$},$

for suitable choice of the parameters. Note that the argument of the proof is
not new and it is analogous as in [MZ97b]. Let us gives its main ideas.

Let us consider
$t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},\eta_{0},C_{0}$ such that
Propositions 3.5 and 3.4 hold. From Proposition 3.4, we have

$\displaystyle\forall(d_{0},d_{1})\in\mathcal{D}_{t_{0},A},\quad
U_{d_{0},d_{1}}(x,t_{0})\in\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},\eta_{0},C_{0},C_{0}^{\prime},t_{0}),$

where $U_{d_{0},d_{1}}(x,t_{0})$ is given by (3.12). Note that
$U_{d_{0},d_{1}}(x,t_{0})\in\mathcal{H}_{a}$ introduced in (1.3). Therefore,
from the local existence theory for the Cauchy problem of (1.1) in
$\mathcal{H}_{a}$, we can define for each
$(d_{0},d_{1})\in\mathcal{D}_{t_{0},A}$ a maximum time
$t_{*}(d_{0},d_{1})\in[t_{0},T)$ such that

$U_{d_{0},d_{1}}(t)\in\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},\eta_{0},C_{0},C_{0}^{\prime},t),\quad\forall
t\in[t_{0},t_{*}).$

If $t_{*}(d_{0},d_{1})=T$ for some $(d_{0},d_{1})\in\mathcal{D}_{t_{0},A}$,
then the proof is complete. Otherwise, we argue by contradiction and assume
that $t_{*}(d_{0},d_{1})<T$ for any $(d_{0},d_{1})\in\mathcal{D}_{t_{0},A}$.
By continuity and the definition of $t_{*}$, the solution $U_{d_{0},d_{1}}(t)$
at time $t=t_{*}$ is on the boundary of $\mathcal{S}^{*}(t_{*})$. From part
$(i)$ of Proposition 3.5, we have

$(Q_{0},Q_{1})(s_{*})\in\partial\hat{\mathcal{V}}_{A}(s_{*})\quad\text{with}\quad
s_{*}=-\log(T-t_{*}).$

Hence, we may define the rescaled flow $\Gamma$ at $s=s_{*}$ for $Q_{0}$ and
$Q_{1}$ as follows:

$\displaystyle\Gamma:\quad\mathcal{D}_{t_{0},A}\quad$
$\displaystyle\mapsto\quad\partial([-1,1]\times[-1,1]^{N})$
$\displaystyle(d_{0},d_{1})\quad$
$\displaystyle\to\quad\left(\frac{s_{*}^{2}}{A}Q_{0}(s_{*}),\frac{s_{*}^{2}}{A}Q_{1}(s_{*})\right).$

It follows from part $(ii)$ of Proposition 3.5 that $\Gamma$ is continuous. If
we manage to prove that the degree of $\Gamma$ on the boundary is different
from zero, then we have a contradiction from the degree theory. Let us prove
that. From part $(i)$ Proposition 3.4, we see that if
$(d_{0},d_{1})\in\partial\mathcal{D}_{t_{0},A}$, then

$Q(s_{0})\in\mathcal{V}_{A,K_{0}}(s_{0})\quad\text{and}\quad(Q_{0},Q_{1})(s_{0})\in\partial\hat{\mathcal{V}}_{A}(s_{0}).$

Using part $(ii)$ of Proposition 3.5, we see that $Q(s)$ must leave
$\mathcal{V}_{A,K_{0}}(s)$ at $s=s_{0}$, hence, $s_{*}(d_{0},d_{1})=s_{0}$.
Using again part $(i)$ of Proposition 3.4, we see that the degree of $\Gamma$
on the boundary must be different from zero. This gives us a contradiction (by
the index theory) and concludes the proof of Proposition 3.3, assuming that
Propositions 3.5 and 3.4 hold. ∎

Let us now give the proof of Theorem 1.1 from Proposition 3.3, assuming that
Propositions 3.4 and 3.5 hold.

###### Proof of Theorem 1.1, assuming Propositions 3.4 and 3.5.

We give in this part the proof of Theorem 1.1. We will present the proofs of
item $(i)$, $(ii)$ and $(iii)$ separately.

$(i)$ The proof of part $(i)$ is equivalent to the proof of (3.3) through the
change of variables (2.17). From Proposition 3.3, we know that equation (1.1)
with the initial data given by (3.12) has the solution
$U(t)\in\mathcal{S}^{*}(t)$ for all $t\in[t_{0},T)$. From part $(i)$ of
Definition 3.1, we have $Q(s)\in\mathcal{V}_{A,K_{0}}(s)$ for all
$s\in[-\log(T-t_{0}),+\infty)$, where $A$ and $K_{0}$ are some fixed large
constants. In Proposition A.1 below, we show that if
$Q(s)\in\mathcal{V}_{A,K_{0}}(s)$, then

$\|Q(s)\|_{W^{1,\infty}(\mathbb{R}^{N})}\leq\frac{C(A,K_{0})}{\sqrt{s}},$
(3.16)

which is the conclusion of (3.3) as well as (1.17).

$(ii)$ Note that part $(i)$ also implies that $U$ and $\nabla U$ blow up in
finite time $T$ at the origin. Indeed, by the definition (1.18) of
$\Phi_{\alpha}$ and part $(i)$ of Theorem 1.1, we have

$U(0,t)\sim-\ln(T-t)\to+\infty\quad\text{as}\quad t\to T,$

hence, $U$ and $e^{U}$ blow up in finite time $T$ at the origin. As for
$\nabla U$, we write from (2.17),

$\nabla W-\nabla\Phi_{\alpha}=\frac{\nabla
Q-\nabla\Phi_{\alpha}Q}{Q+\psi_{\alpha}}.$

From the definition (2.17) of $\psi_{\alpha}$ and (3.16), we see that

$\forall|y|\leq K_{0}\sqrt{s},\quad
Q(y,s)+\psi_{\alpha}(y,s)\geq\psi_{\alpha}(K_{0}\sqrt{s},s)-\|Q(s)\|_{L^{\infty}}>\frac{C}{K_{0}^{2}}.$

We also show in Proposition A.1 that if $Q(s)\in\mathcal{V}_{A,K_{0}}(s)$,
then

$\forall y\in\mathbb{R}^{N},\quad|Q(y,s)|+|\nabla
Q(y,s)|\leq\frac{C(A,K_{0})\ln s}{s^{2}}(|y|^{3}+1).$

Note from the definition (1.18) of $\Phi_{\alpha}$ that
$|\nabla\Phi_{\alpha}(y/\sqrt{s})|\leq\frac{C|y|}{s}$, we have for all
$|y|\leq K_{0}\sqrt{s}$,

$\left|\nabla
W(y,s)-\nabla\Phi_{\alpha}\left(\frac{y}{\sqrt{s}}\right)\right|\leq\frac{C(A,K_{0})\ln
s}{s^{2}}(|y|^{3}+1)+\frac{C(A,K_{0})\ln s}{s^{3}}(|y|^{4}+1).$

From (1.8) and (1.18), we write

$\displaystyle\left|\nabla
W(y,s)-\nabla\Phi_{\alpha}\left(\frac{y}{\sqrt{s}}\right)\right|=\left|\sqrt{T-t}\,\nabla
U(x,t)+\frac{ye^{\Phi_{\alpha}\left(\frac{y}{\sqrt{|\ln(T-t)|}}\right)}}{(2+2\alpha)|\ln(T-t)|}\right|$
$\displaystyle\leq\frac{C(A,K_{0})\ln|\ln(T-t)|}{|\ln(T-t)|^{2}}(|y|^{3}+1)+\frac{C(A,K_{0})\ln|\ln(T-t)|}{|\ln(T-t)|^{3}}(|y|^{4}+1).$

Put $y=y(s)=s^{\frac{1}{4}}\omega=|\ln(T-t)|^{\frac{1}{4}}\omega$ with
$|\omega|=1$, we see that the right hand side of the above estimate is bounded
by
$\frac{\ln|\ln(T-t)|}{|\ln(T-t)|^{\frac{5}{4}}}\ll\frac{1}{|\ln(T-t)|^{\frac{3}{4}}}$
as $t\to T$. Therefore, we have

$\nabla
U(|\ln(T-t)|^{\frac{1}{4}}\sqrt{T-t},t)\sim\frac{C}{|\ln(T-t)|^{\frac{3}{4}}\sqrt{T-t}}\quad\text{as}\quad
t\to T.$

Since $|\ln(T-t)|^{\frac{3}{4}}\sqrt{T-t}\to 0$ as $t\to T$, hence, $\nabla U$
blows up at time $T$ at the origin.

In order to prove that $e^{U}$ and $\nabla U$ blow up only at the origin, we
use the following result:

######  Proposition 3.6 (No blowup under some threshold).

Let $u(\xi,\tau)$ satisfy the following inequality: for all $K>0$, for all
$|\xi|<1$ and $\tau\in[0,1)$,

$\left\\{\begin{array}[]{rl}|\partial_{\tau}u-\Delta u|&\leq K(1+e^{u}+|\nabla
u|^{2}),\\\ |\nabla(\partial_{\tau}u-\Delta u)|&\leq K|\nabla(e^{u}+|\nabla
u|^{2})|.\end{array}\right.$ (3.17)

Assume that there is a constant $\epsilon=\epsilon(K,N)>0$ small enough such
that

$(1-\tau)e^{u(\xi,\tau)}+\sqrt{1-\tau}|\nabla
u(\xi,\tau)|\leq\epsilon,\quad\forall|\xi|<1,\;\tau\in[0,1),$ (3.18)

then,

$e^{u(\xi,\tau)}+|\nabla u(\xi,\tau)|\leq
C\epsilon,\quad\forall|\xi|\leq\frac{1}{8},\;\tau\in[0,1).$ (3.19)

In particular, $e^{u}$ and $\nabla u$ do not blow up at $\xi=0$ and $\tau=1$.

###### Proof.

The proof of this result uses ideas given in Giga and Kohn [GK89], where
(3.17) is considered without the gradient term and the nonlinear source term
$e^{u}$ replaced by $|u|^{p-1}u$. The proof in [GK89] uses a truncation
technique together with the smoothness effect of the heat semigroup
$e^{\tau\Delta}$ and some type of Gronwall’s argument. Although some advanced
parabolic regularities are needed to treat our problem involving the nonlinear
gradient term, but the same argument to those of [GK89] can be extended to our
case without difficulties. Since the proof is long and technical, we give the
proof in Appendix B. ∎

Let us apply Proposition 3.6 to $\mathcal{U}(x_{0},\xi,\tau)$, where
$\mathcal{U}(x_{0},\xi,\tau)$ and $x_{0}$ are defined as in (3.4) and (3.5).
Recall from (3.7) that $\mathcal{U}$ solves the following equations:

$\displaystyle\partial_{\tau}\mathcal{U}$
$\displaystyle=\Delta_{\xi}\mathcal{U}+\alpha|\nabla_{\xi}\mathcal{U}|^{2}+e^{\mathcal{U}},$

hence, (3.17) is satisfied. For the condition (3.18), let us write from the
definition (3.4) of $\mathcal{U}(x_{0},\xi,\tau)$, the definition (3.5) of
$x_{0}$ and part $(i)$ of Theorem 1.1,

$\displaystyle\sup_{|\xi|\leq
1,\tau\in[0,1)}(1-\tau)e^{\mathcal{U}(x_{0},\xi,\tau)}$
$\displaystyle\quad=\sup_{|x-x_{0}|\leq\sqrt{T-t(x_{0})},\tau\in[0,1)}(1-\tau)^{2}(T-t(x_{0}))e^{U(x,t(x_{0})+\tau(T-t(x_{0})))}$
$\displaystyle\qquad\leq\sup_{|x-x_{0}|\leq\frac{|x_{0}|}{2}}(T-t(x_{0}))e^{U(x,t)}\leq
e^{\Phi_{\alpha}\left(\frac{|x_{0}|/2}{\sqrt{(T-t(x_{0}))|\log(T-t(x_{0}))|}}\right)}+\frac{C}{\sqrt{|\log(T-t(x_{0}))|}}$
$\displaystyle\qquad\quad\leq
e^{\Phi_{\alpha}(K_{0}/8)}+\frac{C}{\sqrt{|\ln(T-t(x_{0}))|}}\leq\epsilon_{0}(K_{0},|x-x_{0}|)\to
0,$

as $K_{0}\to+\infty$ and $|x-x_{0}|\to 0$. Similarly, we have

$\displaystyle\sup_{|\xi|\leq
1,\tau\in[0,1)}\sqrt{1-\tau}|\nabla_{\xi}\mathcal{U}(x_{0},\xi,\tau)|$
$\displaystyle\quad\leq\sup_{x\in\mathbb{R}^{N},\tau\in[0,1)}\sqrt{T-t(x_{0})}|\nabla_{x}U(x,t(x_{0})+\tau(T-t(x_{0})))|$
$\displaystyle\qquad\leq\frac{C}{\sqrt{|\log(T-t(x_{0}))|}}\left(\|\zeta
e^{\Phi_{\alpha}(\zeta)}\|_{L^{\infty}(\mathbb{R}^{N})}+1\right)$
$\displaystyle\qquad\quad\leq\epsilon_{0}^{\prime}(|x-x_{0}|)\to
0\quad\text{as $|x-x_{0}|\to 0$}.$

which verifies the condition (3.18) for $\mathcal{U}(x_{0},\xi,\tau)$ and
$\nabla_{\xi}(x_{0},\xi,\tau)$. Hence, we can apply Proposition 3.6 to
$\mathcal{U}(x_{0},\xi,\tau)$ to deduce that $\xi=0$ is not a blowup point of
$e^{\mathcal{U}(x_{0},\xi,\tau)}$ and
$\nabla_{\xi}\mathcal{U}(x_{0},\xi,\tau)$, which means that $x_{0}\neq 0$ is
not a blowup point of $e^{U}$ and $\nabla U$. Let us insist on the fact that
our argument works for any $x_{0}\neq 0$ without any smallness assumptions,
thanks to the adapted definition of $t(x_{0})$ given in (3.5). This proves the
single point blowup result for $e^{U}$ and $\nabla U$, and concludes the proof
of part $(ii)$ of Theorem 1.1.

$(iii)$ We divide the proof into two steps. We first show the existence of the
final profile $U^{*}_{\alpha}$, then we find an equivalent of $U^{*}_{\alpha}$
which concludes the proof of part $(iii)$ of Theorem 1.1. We claim the
following:

######  Proposition 3.7 (Existence of the final blowup profile).

Let $U(t)$ be a solution of equation (1.1) which blows up in finite time $T$
at the origin and verifies the asymptotic behavior (1.17). There exists a
function $U^{*}_{\alpha}(x)\in\mathcal{C}^{1}(\mathbb{R}^{N}\setminus\\{0\\})$
such that $U(x,t)\to U^{*}_{\alpha}(x)$ and $\nabla U(x,t)\to\nabla
U^{*}_{\alpha}(x)$ as $t\to T$, uniformly on compact sets of
$\mathbb{R}^{N}\setminus\\{0\\}$.

###### Proof.

The proof uses the same argument given by Merle [Mer92] treated for equation
(1.2) with $\alpha=0$, which relies on some classical regularity argument of
parabolic problem. In comparison with the work in [Mer92], the only difference
is that we need to extend the no blowup under some threshold result of Giga
and Kohn [GK89] to our equation (1.1), which is Proposition 3.6. Let us denote
by $H=\nabla U$, we write from equation (1.1),

$\displaystyle\partial_{t}U$ $\displaystyle=\Delta U+\alpha|\nabla
U|^{2}+e^{U},$ $\displaystyle\partial_{t}H$ $\displaystyle=\Delta
H+\alpha\nabla(|H|^{2})+He^{U}.$

From Proposition 3.6, we proved in part $(ii)$ that $e^{U}$ and $\nabla U$ are
uniformly bounded on $\Omega\times[0,T)$ for any compact set
$\Omega\subset\mathbb{R}^{N}\setminus\\{0\\}$. By parabolic regularity
techniques, similar to Proposition 3.6, we can show that $\partial_{t}U$ and
$\partial_{t}H$ are also bounded on $\Omega^{\prime}\times[T/2,T)$ for any
$\Omega^{\prime}\subset\mathbb{R}^{N}\setminus\\{0\\}$. Therefore, as in
[Mer92] (see Proposition 2.2, page 269), we conclude that there exists
$U^{*}_{\alpha}$ in $\mathcal{C}^{1}(\mathbb{R}^{N}\setminus\\{0\\})$ such
that $U(x,t)\to U^{*}_{\alpha}(x)$ and $\nabla U(x,t)\to\nabla
U^{*}_{\alpha}(x)$ as $t\to T$, uniformly on each compact set of
$\mathbb{R}^{N}\setminus\\{0\\}$. This concludes the proof of Proposition 3.7.
∎

Let us now find an equivalence of $U^{*}_{\alpha}$ and $\nabla U^{*}_{\alpha}$
in order to complete the proof of part $(iii)$ of Theorem 1.1. To this end, we
use the same argument as in the proof of Proposition 3.7 to show that the
limit of $\mathcal{U}(x,0,\tau)$ and $\nabla_{\xi}\mathcal{U}(x,0,\tau)$ as
$\tau\to 1$ exist for $x$ sufficiently small. Moreover, using part $(ii)$ of
Definition 3.1, we see that

$\lim_{\tau\to
1}\mathcal{U}(x,0,\tau)\sim\hat{\mathcal{U}}_{K_{0}}(1)=-\ln\left(\frac{K_{0}^{2}}{4+4\alpha}\right),$

and

$\lim_{\tau\to
1}|\nabla_{\xi}\mathcal{U}(x,0,\tau)|\leq\frac{C}{\sqrt{|\ln(T-t(x))|}},$

for $|x|$ and $1/K_{0}$ sufficiently small.

From Proposition 3.7, we have that $\lim_{t\to T}U(x,t)=U^{*}_{\alpha}(x)$ and
$\lim_{t\to T}\nabla U(x,t)=\nabla U^{*}_{\alpha}(x)$. Hence, from the
definitions (3.4) and (3.5), we derive

$\displaystyle U^{*}_{\alpha}(x)=\lim_{t\to T}U(x,t)$
$\displaystyle=\lim_{\tau\to
1}\big{[}-\ln(T-t(x))+\mathcal{U}(x,0,\tau)\big{]}$
$\displaystyle\sim-\ln(T-t(x))-\ln\left(\frac{K_{0}^{2}/16}{4+4\alpha}\right)$
$\displaystyle\sim-\ln|x|^{2}+\ln|\ln|x|^{2}|+\ln(K_{0}^{2}/16)-\ln\left(\frac{K_{0}^{2}/16}{4+4\alpha}\right)$
$\displaystyle=\ln\left(\frac{(8+8\alpha)|\ln|x||}{|x|^{2}}\right)\quad\text{as}\quad|x|\to
0,$

and

$\displaystyle|\nabla_{x}U^{*}_{\alpha}(x)|=|\lim_{t\to T}\nabla_{x}U(x,t)|$
$\displaystyle=\left|\lim_{\tau\to
1}\frac{\nabla_{\xi}\mathcal{U}(x,0,\tau)}{\sqrt{T-t(x)}}\right|$
$\displaystyle\leq\frac{C}{\sqrt{|\ln(T-t(x))|(T-t(x))}}\leq\frac{C}{|x|}\quad\text{as}\quad|x|\to
0.$

This concludes the proof of Theorem 1.1 assuming that Propositions 3.4, 3.5
and 3.6 hold. ∎

##  4\. Reduction to a finite dimensional problem.

This section is the heart of our analysis. We aim at proving Proposition 3.5
which reduces the problem to a finite dimensional one. We proceed in two
parts. In the first part, we derive a priori estimates on $U(t)$ in
$\mathcal{S}^{*}(t)$. In the second part, we show that these new bounds are
better than those defined in $\mathcal{S}^{*}(t)$, except for the bounds on
the components $Q_{0}(s)$ and $Q_{1}(s)$. This means that the problem is
reduced to the control of a finite dimensional function $(Q_{0},Q_{1})(s)$
which is the conclusion $(i)$ of Proposition 3.5. The outgoing transversality
property stated in part $(ii)$ of Proposition 3.5 is a direct consequence of
the dynamics of the modes $Q_{0}$ and $Q_{1}$. Let us start with the first
part.

###  4.1. A priori estimates.

We have the following estimates:

######  Proposition 4.1 (A priori estimate in $\mathcal{D}_{1}$).

There exist $K_{0,2}>0$ and $A_{0,2}>0$ such that for all $K_{0}\geq K_{0,2}$,
$\epsilon_{0}>0$, $A\geq A_{0,2}$, $\lambda^{*}>0$, $C_{0,2}>0$, there exists
$t_{0,2}(K_{0},\epsilon_{0},A,\lambda^{*},C_{0,2})$ with the following
property: For all $\delta_{0}\leq\frac{1}{2}|\hat{\mathcal{U}}(1)|$,
$\alpha_{0}>0$, $C_{0}>0$ and $\eta_{0}\leq\eta_{0,2}$ for some
$\eta_{0,2}(\epsilon_{0})>0$, $\lambda\in[0,\lambda^{*}]$ and
$t_{0}\in[t_{0,2},T)$, assume that

  * •

$U(x,t_{0})$ is given by (3.12) and $(d_{0},d_{1})$ is chosen such that
$(\bar{Q}_{0}(s_{0}),\bar{Q}_{1}(s_{0}))\in\hat{\mathcal{V}}_{A}(s_{0})$ where
$s_{0}=\ln(T-t_{0})$ and $\hat{\mathcal{V}}_{A}$ is defined in (3.11).

  * •

for some $\sigma\geq s_{0}$, we have for all
$t\in[T-e^{-\sigma},T-e^{-(\sigma+\lambda)}]$,

$U(x,t)\in\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},C_{0},C_{0}^{\prime},\eta_{0},t).$

Then, we have for all $s\in[\sigma,\sigma+\lambda]$,

  * (i)

(ODEs satisfied by the positive and null modes of $Q$)

$\left|Q_{0}^{\prime}(s)-Q_{0}(s)\right|\leq\frac{C}{s^{2}},$ (4.1) $\forall
i\in\\{1,\cdots,N\\},\quad\left|Q_{1,i}^{\prime}(s)-\frac{1}{2}Q_{1,i}(s)\right|\leq\frac{C}{s^{2}},$
(4.2)

and

$\forall
i,j\in\\{1,\cdots,N\\},\quad\left|Q_{2,ij}^{\prime}(s)+\frac{2}{s}Q_{2,ij}(s)\right|\leq\frac{CA}{s^{3}}.$
(4.3)

  * (ii)

(Control of the negative and outer part of $Q$)
\- For $\sigma\geq s_{0}$:

$\left|Q_{-}(y,s)\right|\leq
C\Big{(}Ae^{-\frac{s-\sigma}{2}}+A^{2}e^{-(s-\sigma)^{2}}+(s-\sigma)\Big{)}s^{-2}(1+|y|^{3}),$
(4.4) $|Q_{e}(y,s)|\leq
C\Big{(}A^{2}e^{-\frac{s-\sigma}{2}}+AK_{0}^{3}e^{s-\sigma}+K_{0}^{3}(s-\sigma+1)\Big{)}s^{-1/2}.$
(4.5)

\- For $\sigma=s_{0}$:

$|Q_{-}(y,s)|\leq C(1+s-\sigma)s^{-2}(1+|y|^{3}),\quad|Q_{e}(y,s)|\leq
CK_{0}^{3}(1+s-\sigma)e^{s-\sigma}s^{-1/2}.$ (4.6)

  * (iii)

(Control of the gradient of $Q$)
\- For $\sigma\geq s_{0}$:

$|(\nabla Q)_{\bot}(y,s)|\leq
C\Big{(}Ae^{-\frac{s-\sigma}{2}}+C(K_{0})C_{0}e^{-(s-\sigma)^{2}}+\sqrt{s-\sigma}+(s-\sigma)\Big{)}s^{-2}(1+|y|^{3}).$
(4.7)

\- For $\sigma=s_{0}$:

$|(\nabla Q)_{\bot}(y,s)|\leq C(1+\sqrt{s-\sigma}+s-\sigma)s^{-2}(1+|y|^{3}).$
(4.8)

###### Proof.

The proof of this proposition is completely the same as in [MZ97a] because our
equation (2.18) and the shrinking set $\mathcal{V}_{A,K_{0}}$ defined in part
$(i)$ of Definition 3.1 are analogous as those defined in that paper. Note
that the coefficient $(\alpha-1)$ apearing in (2.20) and the nonlinear term
$Q^{2}$ in equation (2.18) are replaced by some constants $a>1$ and $Q^{p}$
with $p>1$ in [MZ97a], and that these changes don’t affect to their analysis.
For this reason, we kindly refer the reader to Lemma 3.2 at page 1523 in
[MZ97a] for a similar statement and Appendix B for all details of its proof. ∎

We now turn to the a priori estimates of $U$ in $\mathcal{D}_{2}$. We prove
the following:

######  Proposition 4.2 (A priori estimate in $\mathcal{D}_{2}$).

There exists $K_{0,3}>0$ such that for all $K_{0}\geq K_{0,3}$,
$\delta_{1}\leq 1$, $\xi_{0}\geq 1$ and $C_{0,1}^{*}>0$, $C_{0,2}^{*}$,
$C_{0,3}^{*}$, we have the following property: Assume that $\mathcal{U}$ is a
solution of equation

$\partial_{\tau}\mathcal{U}=\Delta\mathcal{U}+\alpha|\nabla\mathcal{U}|^{2}+e^{\mathcal{U}},$
(4.9)

for $\tau\in[\tau_{1},\tau_{2}]$ with $0\leq\tau_{1}\leq\tau_{2}\leq 1$.
Assume in addition, for all $\tau\in[\tau_{1},\tau_{2}]$,

  * (i)

for all $|\xi|\leq 2\xi_{0}$,
$|\mathcal{U}(\xi,\tau_{1})-\hat{\mathcal{U}}(\tau_{1})|\leq\delta_{1}$ and
$|\nabla\mathcal{U}(\xi,\tau_{1})|\leq\frac{C_{0,1}^{*}}{\xi_{0}}$, where
$\hat{\mathcal{U}}(\tau)$ is given by (3.9),

  * (ii)

for all $|\xi|\leq\frac{7}{4}\xi_{0}$,
$|\nabla\mathcal{U}(\xi,\tau)|\leq\frac{C_{0,2}^{*}}{\xi_{0}}$ and
$|\nabla^{2}\mathcal{U}(\xi,\tau)|\leq C_{0,3}^{*}$,

  * (iii)

for all $|\xi|\leq\frac{7}{4}\xi_{0}$,
$\mathcal{U}(\xi,\tau)\leq\frac{1}{2}\hat{\mathcal{U}}(\tau)$.

Then, for $\xi_{0}\geq\xi_{0,3}(C_{0}^{*})$, there exists
$\epsilon=\epsilon(K_{0},C_{0,2}^{*},C_{0,3}^{*},\delta_{1},\xi_{0})$ such
that for all $|\xi|\leq\xi_{0}$ and $\tau\in[\tau_{1},\tau_{2}]$,

$|\mathcal{U}(\xi,\tau)-\hat{\mathcal{U}}(\tau)|\leq\epsilon,\quad|\nabla\mathcal{U}(\xi,\tau)|\leq\frac{2C_{0,1}^{*}}{\xi_{0}},$

where $\epsilon\to 0$ as $(\delta_{1},\xi_{0})\to(0,+\infty)$.

###### Proof.

We first deal with the gradient estimate. We aim at proving that under the
hypothesis of Proposition 4.2, we have

$\forall|\xi|\leq\frac{5}{4}\xi_{0},\;\;\tau\in[\tau_{1},\tau_{2}],\quad|\nabla\mathcal{U}(\xi,\tau)|\leq\frac{2C_{0,1}^{*}}{\xi_{0}},$
(4.10)

provided that $\xi_{0}\geq\xi_{0,3}(C_{0}^{*})$. To do so, let us denote by
$\theta=|\nabla\mathcal{U}|^{2}$ and write from (4.9),

$\partial_{\tau}\theta\leq\Delta\theta+C(C_{0,3}^{*})\theta,\quad\forall|\xi|\leq\frac{7}{4}\xi_{0},\;\tau\in[\tau_{1},\tau_{2}],$

where we used the fact that
$2\nabla\mathcal{U}\cdot\nabla(\Delta\mathcal{U})\leq\Delta\theta$ and
$2\nabla\mathcal{U}\cdot\nabla(\alpha|\nabla\mathcal{U}|^{2}+e^{\mathcal{U}})=(4\alpha\Delta\mathcal{U}+2e^{\mathcal{U}})\theta\leq
C(C_{0,3}^{*})\theta$.

Let us consider $\rho_{1}\in\mathcal{C}^{\infty}(\mathbb{R}^{N})$ such that
$\rho_{1}\in[0,1]$, $\rho_{1}(\xi)=1$ for $|\xi|\leq\frac{3}{2}\xi_{0}$ and
$\rho_{1}(\xi)=0$ for $|\xi|\geq\frac{7}{4}\xi_{0}$,
$|\nabla\rho_{1}|\leq\frac{C}{\xi_{0}}$ and
$|\Delta\rho_{1}|\leq\frac{C}{\xi_{0}^{2}}$. Then, $\theta_{1}=\rho_{1}\theta$
satisfies the inequality:

$\displaystyle\partial_{\tau}\theta_{1}$
$\displaystyle\leq\Delta\theta_{1}-2\nabla\rho_{1}\cdot\nabla\theta-\Delta\rho_{1}\theta+C(C_{0,3}^{*})\theta_{1}$
$\displaystyle\leq\Delta\theta_{1}+C(C_{0,2}^{*},C_{0,3}^{*})\xi_{0}^{-2}\mathbf{1}_{\\{\frac{3}{2}\xi_{0}\leq|\xi|\leq
2\xi_{0}\\}}+C(C_{0,3}^{*})\theta_{1}.$

Let $\theta_{2}=e^{-C(C_{0,3}^{*})\tau}\theta_{1}$, we have

$\partial_{\tau}\theta_{2}\leq\Delta\theta_{2}+C(C_{0,2}^{*},C_{0,3}^{*})\xi_{0}^{-2}\mathbf{1}_{\\{\frac{3}{2}\xi_{0}\leq|\xi|\leq
2\xi_{0}\\}}\quad\text{and}\quad
0\leq\theta_{2}(\tau_{1})\leq\frac{{C_{0,1}^{*}}^{2}}{\xi_{0}^{2}}.$

Therefore, by the maximum principle, we deduce

$\forall|\xi|\leq\frac{5}{4}\xi_{0},\;\tau\in[\tau_{1},\tau_{2}],\quad\theta(\xi,\tau)\leq\frac{{C_{0,1}^{*}}^{2}}{\xi_{0}^{2}}+C(C_{0,2}^{*},C_{0,3}^{*})^{2}\xi_{0}^{-2}e^{-C^{\prime}\xi_{0}^{2}},$

which yields the conclusion (4.10).

We now turn to the estimate on $\mathcal{U}$. We use here the same argument as
in [MZ97a], and consider $\mathcal{U}_{1}$ a solution to (4.9) such that for
all $|\xi|\leq 2$ and $\tau\in[\tau_{1},\tau_{2}]$,

$|\mathcal{U}_{1}(\xi,\tau_{1})-\hat{\mathcal{U}}(\tau_{1})|\leq\delta_{1},\quad|\nabla\mathcal{U}_{1}(\xi,\tau)|\leq\epsilon.$

Let us show that for $|\xi|\leq 2$ and $\tau\in[\tau_{1},\tau_{2}]$,

$|\mathcal{U}_{1}(\xi,\tau)-\hat{\mathcal{U}}(\tau)|\leq
C(K_{0})\epsilon+\delta_{1}.$

We write for all $\tau\in[\tau_{1},\tau_{2}]$,

$\displaystyle\mathcal{U}_{1}(0,\tau)$
$\displaystyle=\frac{1}{|B_{2}(0)|}\int_{|\xi|\leq
2}\mathcal{U}_{1}(\xi,\tau)d\xi+\mathcal{U}_{2}(\tau),$ $\displaystyle
e^{\mathcal{U}_{1}(0,\tau)}$
$\displaystyle=\frac{1}{|B_{2}(0)|}\int_{|\xi|\leq
2}e^{\mathcal{U}_{1}(\xi,\tau)}d\xi+\mathcal{U}_{3}(\tau),$

where $|B_{2}(0)|$ is the volume of the sphere of radius 2 in
$\mathbb{R}^{N}$, $\|\mathcal{U}_{2}\|_{L^{\infty}}\leq 2\epsilon$ and
$\|\mathcal{U}_{3}\|_{L^{\infty}}\leq C\epsilon.$

For $\epsilon$ small enough, if we consider in the distribution sense,

$\tilde{\mathcal{U}}(\tau)=\frac{1}{|B_{2}(0)|}\int_{|\xi|\leq
2}\mathcal{U}_{1}(\xi,\tau)d\xi,$

then, we have from (4.9),

$e^{\tilde{\mathcal{U}}}-C\epsilon-|\alpha|\epsilon^{2}\leq\frac{d\tilde{\mathcal{U}}}{d\tau}\leq
e^{\tilde{\mathcal{U}}}+C\epsilon+|\alpha|\epsilon^{2},$

and

$|\tilde{\mathcal{U}}(\tau_{1})-\hat{\mathcal{U}}(\tau_{1})|\leq
C\epsilon+\delta_{1}.$

From (4.9), we obtain by classical a priori estimates that for all
$\tau\in[\tau_{1},\tau_{2}]$,
$|\tilde{\mathcal{U}}(\tau)-\hat{\mathcal{U}}(\tau)|\leq
C(K_{0})\epsilon+\delta_{1}$ (since
$C_{1}(K_{0})\leq|\hat{\mathcal{U}}(\tau)|\leq C_{1}^{\prime}(K_{0})$).
Therefore, for $|\xi|\leq 2$ and $\tau\in[\tau_{1},\tau_{2}]$, we have
$|\mathcal{U}_{1}(\xi,\tau)-\hat{\mathcal{U}}(\tau)|\leq
C(K_{0})\epsilon+\delta_{1}$. Applying this result to
$\mathcal{U}_{1}(\xi,\tau)=\mathcal{U}(\xi-\bar{\xi}_{0},\tau)$ with
$|\bar{\xi}_{0}|\in[-\xi_{0}+2,\xi_{0}+2]$ and using the hypothesis, we obtain

$\forall|\xi|\leq\xi_{0},\;\tau\in[\tau_{1},\tau_{2}],\quad|\mathcal{U}(\xi,\tau)-\hat{\mathcal{U}}(\tau)|\leq\epsilon,$

where $\epsilon=\epsilon(\delta_{1},\xi_{0})\to 0$ as
$(\delta_{1},\xi_{0})\to(0,+\infty)$. This concludes the proof of Proposition
4.2. ∎

For the a priori estimates of $U$ in $\mathcal{D}_{3}$, we claim the
following:

######  Proposition 4.3 (A priori estimate in $\mathcal{D}_{3}$).

For all $\epsilon>0$, $\epsilon_{0}>0$, $\sigma_{0}>0$, there exists
$t_{0,4}(\epsilon,\epsilon_{0},\sigma_{0})<T$ such that for all
$t_{0}\in[t_{0,4},T)$, if $U$ is a solution of (1.1) on $[t_{0},t_{*}]$ for
some $t_{*}\in[t_{0},T)$ satisfying

  * (i)

for all $|x|\in\left[\frac{\epsilon_{0}}{6},\frac{\epsilon_{0}}{4}\right]$ and
$t\in[t_{0},t_{*}]$,

$i=0,1,2,\quad|\nabla^{i}U(x,t)|\leq\sigma_{0},$ (4.11)

  * (ii)

For $|x|\geq\frac{\epsilon_{0}}{6}$, $U(x,t_{0})=\hat{U}^{*}(x)$ where
$\hat{U}^{*}$ is defined in (3.14),

then for all $|x|\in\left[\frac{\epsilon_{0}}{4},+\infty\right)$ and
$t\in[t_{0},t_{*}]$,

$|U(x,t)-U(x,t_{0})|+|\nabla U(x,t)-\nabla U(x,t_{0})|\leq\epsilon.$ (4.12)

###### Proof.

We only deal with the estimate on $U$ for $|x|\geq\frac{\epsilon_{0}}{4}$
because the estimate on $\nabla U$ can be obtained similarly. We argue by
contradiction. Let us consider $t_{\epsilon}\in(t_{0},t_{*})$ such that for
all $t\in[t_{0},t_{\epsilon})$,

$\displaystyle\|U(x,t)-U(x,t_{0})\|_{L^{\infty}(|x|\geq\frac{\epsilon_{0}}{4})}\leq\epsilon,\quad\|\nabla
U(x,t)-\nabla
U(x,t_{0})\|_{L^{\infty}(|x|\geq\frac{\epsilon_{0}}{4})}\leq\epsilon,$ (4.13)
$\displaystyle\|U(x,t_{\epsilon})-U(x,t_{0})\|_{L^{\infty}(|x|\geq\frac{\epsilon_{0}}{4})}=\epsilon\quad\|\nabla
U(x,t_{\epsilon})-\nabla
U(x,t_{0})\|_{L^{\infty}(|x|\geq\frac{\epsilon_{0}}{4})}=\epsilon.$ (4.14)

We remark from (3.14) that

$e^{U(x,t_{0})}=e^{\hat{U}^{*}(x)}\leq C(\epsilon_{0}),\quad|\nabla
U(x,t_{0})|^{2}=|\nabla\hat{U}^{*}(x)|^{2}\leq
C(\epsilon_{0}),\quad\forall|x|\geq\frac{\epsilon_{0}}{6}.$

Hence, from (4.13), we have

$|F(U,\nabla U)|=|e^{U(x,t)}+\alpha|\nabla U(x,t)|^{2}|\leq
C(\epsilon_{0}),\quad\forall|x|\geq\frac{\epsilon_{0}}{6},\;t\in[t_{0},t_{\epsilon}).$

Consider $U=\psi+U_{1}$, where $\psi(x)=-\ln(1+a|x|^{2})$ is introduced in
(1.3), we write from (1.1),

$\partial_{t}U_{1}=\Delta U_{1}+F(U,\nabla U)+\Delta\psi.$

From assumption $(i)$, we have in fact for all $t\in[t_{0},t_{\epsilon}]$ and
$|x|\in[\frac{\epsilon_{0}}{6},\frac{\epsilon_{0}}{4}]$, $|F(U,\nabla U)|\leq
C(\sigma_{0})$. We then consider $U_{2}(x,t)=\rho_{2}(x)U_{1}(x,t)$, where
$\rho_{2}(x)\in\mathcal{C}^{\infty}(\mathbb{R}^{N})$, $\rho_{2}(x)=1$ for
$|x|\geq\frac{\epsilon_{0}}{5}$, $\rho_{2}(x)=0$ for
$|x|\leq\frac{\epsilon_{0}}{6}$, $|\nabla\rho_{2}|\leq\frac{C}{\epsilon_{0}}$
and $|\Delta\rho_{2}|\leq\frac{C}{\epsilon_{0}^{2}}$. We then have

$\partial_{t}U_{2}=\Delta U_{2}-2\nabla\rho_{2}\cdot\nabla
U_{1}-\Delta\rho_{2}U_{1}+\rho_{2}F(U,\nabla U)+\rho_{2}\Delta\psi.$

Note from the definitions of $\rho_{2}$ and $\psi$ and assumption $(i)$ that
for all $t\in[t_{0},t_{*}]$,

$|2\nabla\rho_{2}\cdot\nabla U_{1}|+|\Delta\rho_{2}U_{1}|\leq
C(\epsilon_{0},\sigma_{0})\mathbf{1}_{\\{\frac{\epsilon_{0}}{6}\leq|x|\leq\frac{\epsilon_{0}}{5}\\}}.$

We write

$\partial_{t}U_{2}=\Delta U_{2}+\tilde{f}_{1}(x,t)+\rho_{2}F(U,\nabla
U)+\rho_{2}\Delta\psi,$

with $|\tilde{f}_{1}(x,t)|\leq
C(\epsilon_{0},\sigma_{0})\mathbf{1}_{\\{\frac{\epsilon_{0}}{6}\leq|x|\leq\frac{\epsilon_{0}}{5}\\}}.$

We denote by $S(\cdot)$ the linear heat flow, we write for all
$t\in[t_{0},t_{\epsilon})$,

$U_{2}(t)-S(t-t_{0})U_{2}(t_{0})=\int_{t_{0}}^{t}S(t-s)\left[\tilde{f}_{1}+\rho_{2}F(U,\nabla
U)+\rho_{2}\Delta\psi\right]ds.$

Hence,

$\displaystyle\|U_{2}(t)-U_{2}(t_{0})\|_{L^{\infty}(\mathbb{R}^{N})}$
$\displaystyle\leq\|U_{2}(t)-S(t-t_{0})U_{2}(t_{0})\|_{L^{\infty}(\mathbb{R}^{N})}$
$\displaystyle\quad+\|S(t-t_{0})U_{2}(t_{0})-U_{2}(t_{0})\|_{L^{\infty}(\mathbb{R}^{N})}$
$\displaystyle\leq\int_{t_{0}}^{t}\left\|S(t-s)\left[\tilde{f}_{1}+\rho_{2}F(U,\nabla
U)+\rho_{2}\Delta\psi\right]\right\|_{L^{\infty}(\mathbb{R}^{N})}ds$
$\displaystyle\quad+\|S(t-t_{0})\rho_{2}U_{1}(t_{0})-\rho_{2}U_{1}(t_{0})\|_{L^{\infty}(\mathbb{R}^{N})}$
$\displaystyle\leq
C(\epsilon,\sigma_{0})(t-t_{0})+\|S(t-t_{0})\rho_{2}U_{1}(t_{0})-\rho_{2}U_{1}(t_{0})\|_{L^{\infty}(\mathbb{R}^{N})}.$

From assumption $(ii)$ and the definition (3.14) of $\hat{U}^{*}$, we see that

$\|\rho_{2}U_{1}(t_{0})\|_{L^{\infty}(\mathbb{R}^{N})}=\|\rho_{2}(\hat{U}^{*}-\psi)\|_{L^{\infty}(\mathbb{R}^{N})}\leq
C(\epsilon_{0})\mathbf{1}_{\\{\frac{\epsilon_{0}}{6}\leq|x|\leq 1\\}}.$

Hence, if $t_{0}\in[t_{0,4}(\epsilon,\epsilon_{0},\sigma_{0}),T)$, we obtain

$\|U_{2}(t_{\epsilon})-U_{2}(t_{0})\|_{L^{\infty}(\mathbb{R}^{N})}\leq\frac{\epsilon}{2}.$

This follows
$\|U(t_{\epsilon})-U(t_{0})\|_{L^{\infty}(|x|\geq\frac{\epsilon_{0}}{6})}\leq\frac{\epsilon}{2}$,
which is a contradiction to (4.14). This concludes the proof of Proposition
4.3. ∎

###  4.2. Finite dimensional reduction.

In this subsection, we give the proof of Proposition 3.5, which follows from
the a priori estimates obtained in Propositions 4.1, 4.2 and 4.3. We proceed
in two steps: we first show that we can fix $K_{0},\delta_{0}$ and $C_{0}$
independently from $A$, take $A$ sufficiently large and choose
$\epsilon_{0},\alpha_{0}$ and $\eta_{0}$ in terms of $A$, so that all the
bounds given in Definition 3.1 are improved, except for the modes $Q_{0}$ and
$Q_{1}$. This immediately gives the conclusion of part $(i)$ of Proposition
3.5, which reduces the problem to a finite dimensional one. In second step, we
use the dynamics on the modes $Q_{0}$ and $Q_{1}$ given in Proposition 4.1 to
prove part $(ii)$ of Proposition 3.5.

####  4.2.1. Improved controls of $U(t)$ in $\mathcal{S}^{*}(t)$ and
conclusion of part $(i)$ of Proposition 3.5.

We aim at proving that for a suitable choice of the parameters
$t_{0},K_{0},\epsilon_{0},\alpha_{0}$, $A,\delta_{0},C_{0},C_{0}^{\prime}$ and
$\eta_{0}$, the bounds given in Definition 3.1 can be improved. In particular,
we want to prove that under the assumption of Proposition 3.5, the following
estimates hold:
\- (Improved controls in $\mathcal{D}_{1}$) For $s_{*}=-\ln(T-t_{*})$,

$\displaystyle i,j\in\\{1,\cdots,N\\},\;|Q_{2,i,j}(s_{*})|<\frac{A^{2}\ln
s_{*}}{s_{*}^{2}},\quad|Q_{-}(y,s_{*})|\leq\frac{A}{2s_{*}^{2}}(1+|y|^{3}),$
(4.15)
$\displaystyle|Q_{e}(y,s_{*})|\leq\frac{A^{2}}{2\sqrt{s_{*}}},\quad|(\nabla
Q)_{\bot}(y,s_{*})|\leq\frac{A}{2s_{*}^{2}}(1+|y|^{3}).$ (4.16)

\- (Improved controls in $\mathcal{D}_{2}$) For all
$|x|\in\left[\frac{K_{0}}{4}\sqrt{(T-t_{*})|\ln(T-t_{*})|},\epsilon_{0}\right]$
and $|\xi|\leq\alpha_{0}\sqrt{|\ln\theta(x)|}$, where $\theta(x)=T-t(x)$ and
$t(x)$ is defined by (3.5),

$\displaystyle|\mathcal{U}(x,\xi,\tau_{*})-\hat{\mathcal{U}}(\tau_{*})|\leq\frac{\delta_{0}}{2},\quad|\nabla_{\xi}\mathcal{U}(x,\xi,\tau_{*})|\leq\frac{C_{0}}{2\sqrt{|\ln\theta(x)|}},\quad|\nabla^{2}_{\xi}\mathcal{U}(x,\xi,\tau_{*})|\leq\frac{C_{0}^{\prime}}{2},$
(4.17)

where $\tau_{*}=\frac{t_{*}-t(x)}{\theta(x)}$.
\- (Improved controls in $\mathcal{D}_{3}$) For all
$|x|\geq\frac{\epsilon_{0}}{4}$,

$\displaystyle|U(x,t_{*})-U(x,t_{0})|\leq\frac{\eta_{0}}{2},\quad|\nabla
U(x,t_{*})-\nabla U(x,t_{0})|\leq\frac{\eta_{0}}{2}.$ (4.18)

One can see that once these improved estimates are proven, part $(ii)$ of
Proposition 3.5 immediately follows. Let us start with the estimates in
$\mathcal{D}_{1}$.

\- Proof of (4.15) and (4.16): For the estimate on $Q_{2,i,j}$, we argue by
contradiction. We assume that

$\forall s\in[s_{0},s_{*}),\;\;|Q_{2,i,j}(s)|<\frac{A^{2}\ln
s}{s^{2}},\quad|Q_{2,i,j}(s_{*})|=\frac{A^{2}\ln s_{*}}{s_{*}^{2}}.$

Let us consider the case $Q_{2,i,j}(s_{*})>0$ (the case $Q_{2,i,j}(s_{*})<0$
is similar), we have

$Q^{\prime}_{2,i,j}(s_{*})\geq\left.\frac{d}{ds}\left(\frac{A^{2}\ln
s}{s^{2}}\right)\right|_{s=s_{*}}=\frac{A^{2}}{s_{*}^{3}}-\frac{2A^{2}\ln
s^{*}}{s_{*}^{3}},$

on the one hand. On the other hand, we have by (4.3),

$Q^{\prime}_{2,i,j}(s_{*})\leq\frac{CA}{s_{*}^{3}}-\frac{2A^{2}\ln
s_{*}}{s_{*}^{3}},$

and a contradiction follows if $A\geq C+1$. This proves (4.15) for
$Q_{2,i,j}$.

For the improved controls of $Q_{-},Q_{e}$ and $(\nabla Q)_{\bot}$, we
distinguish in two cases:

\- Case 1: $s_{*}-s_{0}\leq\lambda_{1}$ where $\lambda_{1}=\lambda_{1}(A)>0$
is fixed later. We apply Proposition 4.1 with $\lambda^{*}=\lambda_{1}$,
$\lambda=s_{*}-s_{0}$ and $\sigma=s_{0}$ to obtain

$\displaystyle|Q_{-}(y,s_{*})|\leq C(1+\lambda_{1})s_{*}^{-2}(1+|y|^{3}),$
$\displaystyle|Q_{e}(y,s_{*})|\leq
CK_{0}^{3}e^{\lambda_{1}}(1+\lambda_{1})s_{*}^{-1/2},$ $\displaystyle|(\nabla
Q)_{\bot}(y,s_{*})|\leq C(1+\lambda_{1})s_{*}^{-2}(1+|y|^{3}).$

To have (4.15) and (4.16), we need $C(1+\lambda_{1})\leq A/2$ and
$CK_{0}^{3}(1+\lambda_{1})e^{\lambda_{1}}\leq A^{2}/2$, which is possible with
$\lambda_{1}=\frac{3}{2}\ln A$ for $A$ large enough.

\- Case 2: $s_{*}-s_{0}\geq\lambda_{2}$ where
$0<\lambda_{2}=\lambda_{2}(K_{0},A)\leq\lambda_{1}$. We apply Proposition 4.1
with $\lambda^{*}=\lambda=\lambda_{2}$, $\sigma=s_{*}-\lambda_{2}$ to obtain

$\displaystyle|Q_{-}(y,s_{*})|\leq
C(Ae^{-\lambda_{2}/2}+A^{2}e^{-\lambda_{2}^{2}}+\lambda_{2})s_{*}^{-2}(1+|y|^{3}),$
$\displaystyle|Q_{e}(y,s_{*})|\leq
C(A^{2}e^{-\lambda_{2}/2}+AK_{0}^{3}\lambda_{2}e^{\lambda_{2}})s_{*}^{-1/2},$
$\displaystyle|(\nabla Q)_{\bot}(y,s_{*})|\leq
C(Ae^{-\lambda_{2}/2}+C(K_{0})C_{0}e^{-\lambda_{2}^{2}}+\sqrt{\lambda_{2}}+\lambda_{2})s_{*}^{-2}(1+|y|^{3}).$

We now fix $\lambda_{2}$ so that

$C(Ae^{-\lambda_{2}/2}+A^{2}e^{-\lambda_{2}^{2}}+\lambda_{2})\leq\frac{A}{2},\quad
C(A^{2}e^{-\lambda_{2}/2}+AK_{0}^{3}\lambda_{2}e^{\lambda_{2}})\leq\frac{A^{2}}{2},$

and

$C(Ae^{-\lambda_{2}/2}+C(K_{0})C_{0}e^{-\lambda_{2}^{2}}+\sqrt{\lambda_{2}}+\lambda_{2})\leq\frac{A}{2},$

which is possible with $\lambda_{2}=\ln(A/(8CK_{0}^{3}))$ and $C_{0}\leq
A^{3}$, then the conclusion follows for $A$ large enough. This concludes the
proof of (4.15) and (4.16).

\- Proof of (4.17): We aim at using Proposition 4.2 to prove (4.17). For this
we need to check the hypothesis of Proposition 4.3 under the hypothesis of
Proposition 3.5. This is done thanks to the following result:

######  Lemma 4.4.

Under the hypothesis of Proposition 3.5, we have for

$|x|\in\left[\frac{K_{0}}{4}\sqrt{(T-t_{*})|\ln(T-t_{*})|},\epsilon_{0}\right],$

  * (i)

For all $|\xi|\leq\frac{7}{4}\alpha_{0}\sqrt{|\ln\theta(x)|}$ and
$\tau\in\left[\max\left\\{0,\frac{t_{0}-t(x)}{\theta(x)}\right\\},\frac{t_{*}-t(x)}{\theta(x)}\right]$,

$\displaystyle|\nabla_{\xi}\mathcal{U}(x,\xi,\tau)|\leq\frac{2C_{0}}{\sqrt{|\ln\theta(x)|}},\quad|\nabla_{\xi}^{2}\mathcal{U}(x,\xi,\tau)|\leq
2C_{0}^{\prime},\quad\mathcal{U}(x,\xi,\tau)\geq\frac{1}{2}\hat{\mathcal{U}}(\tau).$

  * (ii)

For all $|\xi|\leq 2\alpha_{0}\sqrt{|\ln\theta(x)|}$ and
$\tau=\max\left\\{0,\frac{t_{0}-t(x)}{\theta(x)}\right\\}$,

$\forall\delta_{1}\leq
1,\;|\mathcal{U}(x,\xi,\tau)-\hat{\mathcal{U}}(\tau)|\leq\delta_{1},\quad|\nabla_{\xi}\mathcal{U}(x,\xi,\tau)|\leq\frac{C_{0}}{4\sqrt{|\ln\theta(x)|}}.$

###### Proof.

Since the proof of Lemma 4.4 is very similar to the one given in [MZ97a] and
no new ideas are needed, we refer the interested reader to see Lemma 2.6 at
page 1515 in that paper for all details of the proof. ∎

From Lemma 4.4, we apply Proposition 4.2 with $C_{0,1}^{*}=\frac{C_{0}}{4}$,
$C_{0,2}^{*}=2C_{0}$, $C_{0,3}^{*}=2C_{0}^{\prime}$,
$\xi_{0}=\alpha_{0}\sqrt{|\ln\theta(\epsilon_{0})|}$ with the choice
$\alpha_{0}\leq 1$ to derive for $|\xi|\leq\alpha_{0}\sqrt{|\ln\theta(x)|}$,

$|\mathcal{U}(x,\xi,\tau_{*})-\hat{\mathcal{U}}(\tau_{*})|\leq\frac{\delta_{0}}{2},\quad|\nabla_{\xi}\mathcal{U}(x,\xi,\tau_{*})|\leq\frac{C_{0}}{2\sqrt{|\ln\theta(x)|}}.$

By a direct parabolic estimate, we see that there exists $t_{0,6}(A)<T$ such
that for all $t_{0}\in[t_{0,6},T)$, if

$U(t_{0})\in\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},C_{0},C_{1},\eta_{0},t_{0})$

and

$U(t)\in\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},C_{0},C_{0}^{\prime},\eta_{0},t)\quad\forall
t\in[t_{0},t^{\prime}],$

then

$U(t^{\prime})\in\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},C_{0},3/4,\eta_{0},t^{\prime}).$

With the choice of $C_{0}^{\prime}\geq 2$ and from Proposition 3.4, we have
$|\nabla_{\xi}^{2}\mathcal{U}(x,\xi,\tau_{*})|\leq\frac{3}{4}\leq\frac{C_{0}^{\prime}}{2}$.
This concludes the proof of (4.17), assuming that Lemma 4.4 holds.

\- Proof of (4.18): We aim at using Proposition 4.3 to improve the estimates
in $\mathcal{D}_{3}$. Let us check the hypothesis of Proposition 4.3. From
$(ii)$ of Definition (3.1), we have for all $t\in[t_{0},t_{*}]$ and
$|x|\in[\frac{\epsilon_{0}}{6},\frac{\epsilon_{0}}{4}]$,

$|\mathcal{U}(x,0,\tau)-\hat{\mathcal{U}}(\tau)|\leq\delta_{0},\quad|\nabla_{\xi}\mathcal{U}(x,0,\tau)|\leq\frac{C_{0}}{\sqrt{|\ln\theta(x)|}},\quad|\nabla^{2}_{\xi}\mathcal{U}(x,0,\tau)|\leq
C_{0}^{\prime},$ (4.19)

where $\tau=\frac{t-t(x)}{\theta(x)}$. From the definition (3.4) of
$\mathcal{U}(x,\xi,\tau)$ and (4.19), we obtain
$|\nabla^{i}U(x,t)|\leq\sigma_{0}(K_{0},\epsilon_{0},C_{0},C_{0}^{\prime})$.
From (3.12), we have $U(x,t_{0})=\hat{U}^{*}(x)$ for
$|x|\geq\frac{\epsilon_{0}}{6}$. Hence, Proposition 4.3 applies with
$\epsilon=\eta_{0}/2$ and we obtain for all $t\in[t_{0},t_{*}]$ and
$|x|\geq\frac{\epsilon_{0}}{4}$,

$|U(x,t)-U(x,t_{0})|+|\nabla U(x,t)-\nabla U(x,t_{0})|\leq\frac{\eta_{0}}{2},$

which concludes the proof of (4.18). This also completes the proof of part
$(i)$ of Proposition 3.5.

####  4.2.2. Outgoing transversality of $U(t)$ on
$\partial\mathcal{S}^{*}(t)$.

We give here the proof of part $(ii)$ of Proposition 3.5. The proof simply
follows from part $(i)$ of Proposition 3.5 and the ODEs (4.1) and (4.2).
Indeed, from part $(i)$, we know that for $\omega=\pm 1$,
$Q_{0}(s_{*})=\frac{\omega A}{s_{*}^{2}}$ and $Q_{1,i}(s_{*})=\frac{\omega
A}{s_{*}^{2}}$ for $i\in\\{1,\cdots,N\\}$. From (4.1) and (4.2), we see that

$\displaystyle\omega Q_{0}^{\prime}(s_{*})\geq\omega
Q_{0}(s_{*})-\frac{C}{s_{*}^{2}}\geq\frac{A-C}{s_{*}^{2}},$
$\displaystyle\forall i\in\\{1,\cdots,N\\},\;\;$ $\displaystyle\omega
Q_{1,i}^{\prime}(s_{*})\geq\frac{1}{2}\omega
Q_{1,i}(s_{*})-\frac{C}{s_{*}^{2}}\geq\frac{A/2-C}{s_{*}^{2}}.$

Taking $A$ large enough gives $\omega Q^{\prime}_{0}(s_{*})>0$ and $\omega
Q^{\prime}_{1,i}(s_{*})>0$, which means that $Q_{0}$ and $Q_{1,i}$ are
traversal outgoing to the bounding curve $s\mapsto\omega As^{-2}$ at
$s=s_{*}$. This concludes the proof of part $(ii)$ and completes the proof of
Proposition 3.5.

##  Appendix A Properties of the set $\mathcal{V}_{A,K_{0}}$ and the initial
data (3.12).

In this appendix, we give some properties of the shrinking set
$\mathcal{V}_{A,K_{0}}$ defined in part $(i)$ of Definition 3.1 as well as the
proof of Proposition 3.4. Let us start with the following proposition:

######  Proposition A.1 (Estimates on $Q$ and $\nabla Q$ in
$\mathcal{V}_{A,K_{0}}$).

For all $K_{0}\geq 1$ and $\epsilon_{0}>0$, there exist
$t_{0,2}(K_{0},\epsilon_{0})$ and $\eta_{0,2}(\epsilon_{0})>0$ such that for
all $t_{0}\in[t_{0,2},T)$, $A\geq 1$, $\alpha_{0}>0$, $C_{0}>0$,
$\delta_{0}\leq\frac{1}{2}|\hat{\mathcal{U}}(1)|$ and
$\eta_{0}\in(0,\eta_{0,2}]$, we have the following properties: Assume that
$U(x,t_{0})$ is given by (3.12) and that for some $t\in[t_{0},T)$,

$U(t)\in\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},\eta_{0},C_{0},C_{0}^{\prime},t),$

then, there exists a positive constant $C=C(K_{0},C_{0})$ such that for all
$y\in\mathbb{R}^{N}$ and $s=-\log(T-t)$,
$(i)$ (Estimates on $Q$)

$|Q(y,s)|\leq\frac{CA^{2}}{\sqrt{s}},\quad|Q(y,s)|\leq\frac{CA^{2}\ln
s}{s^{2}}(|y|^{2}+1)+\frac{CA^{2}}{s^{2}}(|y|^{3}+1).$

$(ii)$ (Estimates on $\nabla Q$)

$\displaystyle|\nabla Q(y,s)|\leq\frac{CA^{2}}{\sqrt{s}},\quad|\nabla
Q(y,s)|\leq\frac{CA^{2}\ln s}{s^{2}}(|y|^{3}+1),\quad|(1-\chi(y,s))\nabla
Q(y,s)|\leq\frac{C}{\sqrt{s}}.$

###### Proof.

$(i)$ From part $(i)$ of Definition 3.1, we have
$Q(s)\in\mathcal{V}_{A,K_{0}}(s)$. From the decomposition (2.31), we have

$\displaystyle|Q(y,s)|$
$\displaystyle\leq\sum_{m=0}^{2}|Q_{m}(s)||h_{m}(y)|+|Q_{-}(y,s)|+|Q_{e}(y,s)|$
$\displaystyle\leq\frac{CA^{2}\ln
s}{s^{2}}(|y^{2}|+1)+\frac{A}{s^{2}}(|y|^{3}+1)+\frac{\|Q_{e}(s)\|_{L^{\infty}}(\mathbb{R}^{N})}{|y|^{3}+1}(|y|^{3}+1)\mathbf{1}_{\\{|y|\geq
K_{0}\sqrt{s}\\}}$ $\displaystyle\leq\frac{CA^{2}\ln
s}{s^{2}}(|y|^{2}+1)+\frac{CA^{2}}{s^{2}}(|y|^{3}+1).$

We also have

$|Q_{b}(y,s)|\leq\left\\{\sum_{m=0}^{2}|Q_{m}(s)||h_{m}(y)|+|Q_{-}(y,s)|\right\\}\mathbf{1}_{\\{|y|\leq
2K_{0}\sqrt{s}\\}}\leq\frac{CA^{2}\ln s}{\sqrt{s}},$

hence,

$\displaystyle|Q(y,s)|\leq\|Q_{b}(s)\|_{L^{\infty}(\mathbb{R}^{N})}+\|Q_{e}(s)\|_{L^{\infty}(\mathbb{R}^{N})}\leq\frac{CA^{2}}{\sqrt{s}},$

which concludes the proof of part $(i)$.

$(ii)$ Arguing similarly as for $(i)$, we obtain from part $(i)$ of Definition
3.1 and (2.32),

$|\chi(y,s)\nabla Q(y,s)|\leq\frac{CA^{2}\ln
s}{s^{2}}(|y|^{3}+1),\quad|\chi(y,s)\nabla
Q(y,s)|\leq\frac{CA^{2}}{\sqrt{s}}.$

Note from (2.17) that $|\nabla\psi_{\alpha}(y,s)|\leq\frac{C}{\sqrt{s}}$ and
$\frac{C}{\sqrt{s}}\leq\frac{C}{s^{2}}(|y|^{3}+1)$ for $|y|\geq K_{0}\sqrt{s}$
and $K_{0}\geq 1$. We have to prove that

$|(1-\chi(y,s))\nabla(Q+\psi_{\alpha})(y,s)|=|(1-\chi(y,s))\nabla
W(y,s)e^{W(y,s)}|\leq\frac{C}{\sqrt{s}}$

in order to conclude the proof. From (1.8), this reduces to show that for all
$t\geq t_{0}$ and $|x|\geq r(t)=K_{0}\sqrt{(T-t)|\log(T-t)|}$,

$|\nabla_{x}U(x,t)e^{U(x,t)}|\leq\frac{C}{(T-t)^{\frac{3}{2}}\sqrt{|\ln(T-t)|}}.$
(A.1)

To prove (A.1), we argue as in [MZ97a] by considering two cases:
\- Case 1: $|x|\in[r(t),\epsilon_{0}]$. In this case, we use the bounds given
in part $(ii)$ of Definition 3.1 to prove (A.1). From (3.4), we have

$U(x,t)=-\ln\theta(x)+\mathcal{U}(x,0,\tau(x,t))$

and

$\nabla_{x}U(x,t)=\theta(x)^{-\frac{1}{2}}\nabla_{\xi}\mathcal{U}(x,0,\tau(x,t)),$

where $\theta(x)=T-t(x)$, $\tau(x,t)=\frac{t-t(x)}{\theta(x)}$ and $t(x)$ is
uniquely determined by (3.5). Therefore,

$|\nabla_{x}U(x,t)e^{U(x,t)}|=\theta(x)^{-\frac{3}{2}}\nabla_{\xi}\mathcal{U}(x,0,\tau(x,t))e^{\mathcal{U}(x,0,\tau(x,t))}.$

Using part $(ii)$ of Definition 3.1, we have for $|x|\in[r(t),\epsilon_{0}]$,

$|\mathcal{U}(x,0,\tau(x,t))-\hat{\mathcal{U}}(\tau(x,t))|\leq\delta_{0},\quad|\nabla_{\xi}\mathcal{U}(x,0,\tau(x,t))|\leq\frac{C_{0}}{\sqrt{|\log(\theta(x))|}}.$

Since
$\delta_{0}\leq\frac{1}{2}|\hat{\mathcal{U}}(1)|\leq\frac{1}{2}|\hat{\mathcal{U}}(\tau)|$,
we have from (3.5),

$|\nabla_{x}U(x,t)e^{U(x,t)}|\leq\frac{C(K_{0},C_{0})}{\theta(x)^{\frac{3}{2}}\sqrt{|\log\theta(x)|}}\leq\frac{C(K_{0},C_{0})}{\theta(r(t))^{\frac{3}{2}}\sqrt{|\ln\theta(r(t))|}}.$

Since $r(t)\to 0$ as $t\to T$, we have from (3.5),

$\theta(r(t))\sim\frac{2}{K_{0}^{2}}\frac{r(t)^{2}}{|\ln
r(t)|}\quad\text{and}\quad\ln\theta(r(t))\sim\ln r(t)\quad\text{as}\;\;t\to
T.$

Recall that $r(t)=K_{0}\sqrt{(T-t)|\ln(T-t)|}$ which gives

$\frac{C(K_{0},C_{0})}{\theta(r(t))^{\frac{3}{2}}\sqrt{|\ln\theta(r(t))|}}\sim\frac{C^{\prime}(K_{0},C_{0})}{(T-t)^{\frac{3}{2}}|\ln(T-t)|}.$

This concludes the proof of (A.1) for $|x|\in[r(t),\epsilon_{0}]$.

\- Case 2: $|x|\geq\epsilon_{0}$. We use here the information contained in
$(iii)$ of Definition 3.1, which asserts that

$|\nabla_{x}^{i}(U(x,t)-U(x,t_{0}))|\leq\eta_{0},\quad
i=0,1,\quad\forall|x|\geq\epsilon_{0}.$

Let

$\eta_{0,2}(\epsilon_{0})=\frac{1}{2}\min\left\\{\min_{|x|\geq\epsilon_{0}}|U(x,t_{0})|,\min_{|x|\geq\epsilon_{0}}|\nabla
U(x,t_{0})|\right\\}.$

From (3.12), we have $\eta_{0,2}(\epsilon_{0})>0$. Hence, for
$\eta_{0}\in(0,\eta_{0,2}]$, we obtain for $|x|\geq\epsilon_{0}$,

$\left|\nabla_{x}U(x,t)e^{U(x,t)}\right|\leq
C\left|\nabla_{x}U(x,t_{0})e^{U(x,t_{0})}\right|\leq
C\left|\nabla_{x}U^{*}(x)e^{U^{*}(x)}\right|,$

where $\hat{U}^{*}$ is defined by (3.14). Note that
$\left|\nabla_{x}U^{*}(x)e^{U^{*}(x)}\right|\leq C(\epsilon_{0})$ for
$|x|\geq\epsilon_{0}$. Therefore, if $t_{0}\in[t_{0,2},T)$ for some
$t_{0,2}<T$, we derive (A.1) for $t=t_{0}$ and $|x|\geq\epsilon_{0}$, hence,
for $t\geq t_{0}$ and $|x|\geq\epsilon_{0}$. This completes the proof of
Proposition A.1. ∎

We now turn to the proof of Proposition 3.4 which is a direct consequence of
the following lemma:

######  Lemma A.2.

There exists $K_{0,1}>0$ such that for each $K_{0}\geq K_{0,1}$ and
$\delta_{0,1}>0$, there exist $\alpha_{0,1}(K_{0},\delta_{0,1})>0$,
$C_{0,1}(K_{0})>0$ and $C_{0,1}^{\prime}(K_{0})>0$ such that for all
$\alpha_{0}\in(0,\alpha_{0,1}]$, there exists
$\epsilon_{0,1}(K_{0},\delta_{0,1},\alpha_{0})>0$ such that for all
$\epsilon_{0}\in(0,\epsilon_{0,1}]$ and $A\geq 1$, there exists
$t_{0,1}(K_{0},\delta_{0,1},\epsilon_{0},A,C_{0,1})<T$ such that for all
$t_{0}\in[t_{0,1},T)$, there exists a subset
$\mathcal{D}_{t_{0},A}\subset\mathbb{R}\times\mathbb{R}^{N}$ with the
following properties. If $U(x,t_{0})$ is defined by (3.12), then:

$(I)$ For all $(d_{0},d_{1})\in\mathcal{D}_{t_{0},A}$, we have
$U_{d_{0},d_{1}}(x,t_{0})$ defined by (3.12) belongs in
$\mathcal{S}^{*}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0,1},\eta_{0}=0,C_{0,1},C_{0,1}^{\prime},t_{0})$.
More precisely,

$(i)$ Estimates in $\mathcal{D}_{1}$:
$\bar{Q}(s_{0})\in\mathcal{V}_{A,K_{0}}(s_{0})$ with strict inequalities,
except for $(\bar{Q}_{0},\bar{Q}_{1})(s_{0})$ in the sense that

$\left|\bar{Q}_{0}(s)-\frac{Ad_{0}}{s_{0}^{2}}\right|\leq
Ce^{-s_{0}},\quad\left|\bar{Q}_{1,i}(s_{0})-\frac{Ad_{1,i}}{s_{0}^{2}}\right|\leq
Ce^{-s_{0}},\quad i\in\\{1,\cdots,N\\},$
$|\bar{Q}_{2,ij}(s_{0})|\leq\frac{\ln(s_{0})}{s_{0}^{2}},\quad\forall
i,j\in\\{1,\cdots,N\\},$
$|\bar{Q}_{-}(y,s_{0})|\leq\frac{1}{s_{0}^{2}}(|y|^{3}+1),\quad|(\nabla_{y}\bar{Q})_{\bot}(y,s_{0})|\leq\frac{1}{s_{0}^{2}}(|y|^{3}+1),\quad\forall
y\in\mathbb{R}^{N},$ $\|\bar{Q}_{e}(s_{0})\|_{L^{\infty}(\mathbb{R}^{N})}=0,$

where $\bar{Q}_{m}$ ($m=0,1,2$), $\bar{Q}_{-}$, $\bar{Q}_{\bot}$ and
$\bar{Q}_{e}$ are defined as in (2.31) and (2.32).

$(ii)$ Estimates in $\mathcal{D}_{2}$: For all
$|x|\in\left[\frac{K_{0}}{4}\sqrt{|\ln(T-t_{0})|(T-t_{0})},\epsilon_{0}\right]$,
$\tau_{0}=\tau_{0}(x,t_{0})=\frac{t_{0}-t(x)}{\theta(x)}$ and
$|\xi|\leq\alpha_{0}\sqrt{\ln\theta(x)}$,

$\left|\mathcal{U}(x,\xi,\tau_{0})-\hat{\mathcal{U}}(\tau_{0})\right|\leq\delta_{0,1},\;\;|\nabla_{\xi}\mathcal{U}(x,\xi,\tau_{0})|\leq\frac{C_{0,1}}{\sqrt{|\ln\theta(x)|}},\;\;|\nabla^{2}_{\xi}\mathcal{U}(x,\xi,\tau_{0})|\leq
C_{0,1},$

where $\mathcal{U}$, $\hat{\mathcal{U}}$, $t(x)$ and $\theta(x)$ are defined
in (3.4), (3.9), (3.5) and (3.6) respectively.

$(II)$

$\displaystyle(d_{0},d_{1})\in\mathcal{D}_{t_{0},A}$
$\displaystyle\Longleftrightarrow(\bar{Q}_{0}(s_{0}),\bar{Q}_{1}(s_{0}))\in\hat{\mathcal{V}}_{A}(s_{0}),$
$\displaystyle(d_{0},d_{1})\in\partial\mathcal{D}_{t_{0},A}$
$\displaystyle\Longleftrightarrow(\bar{Q}_{0}(s_{0}),\bar{Q}_{1}(s_{0}))\in\partial\hat{\mathcal{V}}_{A}(s_{0}).$

###### Proof.

Part $(II)$ directly follows from item $(i)$ of part $(I)$. Since we have
almost the same definition of $\mathcal{V}_{A,K_{0}}$ given part $(i)$ of
Definition 3.1 and the definition of initial data (3.15) as those defined in
[MZ97a], the proof of item $(i)$ is completely analogous as those in [MZ97a].
For this reason, we kindly refer the interested reader to see in particular
pages 1535-1536 in that paper for all details of these computations. Because
our definition (3.12) is different from those defined in [MZ97a] in the region
$\mathcal{D}_{2}$, we only deal with the proof of item $(ii)$.

Let us consider $t_{0}<T$, $K_{0}$, $\epsilon_{0}$, $\alpha_{0}$,
$\delta_{0,1}$, $C_{0,1}$ and $C_{0,1}^{\prime}$, and prove that if these
constants are suitably chosen, then for
$|x|\in\left[r_{0},\epsilon_{0}\right]$ and $|\xi|\leq
2\alpha_{0}\sqrt{|\ln\theta(x)|}$, where
$r_{0}=\frac{K_{0}}{4}\sqrt{\theta_{0}|\ln\theta_{0}|}$, $\theta_{0}=T-t_{0}$,
$\theta(x)=T-t(x)$ and $t(x)$ is defined as in (3.5), we have

$\left|\mathcal{U}(x,\xi,\tau_{0})-\hat{\mathcal{U}}(\tau_{0})\right|\leq\delta_{0,1},\;\;|\nabla_{\xi}\mathcal{U}(x,\xi,\tau_{0})|\leq\frac{C_{0,1}(K_{0})}{\sqrt{|\log\theta(x)|}},\;\;|\nabla_{\xi}^{2}\mathcal{U}(x,\xi,\tau_{0})|\leq
C_{0,1}^{\prime}(K_{0}),$

where $\tau_{0}=\frac{t_{0}-t(x)}{T-t(x)}$ and $\hat{\mathcal{U}}$ is given by
(3.9).

We remark from (3.5) that if $\alpha_{0}\leq\frac{K_{0}}{16}$ and
$\epsilon_{0}\leq\frac{2}{3}C(a_{1},\alpha)$, where $C(a_{1},\alpha)$ is
introduced in (3.14), then for $|x|\in[r_{0},\epsilon_{0}]$ and $|\xi|\leq
2\alpha_{0}\sqrt{|\ln\theta(x)|}$, we have
$|\xi\sqrt{\theta(x)}|\leq\frac{|x|}{2}$, hence

$\frac{r_{0}}{2}\leq\frac{|x|}{2}\leq|x+\xi\sqrt{\theta(x)}|\leq\frac{3|x|}{2}\leq
C(a_{1},\alpha).$

This implies that

$\chi(16(x+\xi\sqrt{\theta(x)})\sqrt{\theta_{0}},-\ln\theta_{0})\chi_{1}(x+\xi\sqrt{\theta(x)},t_{0})=0,$

where $\chi$ and $\chi_{1}$ are defined by (2.26) and (3.13) respectively.
Hence, from (3.12), (3.4), (3.14) and (1.18), we write

$\displaystyle\mathcal{U}(x,\xi,\tau_{0})$
$\displaystyle=\ln\left(\frac{(8+8\alpha)\theta(x)|\ln|x+\xi\sqrt{\theta(x)}||}{|x+\xi\sqrt{\theta(x)}|^{2}}\right)(1-\chi_{1}(x+\xi\sqrt{\theta(x)}),t_{0})$
$\displaystyle-\ln\left[\frac{\theta_{0}}{\theta(x)}\left(1+\frac{|x+\xi\sqrt{\theta(x)}|^{2}}{(4+4\alpha)\theta_{0}|\ln\theta_{0}|}\right)\right]\chi_{1}(x+\xi\sqrt{\theta(x)}),t_{0})$
$\displaystyle-\frac{N}{(2+2\alpha)\ln\theta_{0}}\chi_{1}(x+\xi\sqrt{\theta(x)}),t_{0})$
(A.2)
$\displaystyle:=(I)(1-\chi_{1}(x+\xi\sqrt{\theta(x)}),t_{0}))+(II+III)\chi_{1}(x+\xi\sqrt{\theta(x)}),t_{0}).$

From (3.5), we remark that

$\ln\theta(x)\sim
2\ln|x|,\quad\theta(x)\sim\frac{8}{K_{0}^{2}}\frac{|x|^{2}}{|\ln|x||}\quad\text{as
$|x|\to 0$},$ (A.3)

from which we have for a fixed number $K_{0}>0$,

$\theta(r_{0})\sim\theta_{0},\;\theta(R_{0})\sim\frac{16}{K_{0}^{2}}\theta_{0}|\ln\theta_{0}|,\;\theta(2R_{0})\sim\frac{64}{K_{0}^{2}}\theta_{0}|\ln\theta_{0}|,\;R_{0}=\sqrt{\theta_{0}}|\ln\theta_{0}|.$
(A.4)

\- Estimate on $\mathcal{U}$. By linearity and the definition (3.13) of
$\chi_{1}$, it is enough to prove that for $|x|\in[R_{0},\epsilon_{0}]$ and
$|\xi|\leq 2\alpha_{0}\sqrt{|\ln\theta(x)|}$,

$\displaystyle\left|(I)-\hat{\mathcal{U}}(\tau_{0})\right|\leq\frac{\delta_{0,1}}{2},$
(A.5)

and for $|x|\in[r_{0},2R_{0}]$ and $|\xi|\leq
2\alpha_{0}\sqrt{|\ln\theta(x)|}$,

$\displaystyle\left|(II)-\hat{\mathcal{U}}(\tau_{0})\right|\leq\frac{\delta_{0,1}}{2}.$
(A.6)

We start with (A.5). From (3.5) and (3.9), we write

$\displaystyle|(I)-\hat{\mathcal{U}}(\tau_{0})|$
$\displaystyle=\left|-\ln\left(\frac{(8+8\alpha)|\ln|x+\xi\sqrt{\theta(x)}||}{|\frac{K_{0}}{4}\sqrt{|\ln\theta(x)|}+\xi|^{2}}\right)^{-1}+\ln\left(\frac{\theta_{0}}{\theta(x)}+\frac{K_{0}^{2}/16}{4+4\alpha}\right)\right|$
$\displaystyle\leq
C(K_{0})\left|\frac{\theta_{0}}{\theta(x)}+\frac{K_{0}^{2}}{16(4+4\alpha)}-\frac{|\frac{K_{0}}{4}\sqrt{|\ln\theta(x)|}+\xi|^{2}}{(4+4\alpha)|\ln|x+\xi\sqrt{\theta(x)}|^{2}|}\right|$
$\displaystyle\leq
C(K_{0})\left[\left|\frac{\theta_{0}}{\theta(R_{0})}\right|+\left|\frac{|\frac{K_{0}}{4}\sqrt{|\ln\theta(x)|}+2\alpha_{0}\sqrt{|\ln\theta(x)|}|^{2}}{|\ln|x+2\alpha_{0}\sqrt{\theta(x)|\ln\theta(x)|}|^{2}|}-\frac{K_{0}^{2}}{16}\right|\right]$
$\displaystyle\leq
C(K_{0})\left[\frac{1}{|\ln\theta_{0}|}+\left|\frac{|\ln\theta(x)|}{|\ln|x|^{2}+\ln(1+8\alpha_{0}/K_{0})^{2}|}\left(\frac{K_{0}}{4}+2\alpha_{0}\right)^{2}-\frac{K_{0}^{2}}{16}\right|\right].$

Since $|x|\leq\epsilon_{0}$ and $\ln|x|^{2}\sim\ln\theta(x)$ as $|x|\to 0$, we
find $\alpha_{0,1}(K_{0},\delta_{0,1})$ and $t_{0,1}(K_{0},\delta_{0,1})$ such
that for each $\alpha_{0}\leq\alpha_{0,1}$, there is
$\epsilon_{0,1}(K_{0},\delta_{0,1},\alpha_{0})$ such that for all
$\epsilon_{0}\leq\epsilon_{0,1}$ and $t_{0}\geq t_{0,1}$, the estimate (A.5)
holds.

We now deal with (A.6). From (3.9) and (3.5), we write

$\displaystyle|(II)-\hat{\mathcal{U}}(\tau_{0})|$
$\displaystyle=\left|\ln\left(\frac{\theta_{0}}{\theta(x)}+\frac{K_{0}^{2}/16}{4+4\alpha}\right)-\ln\left(\frac{\theta_{0}}{\theta(x)}+\frac{|x+\xi\sqrt{\theta(x)}|^{2}}{(4+4\alpha)\theta(x)|\ln\theta_{0}|}\right)\right|$
$\displaystyle\quad\leq
C\left|\frac{|x+\xi\sqrt{\theta(x)}|^{2}}{\theta(x)|\ln\theta_{0}|}-\frac{K_{0}^{2}}{16}\right|\leq
CK_{0}^{2}\left|\left|\sqrt{\frac{\ln\theta(x)}{\ln\theta_{0}}}+\frac{4\xi}{K_{0}\sqrt{|\ln\theta_{0}|}}\right|^{2}-1\right|.$

Since $|x|\in[r_{0},2R_{0}]$ and $|\xi|\leq 2\alpha_{0}\sqrt{|\ln\theta(x)|}$,
we have

$\displaystyle\frac{\ln\theta(2R_{0})}{\ln\theta_{0}}\left(1-\frac{8\alpha_{0}}{K_{0}}\right)^{2}-1$
$\displaystyle\leq\left(\sqrt{\frac{\ln\theta(x)}{\ln\theta_{0}}}+\frac{4\xi}{K_{0}\sqrt{|\ln\theta_{0}|}}\right)^{2}-1$
$\displaystyle\qquad\qquad\leq\frac{\ln\theta(r_{0})}{\ln\theta_{0}}\left(1+\frac{8\alpha_{0}}{K_{0}}\right)^{2}-1.$

Together with (A.4), we find $\alpha_{0,2}(K_{0},\delta_{0,1})$ and
$t_{0,2}(K_{0},\delta_{0,1})<T$ such that for all $\alpha_{0}\leq\alpha_{0,2}$
and $t_{0}\geq t_{0,2}$, the estimate (A.6) holds.

\- Estimate on $\nabla_{\xi}\mathcal{U}$. From (A.2), we write

$\displaystyle\nabla_{\xi}\mathcal{U}(x,\xi,\tau_{0})$
$\displaystyle=\nabla_{\xi}(I)(1-\chi_{1}(x+\xi\sqrt{\theta(x)}),t_{0}))$
$\displaystyle\quad+\nabla_{\xi}(II)\chi_{1}(x+\xi\sqrt{\theta(x)}),t_{0})$
$\displaystyle\qquad+(II-I)\sqrt{\theta(x)}\nabla_{x}\chi_{1}(x+\xi\sqrt{\theta(x)}),t_{0}).$

By the definition (3.13) of $\chi_{1}$, it is enough to prove the followings
in order to obtain the estimate on $\nabla_{\xi}\mathcal{U}$:
\- for $|x|\in[R_{0},\epsilon_{0}]$ and $|\xi|\leq
2\alpha_{0}\sqrt{|\ln\theta(x)|}$,

$|\nabla_{\xi}(I)|\leq\frac{C(K_{0})}{\sqrt{|\ln\theta(x)|}},$ (A.7)

\- for $|x|\in[r_{0},2R_{0}]$ and $|\xi|\leq
2\alpha_{0}\sqrt{|\ln\theta(x)|}$,

$|\nabla_{\xi}(II)|\leq\frac{C(K_{0})}{\sqrt{|\ln\theta(x)|}},$ (A.8)

\- for $|x|\in[R_{0},2R_{0}]$ and $|\xi|\leq
2\alpha_{0}\sqrt{|\ln\theta(x)|}$,

$|I-II|\sqrt{\theta(x)}|\nabla_{x}\chi_{1}(x+\xi\sqrt{\theta(x)}),t_{0})|\leq\frac{C(K_{0})}{\sqrt{|\ln\theta(x)|}}.$
(A.9)

For (A.7) and (A.8), a straightforward computation yields

$\displaystyle|\nabla_{\xi}(I)|$
$\displaystyle=\left|\nabla_{\xi}\ln\left(\frac{(8+8\alpha)|\ln|x+\xi\sqrt{\theta(x)}||}{|\frac{K_{0}}{4}\sqrt{|\ln\theta(x)|}+\xi|^{2}}\right)\right|$
$\displaystyle\quad\leq\frac{C\sqrt{\theta(x)}}{|x+\xi\sqrt{\theta(x)}||\ln|x+\xi\sqrt{\theta(x)}||}+\frac{C}{|\frac{K_{0}}{4}\sqrt{|\ln\theta(x)|}+\xi|}\leq\frac{C}{\sqrt{|\ln\theta(x)|}},$

and

$\displaystyle|\nabla_{\xi}(II)|$
$\displaystyle=\left|\nabla_{\xi}\ln\left(1+\frac{|x+\xi\sqrt{\theta(x)}|^{2}}{(4+4\alpha)\theta_{0}|\ln\theta_{0}|}\right)\right|$
$\displaystyle\quad\leq\frac{C\sqrt{\theta(x)}}{|x+\xi\sqrt{\theta(x)}|}\leq\frac{C}{|\frac{K_{0}}{4}\sqrt{|\ln\theta(x)|}+\xi|}\leq\frac{C}{\sqrt{|\ln\theta(x)|}}.$

As for (A.9), we note from (3.13) that

$|\nabla_{x}\chi_{1}(x+\xi\sqrt{\theta(x)}),t_{0})|\leq\frac{C}{\sqrt{\theta_{0}}|\ln\theta_{0}|}.$

From (A.4), we have $\theta(x)\sim C(K_{0})\theta_{0}|\ln\theta_{0}|$ and
$|\ln\theta(x)|\sim|\ln\theta_{0}|$ for $x\in[R_{0},2R_{0}]$. Therefore, the
estimate (A.9) follows once the following is proved: for
$|x|\in[R_{0},2R_{0}]$ and $|\xi|\leq 2\alpha_{0}\sqrt{|\ln\theta(x)|}$,

$|(I)|+|(II)|\leq C.$

To this end, we note from (A.4) and (A.3) that
$|\ln|x+\xi\sqrt{\theta(x)}||=|\ln|x|+\ln(1+\frac{\xi\sqrt{\theta(x)}}{|x|})|\sim\frac{1}{2}|\ln\theta(x)|$,
and that $|x+\xi\sqrt{\theta(x)}|^{2}\sim C(K_{0})\theta(x)|\ln\theta(x)|$.
This follows

$|(I)|=\left|\ln\left(\frac{(8+8\alpha)\theta(x)|\ln|x+\xi\sqrt{\theta(x)}||}{|x+\xi\sqrt{\theta(x)}|^{2}}\right)\right|\sim\left|\ln\left(\frac{(4+4\alpha)\theta(x)|\ln\theta(x)|}{C(K_{0})\theta(x)|\ln\theta(x)|}\right)\right|\leq
C,$

and

$|(II)|\sim\left|\ln\left[\frac{1}{|\ln\theta_{0}|}\left(\frac{C(K_{0})\theta(x)|\ln\theta(x)|}{(4+4\alpha)\theta_{0}|\ln\theta_{0}|}\right)\right]\right|\sim\left|\ln
C(K_{0})\right|\leq C,$

which concludes the proof of (A.9). The expected estimate on
$\nabla_{\xi}\mathcal{U}$ then follows from (A.7), (A.8) and (A.9).

In the same way, we can show that it $t_{0}\geq
t_{0,3}(K_{0},\epsilon_{0},C_{0,1})$, then for $|x|\in[r_{0},\epsilon_{0}]$
and $|\xi|\leq 2\alpha_{0}\sqrt{|\ln\theta(x)|}$, we have
$|\nabla^{2}_{\xi}\mathcal{U}(x,\xi,\tau_{0})|\leq C_{0,1}$. This completes
the proof of Lemma A.2 as well as the proof of Proposition 3.4. ∎

##  Appendix B No blowup under some threshold for equation (1.1).

We give in this appendix the proof of Proposition 3.6 whose proof uses ideas
given in [GK89] treated for equation (1.2) without the gradient term
$(\alpha=0)$. The proof is based on the following integral equations for
localizations of $e^{u}$ and $\nabla u$:

######  Lemma B.1.

Let $r>0$ and $\phi_{r}$ be a smooth function supported on
$B_{r}=\\{x\in\mathbb{R}^{N},|x|<r\\}$ such that $\phi_{r}=1$ on $B_{r/2}$ and
$0\leq\phi_{r}\leq 1$. Let $w_{r}=\phi_{r}e^{u}$ and $g_{r}=\phi_{r}\nabla u$,
where $u$ satisfies equation (3.17). Then, we have the following:

$\displaystyle\|w_{r}(\tau)\|_{L^{\infty}}$
$\displaystyle\leq\|w(0)\|_{L^{\infty}}+C\int_{0}^{\tau}\left(1+\frac{1}{\sqrt{\tau-s}}\right)\left\|e^{u(s)}\right\|_{L^{\infty}(B_{r})}ds$
$\displaystyle+C\int_{0}^{\tau}\left(\left\|e^{u(s)}\right\|_{L^{\infty}(B_{r})}+\|\nabla
u(s)\|_{L^{\infty}(B_{r})}^{2}\right)\|w_{r}(s)\|_{L^{\infty}}ds,$ (B.1)
$\displaystyle\|g_{r}(\tau)\|_{L^{\infty}}$
$\displaystyle\leq\|g(0)\|_{L^{\infty}}+C\int_{0}^{\tau}\left(1+\frac{1}{\sqrt{\tau-s}}\right)\|\nabla
u(s)\|_{L^{\infty}(B_{r})}ds+C\int_{0}^{\tau}\|\nabla
u(s)\|^{2}_{L^{\infty}(B_{r})}ds$
$\displaystyle+C\int_{0}^{\tau}\left(\left\|e^{u(s)}\right\|_{L^{\infty}(B_{r})}+\frac{1}{\sqrt{\tau-s}}\|\nabla
u(s)\|_{L^{\infty}(B_{r})}\right)\|g_{r}(s)\|_{L^{\infty}}ds.$ (B.2)

###### Proof.

We only deal with (B.1) since (B.2) follows similarly. By the definition, we
see that $w_{r}$ satisfies the following equation:

$\partial_{\tau}w_{r}-\Delta
w_{r}=\big{(}e^{u}\Delta\phi_{r}-2\nabla\cdot(e^{u}\nabla\phi_{r})\big{)}+e^{u}\phi_{r}(\partial_{\tau}u-\Delta
u-|\nabla u|^{2}),$

hence, the semigroup representation formula for $w_{r}$ gives

$\displaystyle w_{r}(\tau)=e^{\tau\Delta}w_{r}(0)$
$\displaystyle+\int_{0}^{\tau}e^{(\tau-s)\Delta}\big{(}e^{u}\Delta\phi_{r}-2\nabla\cdot(e^{u}\nabla\phi_{r})\big{)}ds$
$\displaystyle+\int_{0}^{\tau}e^{(\tau-s)\Delta}e^{u}\phi_{r}(\partial_{\tau}u-\Delta
u-|\nabla u|^{2})ds,$ (B.3)

for $0<\tau<1$, where $e^{\theta\Delta}$ is the semigroup associated with the
heat equation in $\mathbb{R}^{N}$ with the following well known properties:

$\|e^{\theta\Delta}f\|_{L^{\infty}}\leq\|f\|_{L^{\infty}},\quad\|\nabla
e^{\theta\Delta}f\|_{L^{\infty}}\leq\frac{1}{\sqrt{\theta}}\|f\|_{L^{\infty}},\quad\forall\theta>0,\;f\in
W^{1,\infty}(\mathbb{R}^{N}).$ (B.4)

Using this regularity effect and the definition of $\phi_{r}$, the middle term
in the right hand side of (B.3) is bounded by

$\displaystyle\int_{0}^{\tau}\left\|e^{u}\Delta\phi_{r}-2\nabla\cdot(e^{u}\nabla\phi_{r})\right\|_{L^{\infty}}ds$
$\displaystyle\leq
C\int_{0}^{\tau}\left(1+\frac{1}{\sqrt{\tau-s}}\right)\left\|e^{u(s)}\right\|_{L^{\infty}(B_{r})}ds.$

From inequality (3.17), the last term is dominated by

$\displaystyle C\int_{0}^{\tau}$
$\displaystyle\|e^{(\tau-s)\Delta}e^{u}\phi_{r}(1+e^{u}+|\nabla
u|^{2})\|_{L^{\infty}}ds$ $\displaystyle\quad\leq
C\int_{0}^{\tau}\left\|e^{u(s)}\right\|_{L^{\infty}(B_{r})}ds$
$\displaystyle\qquad+C\int_{0}^{\tau}\left(\left\|e^{u(s)}\right\|_{L^{\infty}(B_{r})}+\|\nabla
u(s)\|_{L^{\infty}(B_{r})}^{2}\right)\|w_{r}(s)\|_{L^{\infty}}ds.$

This concludes the proof of (B.1) as well as Lemma B.1. ∎

Before going to the proof of Proposition 3.6, it is convenient to recall the
two following lemmas from [GK89] which will be used in the proof. The first
lemma gives estimates on an integration.

######  Lemma B.2.

For $0<\alpha<1$, $\theta>0$ and $\theta<h<1$, the integral

$I(h)=\int_{h}^{1}(s-h)^{-\alpha}s^{-\theta}ds$

satisfies

  * (i)

$I(h)\leq\left(\frac{1}{1-\alpha}+\frac{1}{\alpha+\theta-1}\right)h^{1-\alpha-\theta}\;$
if $\;\alpha+\theta>1$.

  * (ii)

$I(h)\leq\frac{1}{1-\alpha}+|\ln h|\;$ if $\;\alpha+\theta=1$.

  * (iii)

$I(h)\leq\frac{1}{1-\alpha-\theta}\;$ if $\;\alpha+\theta<1$.

###### Proof.

See Lemma 2.2, page 851 in [GK89]. ∎

The second lemma is a version of Gronwall’s inequality.

######  Lemma B.3.

If $y(t),r(t)$ and $q(t)$ are continuous functions defined on $[t_{0},t_{1}]$
such that

$y(t)\leq y_{0}+\int_{t_{0}}^{t}q(s)ds+\int_{t_{0}}^{t}r(s)y(s)ds,\quad
t_{0}\leq t\leq t_{1},$

then

$y(t)\leq
e^{\int_{t_{0}}^{t}r(s)ds}\left[y_{0}+\int_{t_{0}}^{t}q(s)e^{-\int_{t_{0}}^{s}r(\tau)d\tau}ds\right].$

###### Proof.

See Lemma 2.3, page 852 in [GK89]. ∎

We now give the proof of Proposition 3.6.

###### Proof of Proposition 3.6.

We proceed in two steps:
\- Step 1: We first apply Lemma B.1 with $r=1$ and use the assumption (3.18)
to get

$\|w_{1}(\tau)\|_{L^{\infty}}\leq\epsilon+C\epsilon\int_{0}^{\tau}\frac{ds}{(1-s)\sqrt{\tau-s}}+C\epsilon\int_{0}^{\tau}\frac{1}{1-s}\|w_{1}(s)\|_{L^{\infty}}ds,$

and

$\displaystyle\|g_{1}(\tau)\|_{L^{\infty}}$
$\displaystyle\leq\epsilon+C\epsilon\int_{0}^{\tau}\left(\frac{1}{\sqrt{(\tau-s)(1-s)}}+\frac{1}{1-s}\right)ds$
$\displaystyle+C\epsilon\int_{0}^{\tau}\left(\frac{1}{\sqrt{(\tau-s)(1-s)}}+\frac{1}{1-s}\right)\|g_{1}(s)\|_{L^{\infty}}ds.$

Now, applying Lemma B.3 to $\|w_{1}(\tau)\|_{L^{\infty}}$ yields

$\|w_{1}(\tau)\|_{L^{\infty}}\leq(1-\tau)^{-C\epsilon}\left[\epsilon+C\epsilon\int_{0}^{\tau}\frac{ds}{(1-s)^{1-C\epsilon}\sqrt{\tau-s}}\right].$

Using item $(i)$ of Lemma B.2, we have

$\int_{0}^{\tau}\frac{ds}{(1-s)^{1-C\epsilon}\sqrt{\tau-s}}=\int_{1-\tau}^{1}\frac{ds}{s^{1-C\epsilon}\sqrt{s-(1-\tau)}}\leq
C(1-\tau)^{-\frac{1}{2}+C\epsilon}.$

Hence, if $\epsilon$ is small enough such that $C\epsilon<\frac{1}{2}$, we
have

$\|w_{1}(\tau)\|_{L^{\infty}}\leq
C\epsilon(1-\tau)^{-\frac{1}{2}},\quad\forall\tau\in[0,1).$

Similarly, applying Lemma B.3 to $\|g_{1}(\tau)\|_{L^{\infty}}$ and using item
$(ii)$ of Lemma B.2 to estimate the integral
$\int_{0}^{\tau}\frac{ds}{\sqrt{(\tau-s)(1-s)}}\leq C+|\ln(1-\tau)|$, we then
obtain

$\|g_{1}(\tau)\|_{L^{\infty}}\leq
C(1-\tau)^{-C\epsilon}\left[\epsilon+C\epsilon\int_{0}^{\tau}\left(\frac{1}{(1-s)^{1/2-C\epsilon}\sqrt{\tau-s}}+\frac{1}{(1-s)^{1-C\epsilon}}\right)ds\right].$

Using item $(iii)$ of Lemma B.2, we then get

$\|g_{1}(\tau)\|_{L^{\infty}}\leq
C\epsilon(1-\tau)^{-C\epsilon},\quad\forall\tau\in[0,1).$

Therefore, we have by the definition of $\phi_{1}$ that

$e^{u(\xi,\tau)}\leq C\epsilon(1-\tau)^{-\frac{1}{2}}\;\;\text{and}\;\;|\nabla
u(\xi,\tau)|\leq
C\epsilon(1-\tau)^{-C\epsilon},\quad\forall|\xi|<\frac{1}{2},\;\forall\tau\in[0,1).$
(B.5)

\- Step 2: Applying Lemma B.1 with $r=\frac{1}{2}$, and using (B.5), we then
get

$\|w_{1/2}(\tau)\|_{L^{\infty}}\leq
C\epsilon+C\epsilon\int_{0}^{\tau}\frac{ds}{\sqrt{(\tau-s)(1-s)}}+C\int_{0}^{\tau}\frac{1}{\sqrt{1-s}}+\frac{1}{(1-s)^{2C\epsilon}}\|w_{1/2}(s)\|_{L^{\infty}}ds,$

and

$\displaystyle\|g_{1/2}(\tau)\|_{L^{\infty}}\leq C\epsilon$
$\displaystyle+C\epsilon\int_{0}^{\tau}\left(\frac{1}{(1-s)^{C\epsilon}\sqrt{\tau-s}}+\frac{1}{(1-s)^{2C\epsilon}}\right)ds$
$\displaystyle+C\epsilon\int_{0}^{\tau}\left(\frac{1}{\sqrt{1-s}}+\frac{1}{(1-s)^{C\epsilon}\sqrt{\tau-s}}\right)\|g_{1/2}(s)\|_{L^{\infty}}ds.$

From Lemmas B.3 and B.2, we deduce

$\|w_{1/2}(\tau)\|_{L^{\infty}}\leq
C\epsilon(1+|\ln(1-\tau)|)\;\text{and}\;\;\|g_{1/2}(\tau)\|_{L^{\infty}}\leq
C\epsilon,\quad\forall\tau\in[0,1).$

Since $|\ln(1-\tau)|\leq(1-\tau)^{-\epsilon}$ for any $\epsilon>0$, we then
have by the definition of $\phi_{1/2}$

$e^{u(\xi,\tau)}\leq C\epsilon(1-\tau)^{-\epsilon}\;\;\text{and}\;\;|\nabla
u(\xi,\tau)|\leq
C\epsilon,\quad\forall|\xi|<\frac{1}{4},\;\forall\tau\in[0,1).$ (B.6)

Repeat this step one again with $r=\frac{1}{4}$ by using (B.6) and (B.1), we
end up with

$e^{u(\xi,\tau)}+|\nabla u(\xi,\tau)|\leq
C\epsilon,\quad\forall|\xi|<\frac{1}{8},\;\forall\tau\in[0,1).$

This concludes the proof of Proposition 3.6. ∎

## References

  * [BB92] J. Bebernes and S. Bricher.  Final time blowup profiles for semilinear parabolic equations via center manifold theory.  SIAM J. Math. Anal., 23(4):852–869, 1992.
  * [BE89] J. Bebernes and D. Eberly.  Mathematical problems from combustion theory, volume 83 of Applied Mathematical Sciences.  Springer-Verlag, New York, 1989.
  * [BK94] J. Bricmont and A. Kupiainen.  Universality in blow-up for nonlinear heat equations.  Nonlinearity, 7(2):539–575, 1994.
  * [Bre90] A. Bressan.  On the asymptotic shape of blow-up.  Indiana Univ. Math. J., 39(4):947–960, 1990.
  * [Bre92] A. Bressan.  Stable blow-up patterns.  J. Differential Equations, 98(1):57–75, 1992.
  * [CFQ03] M. Chlebík, M. Fila, and P. Quittner.  Blow-up of positive solutions of a semilinear parabolic equation with a gradient term.  Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10(4):525–537, 2003.
  * [CW89] M. Chipot and F. B. Weissler.  Some blowup results for a nonlinear parabolic equation with a gradient term.  SIAM J. Math. Anal., 20(4):886–907, 1989.
  * [CZ13] R. Côte and H. Zaag.  Construction of a multisoliton blowup solution to the semilinear wave equation in one space dimension.  Comm. Pure Appl. Math., 66(10):1541–1581, 2013.
  * [Dol85] J. W. Dold.  Analysis of the early stage of thermal runaway.  The Quarterly Journal of Mechanics and Applied Mathematics, 38(3):361–387, 1985.
  * [Dol89] J. W. Dold.  Analysis of thermal runaway in the ignition process.  SIAM J. Appl. Math., 49(2):459–480, 1989.
  * [EZ11] M. A. Ebde and H. Zaag.  Construction and stability of a blow up solution for a nonlinear heat equation with a gradient term.  S$\vec{\rm e}$MA J., (55):5–21, 2011.
  * [Fil91] M. Fila.  Remarks on blow up for a nonlinear parabolic equation with a gradient term.  Proc. Amer. Math. Soc., 111(3):795–801, 1991.
  * [FK92] S. Filippas and R. V. Kohn.  Refined asymptotics for the blowup of $u_{t}-\Delta u=u^{p}$.  Comm. Pure Appl. Math., 45(7):821–869, 1992.
  * [FM85] A. Friedman and B. McLeod.  Blow-up of positive solutions of semilinear heat equations.  Indiana Univ. Math. J., 34(2):425–447, 1985.
  * [FP08] M. Fila and A. Pulkkinen.  Nonconstant selfsimilar blow-up profile for the exponential reaction-diffusion equation.  Tohoku Math. J. (2), 60(3):303–328, 2008.
  * [GK89] Y. Giga and R. V. Kohn.  Nondegeneracy of blowup for semilinear heat equations.  Comm. Pure Appl. Math., 42(6):845–884, 1989.
  * [GNZ16] T. Ghoul, V. T. Nguyen, and H. Zaag.  Construction and stability of blowup solutions for a non-variational parabolic system.  arXiv:1610.09883, 2016.
  * [GV93] V. A. Galaktionov and J. L. Vázquez.  Regional blow up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation.  SIAM J. Math. Anal., 24(5):1254–1276, 1993.
  * [GV96] V. A. Galaktionov and J. L. Vázquez.  Blow-up for quasilinear heat equations described by means of nonlinear Hamilton-Jacobi equations.  J. Differential Equations, 127(1):1–40, 1996.
  * [HV92] M. A. Herrero and J. J. L. Velázquez.  Generic behaviour of one-dimensional blow up patterns.  Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 19(3):381–450, 1992.
  * [HV93] M. A. Herrero and J. J. L. Velázquez.  Blow-up behaviour of one-dimensional semilinear parabolic equations.  Ann. Inst. H. Poincaré Anal. Non Linéaire, 10(2):131–189, 1993.
  * [Mer92] F. Merle.  Solution of a nonlinear heat equation with arbitrarily given blow-up points.  Comm. Pure Appl. Math., 45(3):263–300, 1992.
  * [MZ97a] F. Merle and H. Zaag.  Reconnection of vortex with the boundary and finite time quenching.  Nonlinearity, 10(6):1497–1550, 1997.
  * [MZ97b] F. Merle and H. Zaag.  Stability of the blow-up profile for equations of the type $u_{t}=\Delta u+|u|^{p-1}u$.  Duke Math. J., 86(1):143–195, 1997.
  * [MZ08] N. Masmoudi and H. Zaag.  Blow-up profile for the complex Ginzburg-Landau equation.  J. Funct. Anal., 255(7):1613–1666, 2008.
  * [NZ15] N. Nouaili and H. Zaag.  Profile for a simultaneously blowing up solution to a complex valued semilinear heat equation.  Comm. Partial Differential Equations, 40(7):1197–1217, 2015.
  * [NZ16a] V. T. Nguyen and H. Zaag.  Construction of a stable blow-up solution for a class of strongly perturbed semilinear heat equations.  Ann. Scuola Norm. Sup. Pisa Cl. Sci., to appear, 2016.
  * [NZ16b] V. T. Nguyen and H. Zaag.  Finite degrees of freedom for the refined blow-up profile for a semilinear heat equation.  Ann. Scient. Éc. Norm. Sup. to appear, 2016.
  * [Pul11] A. Pulkkinen.  Blow-up profiles of solutions for the exponential reaction-diffusion equation.  Math. Methods Appl. Sci., 34(16):2011–2030, 2011.
  * [Sou01] P. Souplet.  Recent results and open problems on parabolic equations with gradient nonlinearities.  Electron. J. Differential Equations, pages No. 10, 19 pp. (electronic), 2001.
  * [ST01] P. Souplet and S. Tayachi.  Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities.  Colloq. Math., 88(1):135–154, 2001.
  * [ST07] S. Snoussi and S. Tayachi.  Large time behavior of solutions for parabolic equations with nonlinear gradient terms.  Hokkaido Math. J., 36(2):311–344, 2007.
  * [STW96] P. Souplet, S. Tayachi, and F. B. Weissler.  Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term.  Indiana Univ. Math. J., 45(3):655–682, 1996.
  * [TZ16] S. Tayachi and H. Zaag.  Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term.  arXiv:1506.08306, 2016.
  * [Vel92] J. J. L. Velázquez.  Higher-dimensional blow up for semilinear parabolic equations.  Comm. Partial Differential Equations, 17(9-10):1567–1596, 1992\.
  * [Vel93] J. J. L. Velázquez.  Classification of singularities for blowing up solutions in higher dimensions.  Trans. Amer. Math. Soc., 338(1):441–464, 1993.
  * [VGH91] J. J. L. Velázquez, V. A. Galaktionov, and M. A. Herrero.  The space structure near a blow-up point for semilinear heat equations: a formal approach.  Zh. Vychisl. Mat. i Mat. Fiz., 31(3):399–411, 1991.