# Refined regularity of the blow-up set linked to refined asymptotic behavior
for the semilinear heat equation

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###### Abstract.

We consider $u(x,t)$, a solution of $\partial_{t}u=\Delta u+|u|^{p-1}u$ which
blows up at some time $T>0$, where $u:\mathbb{R}^{N}\times[0,T)\to\mathbb{R}$,
$p>1$ and $(N-2)p<N+2$. Define $S\subset\mathbb{R}^{N}$ to be the blow-up set
of $u$, that is the set of all blow-up points. Under suitable nondegeneracy
conditions, we show that if $S$ contains a $(N-\ell)$-dimensional continuum
for some $\ell\in\\{1,\dots,N-1\\}$, then $S$ is in fact a $\mathcal{C}^{2}$
manifold. The crucial step is to derive a refined asymptotic behavior of $u$
near blow-up. In order to obtain such a refined behavior, we have to abandon
the explicit profile function as a first order approximation and take a non-
explicit function as a first order description of the singular behavior. This
way we escape logarithmic scales of the variable $(T-t)$ and reach significant
small terms in the polynomial order $(T-t)^{\mu}$ for some $\mu>0$. The
refined asymptotic behavior yields geometric constraints of the blow-up set,
leading to more regularity on $S$.

###### Key words and phrases:

Blowup solution, Blowup set, Blowup profile, Regularity, Semilinear heat
equation

###### 1991 Mathematics Subject Classification:

Primary: 35K50, 35B40; Secondary: 35K55, 35K57.

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 semilinear heat equation:

$\left\\{\begin{array}[]{rcl}\partial_{t}u&=&\Delta u+|u|^{p-1}u,\\\
u(0)&=&u_{0}\in L^{\infty}(\mathbb{R}^{N}),\end{array}\right.$ (1.1)

where $u(t):x\in\mathbb{R}^{N}\to u(x,t)\in\mathbb{R}$, $\Delta$ denotes the
Laplacian in $\mathbb{R}^{N}$, and $p>1$ or $1<p<\frac{N+2}{N-2}$ if $N\geq
3$.

It is well known that for each initial data $u_{0}$ the Cauchy problem (1.1)
has a unique solution $u\in\mathcal{C}([0,T),L^{\infty}(\mathbb{R}^{N}))$ for
some $0<T\leq+\infty$, and that either $T=+\infty$ or

$T<+\infty\quad\text{and}\quad\lim_{t\to T}\|u(t)\|_{L^{\infty}}=+\infty.$

In the latter case we say that the solution blows up in finite time, and $T$
is called the blow-up time. In such a blow-up case, a point
$\hat{a}\in\mathbb{R}^{N}$ is called a blow-up point if $u(x,t)$ is not
locally bounded in some neighborhood of $(\hat{a},T)$, this means that there
exists $(x_{n},t_{n})\to(\hat{a},T)$ such that $|u(x_{n},t_{n})|\to+\infty$
when $n\to+\infty$. We denote by $S$ the blow-up set, that is the set of all
blow-up points of $u$.

Given $\hat{a}\in S$, we know from Velázquez [Vel92] (see also Filippas and
Kohn [FK92], Filippas and Liu [FL93], Herrero and Velázquez [HV93], Merle and
Zaag [MZ00]) that up to replacing $u$ by $-u$, one of following two cases
occurs:

\- Case 1 (non degenerate rate of blow-up): For all $K_{0}>0$, there is an
orthonormal $(N\times N)$ matrix $Q_{\hat{a}}$ and
$\ell_{\hat{a}}\in\\{1,\dots,N\\}$ such that

$\sup_{|\xi|\leq
K_{0}}\left|(T-t)^{\frac{1}{p-1}}u(\hat{a}+Q_{\hat{a}}\xi\sqrt{(T-t)|\log(T-t)|},t)-f_{\ell_{\hat{a}}}(\xi)\right|\to
0\;\;\text{as}\;\;t\to T,$ (1.2)

where

$f_{\ell_{\hat{a}}}(\xi)=\left(p-1+\frac{(p-1)^{2}}{4p}\sum_{i=1}^{\ell_{\hat{a}}}\xi_{i}^{2}\right)^{-\frac{1}{p-1}}.$
(1.3)

\- Case 2 (degenerate rate of blow-up): For all $K_{0}\geq 0$, there exists an
even integer $m\geq 4$ such that

$\sup_{|\xi|\leq
K_{0}}\left|(T-t)^{\frac{1}{p-1}}u(\hat{a}+\xi(T-t)^{\frac{1}{m}},t)-\left(p-1+\sum_{|\alpha|=m}c_{\alpha}\xi^{\alpha}\right)^{-\frac{1}{p-1}}\right|\to
0\;\;\text{as}\;\;t\to T,$ (1.4)

where $\xi^{\alpha}=\prod\limits_{i=1}^{N}\xi_{i}^{\alpha_{i}}$,
$|\alpha|=\sum\limits_{i=1}^{N}\alpha_{i}$ if
$\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{N}^{N}$ and
$\sum\limits_{|\alpha|=m}c_{\alpha}\xi^{\alpha}\geq 0$ for all
$\xi\in\mathbb{R}^{N}$.

According to Velázquez [Vel92], if case 1 occurs with $\ell_{\hat{a}}=N$ or
case 2 occurs with $\sum_{|\alpha|=m}c_{\alpha}\xi^{\alpha}>0$ for all
$\xi\neq 0$, then $\hat{a}$ is an isolated blow-up point. Herrero and
Velázquez [HV92b] and [HV92a] proved that the profile (1.3) with
$\ell_{\hat{a}}=N$ is generic in the case $N=1$, and they announced the same
for $N\geq 2$, but they never published it. Bricmont and Kupiainen [BK94],
Merle and Zaag [MZ97] show the existence of initial data for (1.1) such that
the corresponding solutions blow up in finite time $T$ at only one blow-up
point $\hat{a}$ and verify the behavior (1.2) with $\ell_{\hat{a}}=N$. The
method of [MZ97] also gives the stability of the profile (1.3)
($\ell_{\hat{a}}=N$) with respect to perturbations in the initial data (see
also Fermanian, Merle and Zaag [FMZ00], [FZ00] for other proofs of the
stability). In [EZ11] and [NZ16a], the authors prove the stability of the
profile (1.3) ($\ell_{\hat{a}}=N$) with respect to perturbations in the
initial data and also in the nonlinearity, in some class allowing lower order
terms in the solution and also in the gradient. All the other asymptotic
behaviors are suspected to be unstable.

When

$\ell_{\hat{a}}\leq N-1$

in (1.2), we do not know whether $\hat{a}$ is isolated or not, or whether $S$
is continuous near $\hat{a}$. In this paper, we assume that $\hat{a}$ is a
non-isolated blow-up point and that $S$ is continuous locally near $\hat{a}$,
in a sense that we will precisely define later. Our main concern is the
regularity of $S$ near $\hat{a}$. The first relevant result is due to
Velázquez [Vel93] where the author showed that the Hausdorff measure of $S$ is
less or equal to $N-1$. No further results on the description of $S$ were
known until the contributions of Zaag [Zaa02a], [Zaa02b] and [Zaa06] (see also
[Zaa02c] for a summarized note). In [Zaa02a], the author proves that if $S$ is
locally continuous, then $S$ is a $\mathcal{C}^{1}$ manifold. He also obtains
the first singularity description near $\hat{a}$. More precisely, he shows
that (see Theorems 3 and 4 in [Zaa02a]) for some $t_{0}<T$ and $\delta>0$, for
all $K_{0}>0$, $t\in[t_{0},T)$ and $x\in B(\hat{a},2\delta)$ such that
$d(x,S)\leq K_{0}\sqrt{(T-t)|\log(T-t)|}$,

$\left|(T-t)^{\frac{1}{p-1}}u(x,t)-f_{1}\left(\frac{d(x,S)}{\sqrt{(T-t)|\log(T-t)|}}\right)\right|\leq
C(K_{0})\frac{\log|\log(T-t)|}{|\log(T-t)|},$ (1.5)

where $f_{1}$ is defined in (1.3) ($\ell_{\hat{a}}=1$). Moreover, for all
$x\in\mathbb{R}^{N}\setminus S$, $u(x,t)\to u^{*}(x)$ as $t\to T$ with

$u^{*}(x)\sim U(d(x,S))=\left(\frac{8p}{(p-1)^{2}}\frac{|\log
d(x,S)|}{d^{2}(x,S)}\right)^{\frac{1}{p-1}}\quad\text{as}\;\;d(x,S)\to
0\;\;\text{and}\;\;x\in B(\hat{a},2\delta).$ (1.6)

If

$\ell_{\hat{a}}=1,$

Zaag in [Zaa02b] further refines the asymptotic behavior (1.5) and gets to
error terms of order $(T-t)^{\mu}$ for some $\mu>0$. This way, he obtains more
regularity on the blow-up set $S$. The key idea is to replace the explicit
profile $f_{1}$ in (1.5) by a non-explicit function, say $\tilde{u}(x_{1},t)$,
then to go beyond all logarithmic scales through scaling and matching. In
fact, for $\tilde{u}(x_{1},t)$, Zaag takes a symmetric, one dimensional
solution of (1.1) that blows up at the same time $T$ only at the origin, and
behaves like (1.2) with $\ell_{\hat{a}}=1$. More precisely, he abandons the
explicit profile function $f_{1}$ in (1.5) and chooses a non explicit function
$\tilde{u}_{\sigma}(d(x,S),t)$ as a first order description of the singular
behavior, where $\tilde{u}_{\sigma}$ is defined by

$\tilde{u}_{\sigma}(x_{1},t)=e^{-\frac{\sigma}{p-1}}\tilde{u}\left(e^{-\frac{\sigma}{2}}x_{1},T-e^{-\sigma}(T-t)\right).$
(1.7)

He shows that for each blow-up point $a$ near $\hat{a}$, there is an optimal
scaling parameter $\sigma=\sigma(a)$ so that the difference
$(T-t)^{\frac{1}{p-1}}\left(u(x,t)-\tilde{u}_{\sigma(a)}(d(x,S),t)\right)$
along the normal direction to $S$ at $a$ is minimum. Hence, if the function
$\tilde{u}_{\sigma(a)}(d(x,S),t)$ is chosen as a first order description for
$u(x,t)$ near $(a,T)$, we escape logarithmic scales. More precisely, for all
$t\in[t_{0},T)$ and $x\in B(\hat{a},2\delta)$ such that $d(x,S)\leq
K_{0}\sqrt{(T-t)|\log(T-t)|}$,

$(T-t)^{\frac{1}{p-1}}\left|u(x,t)-\tilde{u}_{\sigma(a)}(d(x,S),t)\right|\leq
C(T-t)^{\mu},$ (1.8)

for some $\mu>0$. Note that any other value of $\sigma\neq\sigma(a)$ in (1.8)
gives an error of logarithmic order of the variable $(T-t)$ (the same as in
(1.5)). Exploiting estimate (1.8) yields geometric constraints on $S$ which
imply the $\mathcal{C}^{1,\frac{1}{2}-\eta}$-regularity of $S$ for all
$\eta>0$. A further refinement of (1.8) given in [Zaa06] yields better
estimates in the expansion of $u(x,t)$ near $(a,T)$. Moreover, some following
terms in the expansion of $u(x,t)$ near $(a,T)$ contain geometrical
descriptions of $S$, resulting in more regularity of $S$, namely the
$\mathcal{C}^{2}$-regularity.

In this work, we want to know whether the $\mathcal{C}^{2}$-regularity near
$\hat{a}$ proven in [Zaa06] for $\ell_{\hat{a}}=1$ would hold in the case
where $u$ behaves like (1.2) near $(\hat{a},T)$ with

$\ell_{\hat{a}}\in\\{2,\dots,N-1\\}.$ (1.9)

Since the author in [Zaa02a] and [Zaa06] obtains the result only when
$\ell_{\hat{a}}=1$, this corresponds to $(N-1)$-dimensional blow-up set (the
codimension of the blow-up set is one, according to [Zaa02a]). In our opinion,
in those papers the major obstacle towards the case (1.9) lays in the fact
that the author could not refine the asymptotic behavior (1.2) with
$\ell_{\hat{a}}\in\\{2,\dots,N-1\\}$ to go beyond all logarithmic scales and
get a smaller error term in polynomial orders of the variable $(T-t)$. It
happens that a similar difficulty was already encountered by Fermanian and
Zaag in [FZ00], when they wanted to find a sharp profile in the case (1.2)
with $\ell_{\hat{a}}=N$, which corresponds to an isolated blow-up point, as we
have pointed out right after estimate (1.4). Such a sharp profile could be
obtained in [FZ00] only when $N=1$ (which corresponds also to
$\ell_{\hat{a}}=1$): no surprise it was $\tilde{u}_{\sigma}(x_{1},t)$, the
dilated version of $\tilde{u}(x_{1},t)$, the one-dimensional blow-up solution
mentioned between estimates (1.6) and (1.7). As a matter of fact, the use of
$\tilde{u}(x_{1},t)$ was first used in [FZ00] for the isolated blow-up point
in one space dimension ($N=1$ and $\ell_{\hat{a}}=1$), then later in higher
dimensions with a $(N-1)$ dimensional blow-up surface ($N\geq 2$ and still
$\ell_{\hat{a}}=1$) in [Zaa02b].

The interest of $\tilde{u}(x_{1},t)$ is that it provides a one-parameter
family of blow-up solutions, thanks to the scaling parameter in (1.7), which
enables to get the sharp profile by suitably choosing the parameter.

Handling the case $\ell_{\hat{a}}\geq 2$ remained open, both for the case of
an isolated point ($\ell_{\hat{a}}=N\geq 2$) and a non-isolated blow-up point
($\ell_{\hat{a}}=2,\cdots,N-1$). From the refinement of the expansion around
the explicit profile in $f_{\ell_{\hat{a}}}$ in (1.2), it appeared than one
needs a $\frac{\ell_{\hat{a}}(\ell_{\hat{a}}+1)}{2}$-parameter family of blow-
up solutions obeying (1.2).

Such a family was constructed by Nguyen and Zaag in [NZ16b], and successfully
used to derive a sharp profile in the case of an isolated blow-up point
($\ell_{\hat{a}}=N\geq 2$), by fine-tuning the
$\frac{\ell_{\hat{a}}(\ell_{\hat{a}}+1)}{2}=\frac{N(N+1)}{2}$ parameters.

In this paper, we aim at using that family to handle the case of a non-
isolated blow-up point ($N\geq 2$ and $\ell_{\hat{a}}=2,\cdots,N-1$), in order
to generalize the results of Zaag in [Zaa02a], [Zaa02b] and [Zaa06], proving
in particular the $C^{2}$ regularity of the blow-up set, under the mere
hypothesis that it is continuous.

The main result in this paper is the following.

######  Theorem 1.1 ($\mathcal{C}^{2}$ regularity of the blow-up set assuming
$\mathcal{C}^{1}$ regularity).

Take $N\geq 2$ and $\ell\in\\{1,\cdots,N-1\\}$. Consider $u$ a solution of
(1.1) that blows up in finite time $T$ on a set $S$ and take $\hat{a}\in S$
where $u$ behaves locally as stated in (1.2) with $\ell_{\hat{a}}=\ell$. If
$S$ is locally a $\mathcal{C}^{1}$ manifold of dimension $N-\ell$, then it is
locally $\mathcal{C}^{2}$.

######  Remark 1.2.

Theorem 1.1 was already proved by Zaag [Zaa06] only when $\ell=1$. Thus, the
novelty of our contribution lays in the case $\ell\in\\{2,\dots,N-1\\}$ and
$N\geq 3$.

Under the hypotheses of Theorem 1.1, Zaag [Zaa02a] already proved that $S$ is
a $\mathcal{C}^{1}$ manifold near $\hat{a}$, assuming that $S$ is continuous.
Therefore, Theorem 1.1 can be restated under a weaker assumption. Before
stating this stronger version, let us first clearly describe our hypotheses
and introduce some terminologies borrowed from [Zaa02a] (see also [Zaa02b] and
[Zaa06]). According to Velázquez [Vel92] (see Theorem 2, page 1571), we know
that for all $\epsilon>0$, there is $\delta(\epsilon)>0$ such that

$S\cap
B(\hat{a},2\delta)\subset\Omega_{\hat{a},\epsilon}\equiv\left\\{x\in\mathbb{R}^{N},\left|P_{\hat{a}}(x-\hat{a})\right|\geq(1-\epsilon)|x-\hat{a}|\right\\},$

where $P_{\hat{a}}$ is the orthogonal projection over $\pi_{\hat{a}}$, where

$\pi_{\hat{a}}=\hat{a}+\text{span}\\{Q_{\hat{a}}^{T}e_{\ell_{\hat{a}}+1},\cdots,Q_{\hat{a}}^{T}e_{N}\\}$

is the so-called ”weak” tangent plane to $S$ at $\hat{a}$. Roughly speaking,
$\Omega_{\hat{a},\epsilon}$ is a cone with vertex $\hat{a}$ and shrinks to
$\pi_{\hat{a}}$ as $\epsilon\to 0$. In some ”weak” sense, $S$ is
$(N-\ell_{\hat{a}})$-dimensional. In fact, here comes our second hypothesis:
we assume there is
$\Gamma\in\mathcal{C}((-1,1)^{N-\ell_{\hat{a}}},\mathbb{R}^{N})$ such that
$\Gamma(0)=\hat{a}$ and $Im\,\Gamma\subset S$, where $Im\,\Gamma$ is at least
$(N-\ell_{\hat{a}})$-dimensional, in the sense that

$\begin{array}[]{lll}&\text{$\forall b\in Im\,\Gamma$, there are
$(N-\ell_{\hat{a}})$ independent vectors $v_{1},\dots,v_{N-\ell_{\hat{a}}}$ in
$\mathbb{R}^{N}$ and}&\\\ &\text{$\Gamma_{1},\dots,\Gamma_{N-\ell_{\hat{a}}}$
functions in $\mathcal{C}^{1}([0,1],S)$ such that $\Gamma_{i}(0)=b$ and
$\Gamma^{\prime}_{i}(0)=v_{i}$.}&\end{array}$ (1.10)

The hypothesis (1.10) means that $b$ is actually non-isolated in
$(N-\ell_{\hat{a}})$ independent directions. We assume in addition that
$\hat{a}$ is not an endpoint in $Im\,\Gamma$ in the sense that

$\begin{array}[]{lll}&\text{$\forall\epsilon>0$, the projection of
$\Gamma((-\epsilon,\epsilon)^{N-\ell_{\hat{a}}})$ on the "weak" tangent plane
$\pi_{\hat{a}}$}&\\\ &\text{at $\hat{a}$ contains an open ball centered at
$\hat{a}$.}&\\\ \end{array}$ (1.11)

This is the stronger version of our result:

Theorem 1.1’. Take $N\geq 2$ and $\ell\in\\{1,\cdots,N-1\\}$. Consider $u$ a
solution of (1.1) that blows up in finite time $T$ on a set $S$ and take
$\hat{a}\in S$ where $u$ behaves locally as stated in (1.2) with
$\ell_{\hat{a}}=\ell$. Consider
$\Gamma\in\mathcal{C}((-1,1)^{N-\ell},\mathbb{R}^{N})$ such that
$\hat{a}=\Gamma(0)\in Im\,\Gamma\subset S$ and $Im\,\Gamma$ is at least
$(N-\ell)$-dimensional (in the sense (1.10)). If $\hat{a}$ is not an endpoint
(in the sense (1.11)), then there are $\delta>0$, $\delta_{1}>0$ and
$\gamma\in\mathcal{C}^{2}((-\delta_{1},\delta_{1})^{N-\ell},\mathbb{R}^{\ell})$
such that

$S_{\delta}=S\cap B(\hat{a},2\delta)=graph(\gamma)\cap
B(\hat{a},2\delta)=Im\,\Gamma\cap B(\hat{a},2\delta),$

and the blow-up set $S$ is a $\mathcal{C}^{2}$-hypersurface locally near
$\hat{a}$.

Let us now briefly give the main ideas of the proof of Theorem 1.1. The proof
is based on techniques developed by Zaag in [Zaa02b] and [Zaa06] for the case
when the solution of equation (1.1) behaves like (1.2) with $\ell=1$. As in
[Zaa02b] and [Zaa06], the proof relies on two arguments:

  * -

The derivation of a sharp blow-up profile of $u(x,t)$ near the singularity, in
the sense that the difference between the solution $u(x,t)$ and this sharp
profile goes beyond all logarithmic scales of the variables $(T-t)$. This is
possible thanks to the recent result in [NZ16b].

  * -

The derivation of a refined asymptotic profile of $u(x,t)$ near the
singularity linked to geometric constraints on the blow-up set. In fact, we
derive an asymptotic profile for $u(x,t)$ in every ball $B(a,K_{0}\sqrt{T-t})$
for some $K_{0}>0$ and $a$ a blow-up point close to $\hat{a}$. Moreover, this
profile is continuous in $a$ and the speed of convergence of $u$ to each one
in the ball $B(a,K_{0}\sqrt{T-t})$ is uniform with respect to $a$. If $a$ and
$b$ are in $S$ and $0<|a-b|\leq K_{0}\sqrt{T-t}$, then the balls
$B(a,K_{0}\sqrt{T-t})$ and $B(b,K_{0}\sqrt{T-t})$ intersect each other,
leading to different profiles for $u(x,t)$ in the intersection. However, these
profiles have to coincide, up to the error terms. This makes a geometric
constraint which gives more regularity for the blow-up set near $\hat{a}$.

Let us explain the difficulty raised in [Zaa02b] and [Zaa06] for the case
$\ell\geq 2$. Consider $a\in S\cap B(\hat{a},2\delta)$ for some $\delta>0$ and
introduce the following self-similar variables:

$W_{a}(y,s)=(T-t)^{\frac{1}{p-1}}u(x,t),\quad y=\frac{x-a}{\sqrt{T-t}},\quad
s=-\log(T-t).$ (1.12)

Then, we see from (1.1) that for all $(y,s)\in\mathbb{R}^{N}\times[-\log
T,+\infty)$,

$\frac{\partial W_{a}}{\partial s}=\Delta W_{a}-\frac{1}{2}y\cdot\nabla
W_{a}-\frac{W_{a}}{p-1}+|W_{a}|^{p-1}W_{a}.$ (1.13)

Under the hypotheses stated in Theorem 1.1, Zaag [Zaa02a] proved in
Proposition 3.1, page 513 and in Section 6.1, pages 530-533 that for all $a\in
S_{\delta}\equiv S\cap B(\hat{a},2\delta)$ for some $\delta>0$ and $s\geq-\log
T$, there exists an $(N\times N)$ orthogonal matrix $Q_{a}$ such that

$\left\|W_{a}(Q_{a}y,s)-\left\\{\kappa+\frac{\kappa}{2ps}\left(\ell-\frac{|\bar{y}|^{2}}{2}\right)\right\\}\right\|_{L^{2}_{\rho}}\leq
C\frac{\log s}{s^{2}},$ (1.14)

where $\kappa=(p-1)^{-\frac{1}{p-1}}$, $\bar{y}=(y_{1},\cdots,y_{\ell_{a}})$,
$Q_{a}$ is continuous in terms of $a$ such that
$\\{Q_{a}^{T}e_{j}|j=\ell+1,\dots,N\\}$ spans the tangent plane $\pi_{a}$ to
$S$ at $a$ and $Q_{a}^{T}e_{i},i=1,\cdots,\ell$ are the normal directions to
$S$ at $a$, $L^{2}_{\rho}$ is the weighted $L^{2}$ space associated with the
weight $\rho=\dfrac{1}{(4\pi)^{N/2}}e^{-\frac{|y|^{2}}{4}}$. Note that the
estimate (1.14) implies (1.5) (see Appendix C in [Zaa02a]).

When $\ell=1$, in order to refine estimate (1.14), the author in [Zaa02b]
subtracts from $W_{a}$ a 1-dimensional solution with the same profile. Let us
do the same when $\ell=2,\cdots,N-1$, and explain how the author succeeds in
handing the case $\ell=1$ and gets stuck when $\ell\geq 2$. To this end, we
consider $\hat{u}(\bar{x},t)$ with $\bar{x}=(x_{1},\cdots,x_{\ell})$ a
radially symmetric solution of (1.1) in $\mathbb{R}^{\ell}$ which blows up at
time $T$ only at the origin with the profile (1.2) with $\ell_{\hat{a}}=\ell$
(see Appendix A.1 in [NZ16b] for the existence of such a solution). If the
$\ell$-dimensional solution $\hat{u}$ is considered in $\mathbb{R}^{N}$, then
it blows up on the $(N-\ell)$ vector space $\\{\bar{x}=0\\}$ in
$\mathbb{R}^{N}$. In particular, if we introduce

$\hat{w}(\bar{y},s)=(T-t)^{\frac{1}{p-1}}\hat{u}(\bar{x},t),\quad\bar{y}=\frac{\bar{x}}{\sqrt{T-t}},\quad
s=-\log(T-t),$ (1.15)

then, $\hat{w}$ is a radially symmetric solution of (1.13) which satisfies

$\left\|\hat{w}(\bar{y},s)-\left\\{\kappa+\frac{\kappa}{2ps}\left(\ell-\frac{|\bar{y}|^{2}}{2}\right)\right\\}\right\|_{L^{2}_{\rho}}\leq
C\frac{\log s}{s^{2}}.$ (1.16)

Noting that $\hat{u}$ and $\hat{w}$ maybe considered as solutions defined for
all $y\in\mathbb{R}^{N}$ (and independent of $y_{\ell+1},\cdots,y_{N}$), and
given that $\hat{w}(\bar{y},s)$ and $W_{a}(Q_{a}y,s)$ have the same behavior
up to the first order (see (1.14) and (1.16)), we may try to use $\hat{w}$ as
a sharper (though non-explicit) profile for $W_{a}(Q_{a}y,s)$. In fact, we
have the following classification (see Corollary 2.2 below):

\- Case 1: There is a symmetric, real $\ell_{a}\times\ell_{a}$ matrix
$\mathcal{B}=\mathcal{B}(a)\neq 0$ such that

$W_{a}(Q_{a}y,s)-\hat{w}(\bar{y},s)=\frac{1}{s^{2}}\left(\frac{1}{2}\bar{y}^{T}\mathcal{B}\bar{y}-tr(\mathcal{B})\right)+o\left(\frac{1}{s^{2}}\right)\;\;\text{as}\;\;s\to+\infty\;\;\text{in}\;L^{2}_{\rho}.$
(1.17)

\- Case 2: There is a positive constant $C_{0}$ such that

$\|W_{a}(Q_{a}y,s)-\hat{w}(\bar{y},s)\|_{L^{2}_{\rho}}=\mathcal{O}\left(e^{-\frac{s}{2}}s^{C_{0}}\right)\;\;\text{as}\;\;s\to+\infty.$
(1.18)

If $\ell=1$ ($\mathcal{B}(a)\in\mathbb{R}$), the author in [Zaa02b] noted the
following property

$\hat{w}(y_{1},s+\sigma_{0})-\hat{w}(y_{1},s)=\frac{2\kappa\sigma_{0}}{ps^{2}}\left(\frac{1}{2}y_{1}^{2}-1\right)+o\left(\frac{1}{s^{2}}\right)\quad\text{in}\;L^{2}_{\rho}.$
(1.19)

Therefore, choosing $\sigma_{0}(a)$ such that
$\frac{2\kappa\sigma_{0}}{p}=\mathcal{B}(a)$, we see from (1.17) and (1.19)
that

$W_{a}(Q_{a}y,s)-\hat{w}(y_{1},s+\sigma_{0}(a))=o\left(\frac{1}{s^{2}}\right)\;\;\text{as}\;s\to+\infty\quad\text{in}\;L^{2}_{\rho}.$

From the classification given in (1.17) and (1.18), only (1.18) holds and

$\|W_{a}(Q_{a}y,s)-\hat{w}(y_{1},s+\sigma_{0}(a))\|_{L^{2}_{\rho}}=\mathcal{O}\left(e^{-\frac{s}{2}}s^{C_{0}}\right)\;\;\text{as}\;\;s\to+\infty.$
(1.20)

If we return to the original variables $u(x,t)$ and $\hat{u}(x_{1},t)$ through
(1.12) and (1.15), then (1.8) follows from the transformation (1.7) together
with estimate (1.20) (see Appendix C in [Zaa02b]). In other words,
$\hat{w}(y_{1},s+\sigma_{0}(a))$ serves as a sharp (though non-explicit)
profile for $W_{a}(Q_{a}y,s)$ in the sense of (1.20). Using estimate (1.20)
together with some geometrical arguments, we are able to prove the
$\mathcal{C}^{1,\frac{1}{2}-\eta}$-regularity of the blow-up set, for any
$\eta>0$. Then, a further refinement of (1.20) up to order of
$\frac{e^{-\frac{s}{2}}}{s}$ together with a geometrical constraint on the
blow-up set $S$ results in more regularity for $S$, which yields the
$\mathcal{C}^{2}$-regularity.

If $\ell\geq 2$, the matrix $\mathcal{B}(a)$ in (1.17) has
$\frac{\ell(\ell+1)}{2}$ real parameters. Therefore, applying the trick of
[Zaa02b] (see (1.19) above) only allows to manage one parameter; there remain
$\frac{\ell(\ell+1)}{2}-1$ real parameters to be handled. This is the major
reason which prevents the author in [Zaa02b] and [Zaa06] from deriving a
similar estimate to (1.20), hence, the refined regularity of the blow-up set.
Fortunately, we could overcome this obstacle thanks to a recent result by
Nguyen and Zaag [NZ16b] (see Proposition 2.4 below) where the authors show
that for all symmetric, real $\ell\times\ell$ matrix $\mathcal{A}$, there is a
solution $w_{\mathcal{A}}$ of equation (1.13) in $\mathbb{R}^{\ell}$ such that

$w_{\mathcal{A}}(\bar{y},s)-\hat{w}(\bar{y},s)=\frac{1}{s^{2}}\left(\frac{1}{2}\bar{y}^{T}\mathcal{A}\bar{y}-tr(\mathcal{A})\right)+o\left(\frac{1}{s^{2}}\right)\quad\text{as}\;s\to+\infty\;\;\text{in}\;L^{2}_{\rho}.$
(1.21)

Hence, choosing $\mathcal{A}=\mathcal{B}(a)$, we see from (1.21), (1.17) and
(1.18) that

$\|W_{a}(Q_{a}y,s)-w_{\mathcal{B}(a)}(\bar{y},s)\|_{L^{2}_{\rho}}\leq
Ce^{-\frac{s}{2}}s^{C_{0}},\quad\text{for $s$ large enough}.$ (1.22)

Exploiting estimate (1.22) and adapting the arguments given in [Zaa02b] and
[Zaa06], we are able to prove the $\mathcal{C}^{2}$-regularity of the blow-up
set.

The next result shows how the $\mathcal{C}^{2}$-regularity is linked to the
refined asymptotic behavior of $W_{a}$. More precisely, we link in the
following theorem the refinement of the asymptotic behavior of $W_{a}$ to the
second fundamental form of the blow-up set at $a$.

######  Theorem 1.3 (Refined asymptotic behaviors linked to the geometrical
description of the blow-up set).

Under the hypotheses of Theorem 1.1, there exists $\tilde{s}_{0}\geq-\log T$
and $\delta>0$ such that for all $a\in S_{\delta}=S\cap B(\hat{a},2\delta)$,
there exists a continuous $(\ell\times\ell)$ symmetric matrix $\mathcal{B}(a)$
such that for all $s\geq\tilde{s}_{0}$,

$\displaystyle\left\|W_{a}(Q_{a}y,s)-w_{\mathcal{B}(a)}(\bar{y},s)-\frac{\kappa
e^{-\frac{s}{2}}}{2ps}\sum_{i=1}^{\ell}y_{i}\sum_{k,j=\ell+1}^{N}\frac{\Lambda^{(i)}_{k,j}(a)}{1+\delta_{k,j}}(y_{k}y_{j}-2\delta_{k,j})\right\|_{L^{2}_{\rho}}\leq
C\frac{e^{-\frac{s}{2}}}{s^{\frac{3}{2}-\nu}},$ (1.23)

for some $\nu\in(0,\frac{1}{2})$, where
$a\to\\{\Lambda^{(i)}_{k,j}(a)\\}_{\ell+1\leq j,k\leq N}$ is a continuous
symmetric matrix representing the second fundamental form of the blow-up set
at the blow-up point $a$ along the unitary normal vector $Q_{a}^{T}e_{i}$.
Moreover,

$\Lambda^{(i)}_{k,j}(a)=\frac{p}{4\kappa}\lim_{s\to+\infty}se^{\frac{s}{2}}\int_{\mathbb{R}^{N}}W_{a}(Q_{a}y,s)y_{i}(y_{k}y_{j}-2\delta_{k,j})\rho(y)dy.$
(1.24)

In Section 2, we give the main steps of the proof of Theorems 1.1 and 1.3. We
leave all long and technical proofs to Section 3.

##  2\. Setting of the problem and strategy of the proof of the
$\mathcal{C}^{2}$-regularity of the blow-up set.

In this section we give the main steps of the proof of Theorems 1.1 and 1.3.
All long and technical proofs will be left to the next section. We proceed in
3 parts corresponding to 3 separate subsections. For the reader’s convenience,
we briefly describe these parts as follows:

  * •

Part 1: We derive a sharp blow-up behavior for solutions of equation (1.1)
having the profile (1.2) with $\ell_{\hat{a}}\in\\{1,\cdots,N-1\\}$ such that
the difference between the solution and this sharp blow-up behavior goes
beyond all logarithmic scales of the variable $T-t$. The main result in this
step is stated in Proposition 2.5.

  * •

Part 2: Through the introduction of a local chart, we give a geometrical
constraint on the expansion of the solution linked to the asymptotic behavior
(see Proposition 2.7 below). This geometrical constraint is a crucial point
which is the bridge between the asymptotic behavior and the regularity of the
blow-up set.

  * •

Part 3: Using the sharp blow-up behavior derived in Part 1, we first get the
$\mathcal{C}^{1,\frac{1}{2}-\eta}$\- regularity of the blow-up set $S$ (see
Proposition 2.8 below), then together with the geometrical constraint, we
achieve the $\mathcal{C}^{1,1-\eta}$-regularity of $S$ (see Proposition 2.9
below). With this better regularity and the geometric constraint, we further
refine the asymptotic behavior (see Proposition 2.10 below) and use again the
geometric constraint to get $\mathcal{C}^{2}$ -regularity of $S$, which yields
the conclusion of Theorems 1.1 and 1.3.

The reader should be noticed that Parts 1 and 2 are independent, whereas Part
3 is a combination of the first two parts. Throughout this paper, we work
under the hypotheses of Theorem 1.1. Since $S$ is locally near $\hat{a}$ a
manifold of dimension $N-\ell$, we may assume that there is a
$\mathcal{C}^{1}$ function $\gamma$ such that

$S_{\delta}\equiv S\cap B(\hat{a},2\delta)=\text{graph}(\gamma)\cap
B(\hat{a},2\delta),$ (2.1)

for some $\delta>0$ and
$\gamma\in\mathcal{C}^{1}((-\delta_{1},\delta_{1})^{N-\ell},\mathbb{R}^{\ell})$
with $\delta_{1}>0$.

In what follows, $\ell\in\\{1,\cdots,N-1\\}$ is fixed, and for all
$z=(z_{1},\cdots,z_{N})\in\mathbb{R}^{N}$, we denote by $\bar{z}$ the first
$\ell$ coordinates of $z$, namely $\bar{z}=(z_{1},\cdots,z_{\ell})$, and by
$\tilde{z}$ the last $(N-\ell)$ coordinates of $z$, namely
$\tilde{z}=(z_{\ell+1},\cdots,z_{N})$. We usually use indices $i$, $m$ for the
range $1,\cdots,\ell$ and indices $j$, $k$, $n$ for the range
$\ell+1,\cdots,N$.

###  2.1. Part 1: Blow-up behavior beyond all logarithmic scales of $(T-t)$.

In this subsection, we use the ideas given by Fermanian and Zaag [FZ00]
together with a recent result by Nguyen and Zaag in [NZ16b] in order to derive
a sharp (though non-explicit) profile for blow-up solutions of (1.1) in the
sense that the first order in the expansion of the solution around this sharp
profile goes beyond all logarithmic scales of $(T-t)$ and reaches to
polynomial scales of $(T-t)$. In fact, we replace the 1-scaling parameter
$\sigma$ in (1.8) by a $\frac{\ell(\ell+1)}{2}$-parameters family, which
generates a substitution for $\tilde{u}_{\sigma}$ (1.7) and serves as a sharp
profile for solutions having the behavior (1.2) with
$\ell_{\hat{a}}\in\\{1,\dots,N-1\\}$. The main result in this part is
Proposition 2.5 below.

Consider $a\in S_{\delta}$. If $W_{a}(y,s)$ and $\hat{w}(\bar{y},s)$ are
defined as in (1.12) and (1.15), then we know from [Zaa02a] that

$\left\|W_{a}(Q_{a}y,s)-\left\\{\kappa+\frac{\kappa}{2ps}\left(\ell-\frac{|\bar{y}|^{2}}{2}\right)\right\\}\right\|_{L^{2}_{\rho}}\leq
C\frac{\log s}{s^{2}},$ (2.2)

and

$\left\|\hat{w}(\bar{y},s)-\left\\{\kappa+\frac{\kappa}{2ps}\left(\ell-\frac{|\bar{y}|^{2}}{2}\right)\right\\}\right\|_{L^{2}_{\rho}}\leq
C\frac{\log s}{s^{2}}.$ (2.3)

The first step is to classify all possible asymptotic behaviors of
$W_{a}(Q_{a}y,s)-\hat{w}(\bar{y},s)$ as $s$ goes to infinity. To do so, we
shall use the following result which is inspired by Fermanian and Zaag [FZ00]:

######  Proposition 2.1 (Classification of the difference between two
solutions of (1.13) having the same profile).

Assume that $W_{1}$ and $W_{2}$ are two solutions of (1.13) verifying

$i=1,2,\quad\left\|W_{i}(y,s)-\left\\{\kappa+\frac{\kappa}{2ps}\left(\ell-\frac{|\bar{y}|^{2}}{2}\right)\right\\}\right\|_{L^{2}_{\rho}}\leq
C\frac{\log s}{s^{2}},$ (2.4)

where $\bar{y}=(y_{1},\cdots,y_{\ell})$ for some $\ell\in\\{1,\cdots,N-1\\}$.
Then, one of the two following cases occurs:
\- Case 1: there is a symmetric, real $(\ell\times\ell)$ matrix
$\mathcal{B}\neq 0$ such that

$W_{1}(y,s)-W_{2}(y,s)=\frac{1}{s^{2}}\left(\frac{1}{2}\bar{y}^{T}\mathcal{B}\bar{y}-tr(\mathcal{B})\right)+o\left(\frac{1}{s^{2}}\right)\quad\text{as}\;\;s\to+\infty\;\;\text{in}\;L^{2}_{\rho}.$
(2.5)

\- Case 2: there is $C_{0}>0$ such that

$\|W_{1}(y,s)-W_{2}(y,s)\|_{L^{2}_{\rho}}=\mathcal{O}\left(e^{-\frac{s}{2}}s^{C_{0}}\right)\quad\text{as}\;s\to+\infty.$
(2.6)

###### Proof.

The proof follows from the strategy given in [FZ00] for the difference of two
solutions with the radial profile $(\ell=N)$. Note that the case when $\ell=1$
was treated in [Zaa02b]. Since some technical details are straightforward, we
briefly give the main steps of the proof in Section 3.1 and just emphasize on
the novelties. ∎

An application of Proposition 2.1 with $W_{1}(y,s)=W_{a}(Q_{a}y,s)$ and
$W_{2}(y,s)=\hat{w}(\bar{y},s)$ yields the following corollary directly:

######  Corollary 2.2.

As $s$ goes to infinity, one of the two following cases occurs:
\- Case 1: there is a symmetric, real $(\ell\times\ell)$ matrix
$\mathcal{B}=\mathcal{B}(a)\neq 0$ continuous as a function of $a$ such that

$W_{a}(Q_{a}y,s)-\hat{w}(\bar{y},s)=\frac{1}{s^{2}}\left(\frac{1}{2}\bar{y}^{T}\mathcal{B}\bar{y}-tr(\mathcal{B})\right)+o\left(\frac{1}{s^{2}}\right)\quad\text{in}\;L^{2}_{\rho}.$
(2.7)

\- Case 2: there is $C_{0}>0$ such that

$\|W_{a}(Q_{a}y,s)-\hat{w}(\bar{y},s)\|_{L^{2}_{\rho}}=\mathcal{O}\left(e^{-\frac{s}{2}}s^{C_{0}}\right).$
(2.8)

######  Remark 2.3.

Note that the continuity of $\mathcal{B}$ comes from the continuity of $W_{a}$
with respect to $a$, where $W_{a}$ behaves as in (2.2). In particular, Zaag
[Zaa02a] showed the stability of the blow-up behavior (2.2) with respect to
blow-up points (see Proposition 3.1 and Section 6.1 in [Zaa02a]).

In the next step, we recall the recent result by Nguyen and Zaag [NZ16b],
which gives the construction of solutions for equation (1.13) with some
prescribed behavior.

######  Proposition 2.4 (Construction of solutions for (1.13) with some
prescribed behavior).

Consider $\ell\in\\{1,\cdots,N-1\\}$. For all
$\mathcal{A}\in\mathbb{M}_{\ell}(\mathbb{R})$, where
$\mathbb{M}_{\ell}(\mathbb{R})$ is the set of all symmetric, real
$(\ell\times\ell)$-matrix, there exists a solution $w_{\mathcal{A}}(y,s)$ of
(1.13) defined on $\mathbb{R}^{N}\times[s_{0}(\mathcal{A}),+\infty)$ such that

$w_{\mathcal{A}}(\bar{y},s)-\hat{w}(\bar{y},s)=\frac{1}{s^{2}}\left(\frac{1}{2}\bar{y}^{T}\mathcal{A}\bar{y}-tr(\mathcal{A})\right)+o\left(\frac{1}{s^{2}}\right)\quad\text{as}\;s\to+\infty\;\;\text{in}\;\;L^{2}_{\rho},$
(2.9)

where $\hat{w}$ is the radially symmetric, $\ell$-dimensional solution of
(1.13) satisfying (2.3).

###### Proof.

See Theorem 3 in [NZ16b]. Although that result is stated for the case
$\ell=N$, we can extend it to the case when $\ell\leq N-1$ by considering
solutions of (1.13) as $\ell$-dimensional solutions, those artificially
generated by adding irrelevant space variables $(y_{\ell+1},\cdots,y_{N})$ to
the domain of definition of the solutions. ∎

The following result is a direct consequence of Corollary 2.2 and Proposition
2.4:

######  Proposition 2.5 (Sharp (non-explicit) profile for solutions of (1.1)
having the behavior (1.2) with $\ell\leq N-1$).

There exist $s_{0}>0$ and a continuous matrix $\mathcal{B}:\;S_{\delta}\to
M_{\ell}(\mathbb{R})$, such that for all $a\in S_{\delta}$ and $s\geq s_{0}$,

$\left\|W_{a}(Q_{a}y,s)-w_{\mathcal{B}(a)}(\bar{y},s)\right\|_{L^{2}_{\rho}}\leq
Ce^{-\frac{s}{2}}s^{C_{0}},$ (2.10)

where $w_{\mathcal{B}}$ is the solution constructed as in Proposition 2.4,
$C_{0}>0$ is given in Proposition 2.1. Moreover, we have
$(i)\;\;$ For all $s\geq s_{0}+1$,

$\sup_{|y|\leq
K\sqrt{s}}\left|W_{a}(y,s)-w_{\mathcal{B}(a)}(\bar{y}_{a},s)\right|\leq
C(K)e^{-\frac{s}{2}}s^{\frac{3}{2}+C_{0}},$ (2.11)

where $\bar{y}_{a}=(y\cdot Q_{a}e_{1},\cdots,y\cdot Q_{a}e_{\ell})$.
$(ii)\;$ For all $t\in[T-e^{-s_{0}-1},T)$,

$\displaystyle\sup_{|x-a|\leq K\sqrt{(T-t)|\log(T-t)|}}$
$\displaystyle\left|(T-t)^{\frac{1}{p-1}}u(x,t)-w_{\mathcal{B}(a)}(\bar{y}_{a,x},-\log(T-t))\right|$
$\displaystyle\qquad\qquad\qquad\leq
C(K)(T-t)^{\frac{1}{2}}|\log(T-t)|^{\frac{3}{2}+C_{0}},$ (2.12)

where $\bar{y}_{a,x}=\frac{1}{\sqrt{T-t}}((x-a)\cdot
Q_{a}e_{1},\cdots,(x-a)\cdot Q_{a}e_{\ell})$.

###### Proof.

From (2.7) and (2.9), we have for any $\ell\times\ell$ symmetric matrix
$\mathcal{A}$,

$W_{a}(Q_{a}y,s)-w_{\mathcal{A}}(\bar{y},s)=\frac{1}{s^{2}}\left(\frac{1}{2}\bar{y}^{T}(\mathcal{B}-\mathcal{A})\bar{y}-tr(\mathcal{B}-\mathcal{A})\right)+o\left(\frac{1}{s^{2}}\right)\quad\text{in}\;\;L^{2}_{\rho}.$

Choosing $\mathcal{A}=\mathcal{B}(a)$, we get

$\left\|W_{a}(Q_{a}y,s)-w_{\mathcal{B}(a)}(\bar{y},s)\right\|_{L^{2}_{\rho}}=o\left(\frac{1}{s^{2}}\right),\quad\text{as}\;s\to+\infty.$
(2.13)

Note that an alternative application of Proposition 2.1 with $W_{1}=W_{a}$ and
$W_{2}=w_{\mathcal{B}(a)}$ yields either (2.7) or (2.8). However, the case
(2.7) is excluded by (2.13). Hence, (2.10) follows. Since we showed in
Corollary 2.2 that $a\mapsto\mathcal{B}(a)$ is continuous, the same holds for
$a\mapsto\mathcal{A}(a)$.

As for (2.11), it is a direct consequence of the following lemma which allows
us to carry estimate (2.10) from compact sets $|y|\leq K$ to sets $|y|\leq
K\sqrt{s}$:

######  Lemma 2.6 (Extension of the convergence from compact sets to sets
$|y|\leq K\sqrt{s}$).

Assume that $Z$ satisfies

$\partial_{s}Z\leq\Delta Z-\frac{1}{2}y\cdot\nabla Z+Z+\frac{C_{1}}{s}Z,\quad
0\leq Z(y,s)\leq
C_{1},\quad\forall(y,s)\in\mathbb{R}^{N}\times[\hat{s},+\infty),$ (2.14)

for some $C_{1}>0$. Then for all $s^{\prime}\geq\hat{s}$ and $s\geq
s^{\prime}+1$ such that $e^{\frac{s-s^{\prime}}{2}}=\sqrt{s}$, we have

$\sup_{|y|\leq K\sqrt{s}}Z(y,s)\leq
C(C_{1},K)e^{s-s^{\prime}}\|Z(s^{\prime})\|_{L^{2}_{\rho}}.$

###### Proof.

This lemma is a corollary of Proposition 2.1 in Velázquez [Vel92] and it is
proved in the course of the proof of Proposition 2.13 in [FZ00] (in
particular, pp. 1203-1205). ∎

Let us derive (2.11) from Lemma 2.6. If we define
$G(y,s)=W_{a}(Q_{a}y,s)-w_{\mathcal{B}(a)}(\bar{y},s)$, straightforward
calculations based on (1.13) yield

$\partial_{s}G=\Delta G-\frac{1}{2}y\cdot\nabla G+G+\alpha
G,\quad\forall(y,s)\in\mathbb{R}^{N}\times[-\log T,+\infty),$ (2.15)

where

$\alpha(y,s)=\frac{|W_{a}|^{p-1}W_{a}-|w_{\mathcal{B}}|^{p-1}w_{\mathcal{B}}}{W_{a}-w_{\mathcal{B}}}-\frac{p}{p-1}=p|\tilde{w}(y,s)|^{p-1}-\frac{p}{p-1}\quad\text{if}\;\;W_{a}\neq
w_{\mathcal{B}},$

for some
$\tilde{w}(y,s)\in\left(W_{a}(Q_{a}y,s),w_{\mathcal{B}(a)}(\bar{y},s)\right)$.

From Merle and Zaag [MZ98] (Theorem 1), we know that for $s$ large enough,

$\|\tilde{w}(s)\|_{L^{\infty}}\leq\kappa+\frac{C}{s},$

which follows

$\alpha(y,s)\leq
p\left(\kappa+\frac{C}{s}\right)^{p-1}-\frac{p}{p-1}\leq\frac{C_{1}}{s}.$
(2.16)

If $Z=|G|$, then we use the Kato’s inequality $\Delta
G\cdot\text{sgn}(G)\leq\Delta(|G|)$ to derive equation (2.14) from (2.15) and
(2.16). Applying Lemma 2.6 together with estimate (2.10) yields for all
$s^{\prime}\geq s_{1}$ and $s\geq s^{\prime}+1$ for some $s_{1}>0$ large such
that $e^{\frac{s-s^{\prime}}{2}}=\sqrt{s}$,

$\sup_{|y|\leq K\sqrt{s}}Z(y,s)\leq
Ce^{s-s^{\prime}}e^{-\frac{s^{\prime}}{2}}(s^{\prime})^{C_{0}}\leq
Ce^{-\frac{s}{2}}s^{\frac{3}{2}+C_{0}},$

which yields (2.11). The estimate (2.12) directly follows from (2.11) by the
transformation (1.12). This ends the proof of Proposition 2.5. ∎

###  2.2. Part 2: A geometric constraint linked to the asymptotic behaviors.

In this subsection, we follow the idea of [Zaa06] to introduce local
$\mathcal{C}^{1,\alpha^{*}}$-charts of the blow-up set, and get a geometric
constraint mechanism on the blow-up set (see Proposition 2.7 below) which is a
crucial step in linking refined asymptotic behaviors of the solution to
geometric descriptions of the blow-up set.

Consider $a\in S_{\delta}$ and $\ell\in\\{1,\cdots,N-1\\}$, we introduce the
local $\mathcal{C}^{1,\alpha^{*}}$-chart of the blow-up set at the point $a$
as follows:

$\displaystyle\mathbb{R}^{N-\ell}\;\;$
$\displaystyle\to\quad\quad\mathbb{R}^{N}$
$\displaystyle\tilde{\xi}\;\;\;\;\;$
$\displaystyle\mapsto\;\;(\gamma_{a,1}(\tilde{\xi}),\cdots,\gamma_{a,\ell}(\tilde{\xi}),\tilde{\xi}),$

where $\tilde{\xi}=(\xi_{\ell+1},\cdots,\xi_{N})$ and
$\gamma_{a,i}\in\mathcal{C}^{1,\alpha^{*}}((-\epsilon_{a},\epsilon_{a})^{N-\ell})$
for some $\alpha^{*}\in\left(0,\frac{1}{2}\right)$ and $\epsilon_{a}>0$, then
the set $S_{\delta}$ is locally near $a$ defined by

$\left\\{a+\sum_{i=1}^{\ell}\gamma_{a,i}(\tilde{\xi})\eta_{i}(a)+\sum_{j=\ell+1}^{N}\xi_{k}\tau_{k}(a)\;\big{|}\;|\tilde{\xi}|<\epsilon_{a}\right\\},$
(2.17)

where $\eta_{1}(a),\cdots,\eta_{\ell}(a)$ and
$\tau_{\ell+1}(a),\cdots,\tau_{N}(a)$ are of norm 1, and respectively, normal
and tangent to $S_{\delta}$ at $a$. By definition, we have

$\gamma_{a,i}(0)=0\quad\text{and}\quad\nabla\gamma_{a,i}(0)=0,\;\forall
i=1,\cdots,\ell.$

Let $Q_{a}$ be the orthogonal matrix whose columns are $\eta_{i}(a)$ and
$\tau_{j}(a)$, namely that

$\eta_{i}(a)=Q_{a}e_{i}\quad\text{and}\quad\tau_{j}(a)=Q_{a}e_{j},$ (2.18)

and define

$w_{a}(y,s)=(T-t)^{\frac{1}{p-1}}u(x,t),\quad
y=Q_{a}^{T}\left(\frac{x-a}{\sqrt{T-t}}\right),\quad s=-\log(T-t),$ (2.19)

then we see from (1.12) that $w_{a}$ satisfies (1.13) and

$w_{a}(y,s)=W_{a}(Q_{a}y,s),\quad\forall(y,s)\in\mathbb{R}^{N}\times[-\log
T,+\infty).$ (2.20)

Note from (2.18) that the point $(y,s)$ in the domain of $w_{a}$ becomes the
point $(x,t)$ in the domain of $u$, where

$x=a+e^{-\frac{s}{2}}Q_{a}y=a+e^{-\frac{s}{2}}\left(\sum_{i=1}^{\ell}y_{i}\eta_{i}(a)+\sum_{j=\ell+1}^{N}y_{j}\tau_{j}(a)\right),\quad
t=T-e^{-s}.$

Now, fix $a\in S_{\delta}$ and consider an arbitrary $b\in S_{\delta}$. From
(2.19), we have

$w_{a}(y,s)=w_{b}(Y,s),\quad\text{where}\;\;\;Y=Q_{b}^{T}\left(Q_{a}y+e^{\frac{s}{2}}(a-b)\right).$
(2.21)

If we differentiate (2.21) with respect to $y_{k}$ with
$k\in\\{\ell+1,\cdots,N\\}$, we get

$\displaystyle(T-t)^{\frac{1}{p-1}+\frac{1}{2}}\frac{\partial
u}{\partial\tau_{k}(a)}(x,t)$ $\displaystyle=\frac{\partial w_{a}}{\partial
y_{k}}(y,s)=\sum_{i=1}^{\ell}\tau_{k}(a)\cdot\eta_{i}(b)\frac{\partial
w_{b}}{\partial
y_{i}}(Y,s)+\sum_{j=\ell+1}^{N}\tau_{k}(a)\cdot\tau_{j}(b)\frac{\partial
w_{b}}{\partial y_{j}}(Y,s).$ (2.22)

If we fix $b$ as the projection of $x=a+e^{-\frac{s}{2}}Q_{a}y$ on the blow-up
set in the orthogonal direction to the tangent space to the blow-up set at
$a$, then $b$ has the same components on the tangent space spanned by
$\\{\tau_{\ell+1}(a),\cdots,\tau_{N}(a)\\}$ as $x$. In particular,

$b=b(a,y,s)=a+\sum_{i=1}^{\ell}\gamma_{a,i}(e^{-\frac{s}{2}}\tilde{y})\eta_{i}(a)+\sum_{j=\ell+1}^{N}e^{-\frac{s}{2}}y_{j}\tau_{j}(a),\quad\tilde{y}=(y_{\ell+1},\cdots,y_{N}).$
(2.23)

The following proposition gives a geometric constraint on the expansion of
$w_{a}$, which is the bridge linking the refined asymptotic behavior to the
refined regularity of the blow-up set.

######  Proposition 2.7 (A geometric constraint on the expansion of $w_{a}$).

Assume that

$\gamma_{a}\in\mathcal{C}^{1,\alpha^{*}}((-\epsilon_{a},\epsilon_{a})^{N-\ell},\mathbb{R}^{\ell})\quad\text{for
some}\;\;\alpha^{*}\in\left(0,\frac{1}{2}\right)\;\;\text{and}\;\;\epsilon_{a}>0.$

Then, there exists $s_{1}\geq\max\\{-\log T,s_{0}\\}$ ($s_{0}$ is introduced
in Proposition 2.5) such that for all $a\in S_{\delta}$, $|y|\leq 1$, $s\geq
s_{1}$ and $k=\ell+1,\cdots,N$, it holds that

$\displaystyle\left|\frac{\partial w_{a}}{\partial
y_{k}}(y,s)-\left\\{\frac{\partial w_{b}}{\partial
y_{k}}(\bar{y},0,\cdots,0,s)+\frac{\kappa}{2ps}\sum_{i=1}^{\ell}\frac{\partial\gamma_{a,i}}{\partial\xi_{k}}(e^{-\frac{s}{2}}\tilde{y})y_{i}\right\\}\right|$
$\displaystyle\leq
C\sum_{i=1}^{\ell}\left|\frac{\partial\gamma_{a,i}}{\partial\xi_{k}}(e^{-\frac{s}{2}}\tilde{y})\right|\left[|\bar{y}|\frac{\log
s}{s^{2}}+\frac{1}{s}e^{-\frac{\alpha^{*}s}{2}}+e^{-\frac{s}{2}}s^{C_{0}}\right]+Ce^{-\frac{(1+\alpha^{*})s}{2}}s^{C_{0}},$
(2.24)

where $\bar{y}=(y_{1},\cdots,y_{\ell})$, $\tilde{y}=(y_{\ell+1},\cdots,y_{N})$
and $b$ is defined by (2.23).

###### Proof.

Note that the proof of Proposition 2.7 was given in [Zaa06] only when
$\ell=1$. Of course, that proof naturally extends to the case when
$\ell\in\\{2,\cdots,N-1\\}$. Since our paper is relevant only when $\ell\geq
2$ and Proposition 2.7 presents an essential link between the asymptotic
behavior of the solution and a geometric constraint of the blow up set, we
felt we should give the proof of this proposition for the completeness and for
the reader’s convenience. As said earlier, this section just gives the main
steps of the proof of Theorem 1.1, and because the proof is long and
technical, we leave it to Section 3.3. ∎

###  2.3. Part 3: Refined regularity of the blow-up set and conclusion of
Theorem 1.1.

In this subsection, we give the proof of the $\mathcal{C}^{2}$-regularity of
the blow-up set (Theorems 1.1 and 1.3). We proceed in 2 steps:

  * •

Step 1: We derive from Proposition 2.5 that $\gamma_{a}$ is
$\mathcal{C}^{1,\frac{1}{2}-\eta}$ for all $\eta>0$. Then we apply Proposition
2.7 with $\alpha^{*}=\alpha\in(0,\frac{1}{2})$ to improve the regularity of
$\gamma_{a}$ which reaches to $\mathcal{C}^{1,1-\eta}$ for all $\eta>0$.

  * •

Step 2: Using the $\mathcal{C}^{1,1-\eta}$-regularity and the geometric
constraint in Proposition 2.7, we refine the asymptotic behavior given in
Proposition 2.5, which involves terms of order $\frac{1}{s}e^{-\frac{s}{2}}$.
Exploiting this refined asymptotic behavior together with the geometric
constraint (2.24), we derive that $\gamma_{a}$ is of class $\mathcal{C}^{2}$,
which is the conclusion of Theorem 1.1. From the information obtained on the
$\mathcal{C}^{2}$-regularity, we calculate the second fundamental form of the
blow-up set, which concludes the proof of Theorem 1.3.

#### Step 1: Deriving $\mathcal{C}^{1,1-\eta}$-regularity of the blow-up set.

We first derive the $\mathcal{C}^{1,\frac{1}{2}-\eta}$-regularity of the blow-
up set for all $\eta>0$ from Proposition 2.5. Then we apply Proposition 2.7
with $\alpha^{*}=\alpha\in(0,\frac{1}{2})$ to get
$\mathcal{C}^{1,1-\eta}$-regularity for all $\eta>0$. In particular, we claim
the following:

######  Proposition 2.8 ($C^{1,\frac{1}{2}-\eta}$-regularity for $S$).

Under the hypotheses of Theorem 1.1, $S$ is the graph of a vector function
$\gamma\in\mathcal{C}^{1,\frac{1}{2}-\eta}((-\delta_{1},\delta_{1})^{N-\ell},\mathbb{R}^{\ell})$
for any $\eta>0$, locally near $\hat{a}$. More precisely, there is an
$h_{0}>0$ such that for all $|\tilde{\xi}|<\delta_{1}$ and $|\tilde{h}|<h_{0}$
such that $|\tilde{\xi}+\tilde{h}|<\delta_{1}$, we have for all
$i\in\\{1,\cdots,\ell\\}$,

$|\gamma_{i}(\tilde{\xi}+\tilde{h})-\gamma_{i}(\tilde{\xi})-\tilde{h}\cdot\nabla\gamma_{i}(\tilde{\xi})|\leq
C|\tilde{h}|^{\frac{3}{2}}|\log|\tilde{h}||^{\frac{1}{2}+\frac{C_{0}}{2}}.$
(2.25)

###### Proof.

The proof is mainly based on the derivation of the sharp asymptotic profile
given in Proposition 2.5. In fact, we exploit the estimate (2.12) to find out
a geometric constraint on the blow-up set $S$, which implies some more
regularity on $S$. Since the argument follows the same lines as in Section 4,
[Zaa02b] for the case $\ell=1$, and no new ideas are needed for the case
$\ell\geq 2$, we will just sketch the proof by underlying the most relevant
aspects in Section 3.2 for the reader’s sake. ∎

The next proposition shows the $\mathcal{C}^{1,1-\eta}$-regularity of the
blow-up set.

######  Proposition 2.9 ($C^{1,1-\eta}$-regularity for $S_{\delta}$).

There exists $\xi_{0}>0$ such that for each $a\in S_{\delta}$, the local chart
defined in (2.17) satisfies for all $k=\ell+1,\cdots,N$ and
$|\tilde{\xi}|<\xi_{0}$,

$\sum_{i=1}^{\ell}\left|\frac{\partial\gamma_{a,i}}{\partial\xi_{k}}(\tilde{\xi})\right|\leq
C|\tilde{\xi}||\log|\tilde{\xi}||^{1+\mu}\quad\text{for some}\;\;\mu>0.$

###### Proof.

Note that the case $\ell=1$ was already proven in [Zaa06] (see Lemma 3.4, page
516). Here we use again the argument of [Zaa06] for the case $\ell\geq 2$.
Using the estimate given in Proposition 2.5 and parabolic regularity, we see
that for all $k\geq\ell+1$ and $s\geq s_{0}+1$,

$\sup_{a\in S_{\delta},|y|<2}\left|\frac{\partial w_{a}}{\partial
y_{k}}(y,s)\right|\leq Ce^{-\frac{s}{2}}s^{\mu}\quad\text{for some
}\;\;\mu>0.$

Consider $a\in S_{\delta}$ and $y=(\bar{y},\tilde{y})$, where
$\bar{y}=(y_{1},\cdots,y_{\ell})$ is such that $y_{i_{*}}=1$ for some
$i_{*}\in\\{1,\cdots,\ell\\}$, $y_{j}=0$ for $1\leq j\neq i_{*}\leq\ell$, and
$\tilde{y}=(y_{\ell+1},\cdots,y_{N})$ is arbitrary in $\partial
B_{N-\ell}(0,1)$. For $s\geq\max\\{s_{0}+1,s_{1}\\}$, we consider $b=b(a,y,s)$
defined as in (2.23). Since $\gamma_{a}$ is $\mathcal{C}^{1,\frac{1}{2}-\eta}$
for any $\eta>0$, we use (2.24) with $\alpha^{*}=\alpha\in(0,\frac{1}{2})$ to
write for $k\in\\{\ell+1,\cdots,N\\}$,

$\frac{\kappa}{2ps}\left|\frac{\partial\gamma_{a,i_{*}}}{\partial\xi_{k}}(e^{-\frac{s}{2}}\tilde{y})\right|\leq
C\frac{\log
s}{s^{2}}\sum_{i=1}^{\ell}\left|\frac{\partial\gamma_{a,i}}{\partial\xi_{k}}(e^{-\frac{s}{2}}\tilde{y})\right|+Ce^{-\frac{s}{2}}s^{\mu}.$

Since $i_{*}$ is arbitrary in $\\{1,\cdots,\ell\\}$, we get

$\frac{\kappa}{2ps}\sum_{i=1}^{\ell}\left|\frac{\partial\gamma_{a,i}}{\partial\xi_{k}}(e^{-\frac{s}{2}}\tilde{y})\right|\leq
C\frac{\log
s}{s^{2}}\sum_{i=1}^{\ell}\left|\frac{\partial\gamma_{a,i}}{\partial\xi_{k}}(e^{-\frac{s}{2}}\tilde{y})\right|+Ce^{-\frac{s}{2}}s^{\mu},$

which gives

$\sum_{i=1}^{\ell}\left|\frac{\partial\gamma_{a,i}}{\partial\xi_{k}}(e^{-\frac{s}{2}}\tilde{y})\right|\leq
Ce^{-\frac{s}{2}}s^{1+\mu}.$

If $\tilde{\xi}=e^{-\frac{s}{2}}\tilde{y}$, then
$|\tilde{\xi}|=e^{-\frac{s}{2}}$ and $|\log|\tilde{\xi}||=\frac{s}{2}$ since
$|\tilde{y}|=1$. Therefore,

$\sum_{i=1}^{\ell}\left|\frac{\partial\gamma_{a,i}}{\partial\xi_{k}}(e^{-\frac{s}{2}}\tilde{y})\right|\leq
C|\tilde{\xi}||\log|\tilde{\xi}||^{1+\mu}.$

Since $\tilde{y}$ is arbitrary in $\partial B_{N-\ell}(0,1)$,
$\tilde{\xi}=e^{-\frac{s}{2}}\tilde{y}$ covers a whole neighborhood of $0$,
namely $B(0,\xi_{0})$ where $\xi_{0}=e^{-\frac{1}{2}\max\\{s_{0}+1,s_{1}\\}}$,
this concludes the proof of Proposition 2.9. ∎

#### Step 2: Further refined asymptotic behavior and deriving
$\mathcal{C}^{2}$-regularity of $S$.

In this part, we shall use the $\mathcal{C}^{1,1-\eta}$-regularity of the
blow-up set together with the geometric constraint (2.24) in order to refine
further the asymptotic behavior (2.10). In particular, we claim the following:

######  Proposition 2.10 (Further refined asymptotic behavior (2.10)).

There exist $s_{2}>0$, $d\in(0,\frac{1}{2})$ and continuous functions
$a\to\lambda_{\beta}(a)$ for all $\beta\in\mathbb{N}^{N}$ with $|\beta|=3$ and
$|\bar{\beta}|=1$, where $\bar{\beta}=(\beta_{1},\cdots,\beta_{\ell})$,
$|\bar{\beta}|=\sum\limits_{i=1}^{\ell}\beta_{i}$, such that for all $a\in
S_{\delta}$ and $s\geq s_{2}$,

$\left\|W_{a}(Q_{a}y,s)-w_{\mathcal{B}(a)}(\bar{y},s)-\frac{e^{-\frac{s}{2}}}{s}\sum_{|\beta|=3,|\bar{\beta}|=1}\lambda_{\beta}(a)h_{\beta}(y)\right\|_{L^{2}_{\rho}}\leq
Ce^{-\frac{s}{2}}s^{d-\frac{3}{2}},$ (2.26)

where $h_{\beta}$ is defined in (3.4).

###### Proof.

The proof of this proposition is based on ideas of [Zaa06] where the case
$\ell=1$ was treated. As in [Zaa06], the geometric constraint given in
Proposition 2.7 plays an important role in deriving (2.26). Since the proof is
long and technical, we leave it to Section 3.4. ∎

Let us derive Theorem 1.1 from Propositions 2.10 and 2.7. In particular,
Theorem 1.1 is a direct consequence of the following:

######  Proposition 2.11.

For all $a\in S_{\delta}$, we have for all $i\in\\{1,\cdots,\ell\\}$,
$j,k\in\\{\ell+1,\cdots,N\\}$,

$\Lambda^{(i)}_{j,k}(a)=\frac{\partial^{2}\gamma_{a,i}}{\partial\xi_{j}\xi_{k}}(0)=\frac{2p}{\kappa}(1+\delta_{j,k})\lambda_{e_{i}+e_{j}+e_{k}}(a),$

where $a\to\lambda_{\beta}(a)$ is introduced in Proposition 2.10, $e_{i}$ is
the $i$-th vector of canonical base of $\mathbb{R}^{N}$, and $\delta_{i,k}$ is
the Kronecker symbol.

###### Proof.

From (2.20), (2.26) and the fact that estimate (2.26) also holds in
$W^{2,\infty}(|y|<2)$ by parabolic regularity, we derive for all $k\geq\ell+1$
and $s\geq s_{2}+1$,

$\sup_{a\in S_{\delta},|y|<2}\left|\frac{\partial w_{a}}{\partial
y_{k}}(y,s)-\frac{e^{-\frac{s}{2}}}{s}\sum_{|\beta|=3,|\bar{\beta}|=1}\lambda_{\beta}(a)\frac{\partial
h_{\beta}}{\partial y_{k}}(y)\right|\leq Ce^{-\frac{s}{2}}s^{d-\frac{3}{2}},$
(2.27)

for some $d\in(0,\frac{1}{2})$.
Note that if $|\bar{\beta}|=1$, then there is a unique index
$i^{*}\in\\{1,\cdots,\ell\\}$ such that $\beta_{i^{*}}=1$ and $\beta_{m}=0$
for $m\in\\{1,\cdots,\ell\\}$, $m\neq i^{*}$. Note also from the definition of
$h_{\beta}$ (see (3.4) below)that

$\frac{\partial h_{\beta}}{\partial
y_{k}}(y)=\beta_{k}h_{\beta_{k}-1}(y_{k})\prod_{j=1,j\neq
k}^{N}h_{\beta_{j}}(y_{j}),$

and that $h_{0}=1$. Therefore, (2.27) yields

$\left|\frac{\partial w_{a}}{\partial
y_{k}}(y,s)-\frac{e^{-\frac{s}{2}}}{s}\sum_{i=1}^{\ell}\sum_{|\beta|=3,\beta_{i}=1}\lambda_{\beta}(a)h_{1}(y_{i})\beta_{k}h_{\beta_{k}-1}(y_{k})\prod_{j=\ell_{a}+1,j\neq
k}h_{\beta_{j}}(y_{j})\right|\leq Ce^{-\frac{s}{2}}s^{d-\frac{3}{2}}.$

Take $i^{*}\in\\{1,\cdots,\ell\\}$ arbitrarily and $y=e_{i^{*}}+\epsilon
e_{j}$ where $\epsilon=\pm 1$ and $j\geq\ell+1$, and note that $h_{m}(0)=0$ if
$m$ is odd, and that if $|\beta|=3,\beta_{i^{*}}=1$, then either
$\beta=e_{i^{*}}+e_{j^{*}}+e_{k^{*}}$ or $\beta=e_{i^{*}}+2e_{j^{*}}$ for some
$j^{*},k^{*}\in\\{\ell+1,\cdots,N\\}$, the above identity yields

$\left|\frac{\partial w_{a}}{\partial y_{k}}(e_{i^{*}}+\epsilon
e_{j},s)-\epsilon\frac{e^{-\frac{s}{2}}}{s}(1+\delta_{k,j})\lambda_{e_{i^{*}}+e_{k}+e_{j}}(a)\right|\leq
Ce^{-\frac{s}{2}}s^{d-\frac{3}{2}}.$ (2.28)

Similarly, we have

$\left|\frac{\partial w_{a}}{\partial y_{k}}(e_{i^{*}},s)\right|\leq
Ce^{-\frac{s}{2}}s^{d-\frac{3}{2}}.$ (2.29)

Now using Proposition 2.7, we write for $y=e_{i^{*}}+\epsilon e_{j}$ and
$s\geq\max\\{s_{2}+1,s_{1}\\}$,

$\displaystyle\left|\frac{\partial w_{a}}{\partial y_{k}}(e_{i^{*}}+\epsilon
e_{j},s)-\frac{\partial w_{a}}{\partial
y_{k}}(e_{i^{*}},s)-\frac{\kappa}{2ps}\frac{\partial\gamma_{a,i^{*}}}{\partial\xi_{k}}(e^{-\frac{s}{2}}\epsilon
e_{j})\right|$ $\displaystyle\leq C\frac{\log
s}{s^{2}}\sum_{i=1}^{\ell}\left|\frac{\partial\gamma_{a,i}}{\partial\xi_{k}}(e^{-\frac{s}{2}}\epsilon
e_{j})\right|+Ce^{-\frac{(1+\alpha^{*})s}{2}}s^{C_{0}}+Ce^{-s}s^{C_{0}+1}.$

Using this estimate together with (2.28) and (2.29), we obtain

$\displaystyle\left|\epsilon
e^{-\frac{s}{2}}(1+\delta_{k,j})\lambda_{e_{i^{*}}+e_{k}+e_{j}}(a)-\frac{\kappa}{2p}\frac{\partial\gamma_{a,i^{*}}}{\partial\xi_{k}}(e^{-\frac{s}{2}}\epsilon
e_{j})\right|$ $\displaystyle\leq C\frac{\log
s}{s}\sum_{i=1}^{\ell}\left|\frac{\partial\gamma_{a,i}}{\partial\xi_{k}}(e^{-\frac{s}{2}}\epsilon
e_{j})\right|+Ce^{-\frac{s}{2}}s^{d-\frac{1}{2}}.$ (2.30)

From Proposition 2.10, we see that

$\forall s\geq
s_{2},\quad\|W_{a}(Q_{a}y,s)-w_{\mathcal{B}(a)}(\bar{y},s)\|_{L^{2}_{\rho}}\leq
Cs^{-1}e^{-\frac{s}{2}}.$

Using this estimate and noticing that the same proof of Proposition 2.9 holds
with $\mu=-1$, we derive

$\sum_{i=1}^{\ell}\left|\frac{\partial\gamma_{a,i}}{\partial\xi_{k}}(e^{-\frac{s}{2}}\epsilon
e_{j})\right|\leq Ce^{-\frac{s}{2}}.$

Putting this estimate into (2.30) and note that
$\frac{\partial\gamma_{a,i^{*}}}{\partial\xi_{k}}(0)=0$, we find that

$\frac{\partial^{2}\gamma_{a,i^{*}}}{\partial\xi_{k}\partial\xi_{j}}(0)=\lim_{s\to+\infty}\frac{\frac{\partial\gamma_{a,i^{*}}}{\partial\xi_{k}}(e^{-\frac{s}{2}}\epsilon
e_{j})}{\epsilon
e^{-\frac{s}{2}}}=\frac{2p}{\kappa}(1+\delta_{k,j})\lambda_{e_{i^{*}}+e_{k}+e_{j}}(a).$
(2.31)

Since $i^{*}$ is taken arbitrarily belonging to $\\{1,\cdots,\ell\\}$,
identity (2.31) holds for all $i^{*}\in\\{1,\cdots,\ell\\}$. This concludes
the proof of Proposition 2.11. ∎

###### Proof of Theorem 1.1.

From the definition of the local chart (2.17), we have for all
$i\in\\{1,\cdots,\ell\\}$, $\gamma_{a,i}(0)=\nabla\gamma_{a,i}(0)=0$. Hence,
we deduce from (2.31) the expression of the second fundamental form of the
blow-up set at the point $a$ along the unitary basic vector $Q_{a}^{T}e_{i}$:
for all $k,j\in\\{\ell+1,\cdots,N\\}$,

$\Lambda_{k,j}^{(i)}(a)=\frac{\partial^{2}\gamma_{a,i}}{\partial\xi_{k}\partial\xi_{j}}(0)=\frac{2p}{\kappa}(1+\delta_{k,j})\lambda_{e_{i}+e_{k}+e_{j}}(a).$
(2.32)

In addition, since $a\to\lambda_{\beta}(a)$ is continuous, we conclude that
the blow-up set is of class $\mathcal{C}^{2}$. This completes the proof of
Theorem 1.1. ∎

###### Proof of Theorem 1.3.

The estimate (1.23) directly follows from Propositions 2.10 and 2.11. Indeed,
the sum in estimate (2.26) can be indexed as

$\\{\beta\in\mathbb{N}^{N},|\beta|=3,|\bar{\beta}|=1\\}=\\{e_{i}+e_{j}+e_{k},1\leq
i\leq\ell,\ell+1\leq j,k\leq N\\},$

where $e_{k}$ is the $k$-th canonical basis vector of $\mathbb{R}^{N}$. From
(2.32) and the definition of $h_{\beta}$ (see (3.4) below), we write

$\displaystyle\sum_{|\beta|=3,|\bar{\beta}|=1}\lambda_{\beta}(a)h_{\beta}(y)$
$\displaystyle=\sum_{i=1}^{\ell}\sum_{j,k=\ell+1}^{N}\lambda_{e_{i}+e_{j}+e_{k}}h_{e_{i}+e_{j}+e_{k}}(y)$
$\displaystyle=\frac{\kappa}{2p}\sum_{i=1}^{\ell}y_{i}\sum_{j,k=\ell+1}^{N}\frac{\Lambda^{(i)}_{j,k}(a)}{1+\delta_{j,k}}(y_{j}y_{k}-2\delta_{j,k}),$

which yields (1.23).

As for (1.24), we note from (2.26) that for all $|\beta|=3$ with
$|\bar{\beta}|=1$ that (recall that
$g_{a}(y,s)=W_{a}(Q_{a}y,s)-w_{\mathcal{B}(a)}(\bar{y},s)$),

$\left|g_{a,\beta}(s)-\frac{e^{-\frac{s}{2}}}{s}\lambda_{\beta}(s)\right|\leq
Ce^{-\frac{s}{2}}s^{d-\frac{3}{2}}.$

Hence, we write from (2.32),

$\displaystyle\Lambda^{(i)}_{j,k}(a)$
$\displaystyle=\frac{2p}{\kappa}(1+\delta_{j,k})\lambda_{e_{i}+e_{j}+e_{k}}(a)$
$\displaystyle=\frac{2p}{\kappa}(1+\delta_{j,k})\lim_{s\to+\infty}se^{\frac{s}{2}}g_{a,e_{i}+e_{j}+e_{k}}(s)$
$\displaystyle=\frac{2p}{\kappa}(1+\delta_{j,k})\lim_{s\to+\infty}se^{\frac{s}{2}}\int_{\mathbb{R}^{N}}g_{a}(y,s)\frac{h_{e_{i}+e_{j}+e_{k}}(y)}{\|h_{e_{i}+e_{j}+e_{k}}\|^{2}_{L^{2}_{\rho}}}\rho(y)dy$

Using again the definition of $h_{\beta}$ (see (3.4) below), we see that
$h_{e_{i}+e_{j}+e_{k}}=y_{i}(y_{j}y_{k}-\delta_{j,k})$ and
$\|h_{e_{i}+e_{j}+e_{k}}\|^{2}_{L^{2}_{\rho}}=8(1+\delta_{j,k})$. Recall that
$w_{\mathcal{A}}$ does not depend on $y_{j}$ for $j\geq\ell+1$. Hence, for all
$j,k\geq\ell+1$,

$\Lambda^{(i)}_{j,k}(a)=\frac{p}{4\kappa}\lim_{s\to+\infty}se^{\frac{s}{2}}\int_{\mathbb{R}^{N}}W_{a}(Q_{a}y,s)y_{i}(y_{j}y_{k}-2\delta_{j,k})\rho(y)dy,$

which is (1.24). This concludes the proof of Theorem 1.3. ∎

##  3\. Proof of Propositions 2.1, 2.7, 2.8 and 2.10.

###  3.1. Classification of the difference of two solutions of (1.13) having
the same asymptotic behavior.

In this subsection, we give the proof of Proposition 2.1. The formulation is
the same as given in [FZ00] for the difference of two solutions with the
radial profile ($\ell=N$). Therefore, we sketch the proof and emphasize only
on the novelties. Note also that the case $\ell=1$ was treated in [Zaa02b].

Let us define

$g(y,s)=W_{1}(y,s)-W_{2}(y,s),$ (3.1)

where $W_{i},\;i=1,2$ are the solutions of equation (1.13) and behave like
(2.4). We see from (1.13) and (2.4) that for all
$(y,s)\in\mathbb{R}^{N}\times[-\log T,+\infty)$,

$\left\\{\begin{array}[]{l}\partial_{s}g=\mathscr{L}g+\alpha g,\\\
\|g(s)\|_{L^{2}_{\rho}}\leq C\frac{\log s}{s^{2}},\end{array}\right.$ (3.2)

where

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

and

$\alpha(y,s)=\frac{|W_{1}|^{p-1}W_{1}-|W_{2}|^{p-1}W_{2}}{W_{1}-W_{2}}-\frac{p}{p-1}\quad\text{if}\;\;W_{1}\neq
W_{2},$

in particular,

$\alpha(y,s)=p|W_{0}(y,s)|^{p-1}-\frac{p}{p-1},\quad\text{for
some}\;\;W_{0}(y,s)\in(W_{1}(y,s),W_{2}(y,s)).$ (3.3)

The operator $\mathscr{L}$ is self-adjoint on $\mathcal{D}(\mathscr{L})\subset
L^{2}_{\rho}(\mathbb{R}^{N})$. Its spectrum consists of eigenvalues

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

The eigenfunctions corresponding to $1-\frac{n}{2}$ are

$h_{\beta}(y)=h_{\beta_{1}}(y_{1})\cdots
h_{\beta_{N}}(y_{N}),\quad\beta_{1}+\cdots+\beta_{N}=|\beta|=n,$ (3.4)

where

$h_{m}(\xi)=\sum_{i=0}^{[m/2]}\frac{m!}{i!(m-2i)!}(-1)^{i}\xi^{m-2i}\quad\text{for}\;\;m\in\mathbb{N},$

satisfy

$\int_{\mathbb{R}}h_{m}(\xi)h_{n}(\xi)\rho(\xi)d\xi=2^{m}m!\delta_{m,n}.$

The component of $g$ on $h_{\beta}$ is given by

$g_{\beta}(s)=\int_{\mathbb{R}^{N}}k_{\beta}(y)g(y,s)\rho(y)dy,\quad\text{where}\quad
k_{\beta}(y)=\frac{h_{\beta}(y)}{\|h_{\beta}\|_{L^{2}_{\rho}}^{2}}.$

If we denote by $P_{n}$ the orthogonal projector of $L^{2}_{\rho}$ over the
eigenspace of $\mathscr{L}$ corresponding to the eigenvalue $1-\frac{n}{2}$,
then

$P_{n}g(y,s)=\sum_{|\beta|=n}g_{\beta}(s)h_{\beta}(y).$

Since the eigeinfunctions of $\mathscr{L}$ span the whole space
$L^{2}_{\rho}$, we can write

$g(y,s)=\sum_{n\in
N}P_{n}g(y,s)=\sum_{\beta\in\mathbb{N}^{N}}g_{\beta}(s)h_{\beta}(y)=\sum_{\beta\in\mathbb{N}^{N},|\beta|\leq
k}g_{\beta}(s)h_{\beta}(y)+R_{k+1}g(y,s),$

where $R_{k}g=\sum\limits_{n\geq k}P_{n}g$. We also denote

$I(s)^{2}=\|g(s)\|_{L^{2}_{\rho}}^{2}=\sum_{n\in\mathbb{N}}l_{n}^{2}(s)=\sum_{n\leq
k}l_{n}^{2}(s)+r_{k+1}^{2}(s),$ (3.5)

where

$l_{n}(s)=\|P_{n}g(s)\|_{L^{2}_{\rho}},\quad
r_{k}(s)=\|R_{k}g(s)\|_{L^{2}_{\rho}}.$ (3.6)

As for $\alpha$, we have the following estimates:

######  Lemma 3.1 (Estimates on $\alpha$).

For all $y\in\mathbb{R}^{N}$ and $s\geq-\log T$, we have

$\alpha(y,s)\leq\frac{C}{s},\quad|\alpha(y,s)|\leq\frac{C}{s}(1+|y|^{2}),$

and

$\left|\alpha(y,s)+\frac{1}{4s}\sum_{i=1}^{\ell}h_{2}(y_{i})\right|\leq\frac{C}{s^{\frac{3}{2}}}(1+|y|^{3}).$
(3.7)

###### Proof.

The proof follows the same lines as the proof of Lemma 2.5 in [FZ00] where the
case $\ell=N$ was treated. ∎

In the following lemma, we project equation (3.2) on the different modes to
get estimates for $I(s)$, $l_{n}(s)$ and $r_{n}(s)$. More precisely, we claim
the following:

######  Lemma 3.2 (Evolution of $I(s)$, $l_{n}(s)$ and $r_{n}(s)$).

There exist $s_{3}\geq-\log T$ and $s_{*}>0$ such that for all $s\geq s_{3}$,
$n\in\mathbb{N}$ and $\beta\in\mathbb{N}^{N}$, we have

  * i)

$\left|l^{\prime}_{n}(s)+\left(\frac{n}{2}-1\right)l_{n}(s)\right|\leq
C(n)\dfrac{I(s)}{s}$.

  * ii)

$I^{\prime}(s)\leq\left(1-\frac{n+1}{2}+\frac{C_{0}}{s}\right)I(s)+\sum\limits_{k=0}^{n}\frac{1}{2}(n+1-k)l_{k}(s)$.

  * iii)

$\left|g^{\prime}_{\beta}(s)+\left(-1+\frac{|\beta|}{2}+\frac{1}{s}\sum\limits_{i=1}^{\ell}\beta_{i}\right)g_{\beta}(s)\right|\leq
C(\beta)\left(\dfrac{1}{s^{\frac{3}{2}}}I(s)+\dfrac{1}{s}\left(l_{|\beta|-2}(s)+l_{|\beta|+2}\right)\right)$.

  * iv)

$r^{\prime}_{n}(s)\leq\left(1-\frac{n}{2}\right)r_{n}(s)+\frac{C}{s}I(s-s_{*})$.

###### Proof.

For $(i)$ and $(ii)$, see Lemma 2.7, page 1197 in [FZ00]. For $(iii)$, see
Appendix B.1, page 545 in [Zaa02b] for a similar calculations. For $(iv)$, see
page 523 in [Zaa06], where the calculation is mainly based on the following
regularizing property of equation (3.2) by Herrero and Velázquez [HV93]
(control of the $L^{4}_{\rho}$-norm by the $L^{2}_{\rho}$-norm up to some
delay in time, see Lemma 2.3 in [HV93]):

$\left(\int g^{4}(y,s)\rho dy\right)^{1/4}\leq C\left(\int
g^{2}(y,s-s_{*})\rho dy\right)^{\frac{1}{2}}\quad\text{for some}\;\;s_{*}>0.$

This ends the proof of Lemma 3.2. ∎

In the next step, we use Lemma 3.2 to show that either the null mode or a
negative mode of $\mathscr{L}$ will dominate as $s\to+\infty$. In particular,
we have the following:

######  Proposition 3.3 (Dominance of a mode and its description).

We have
$i)\;$ Either for all $n\in\mathbb{N}$,
$l_{n}(s)=\mathcal{O}\left(\dfrac{I(s)}{s}\right)$ and there exist
$\sigma_{n}$, $C_{n}>0$ and $C_{n}^{\prime}>0$ such that

$\forall s\geq\sigma_{n},\quad I(s)\leq
C_{n}s^{C^{\prime}_{n}}exp\left((1-n/2)s\right).$

$ii)$ Or there is $n_{0}\geq 2$ such that

$I(s)\sim l_{n_{0}}(s)\;\;\text{and}\;\;\forall n\neq
n_{0},\;\;l_{n}(s)=\mathcal{O}\left(\dfrac{I(s)}{s}\right)\quad\text{as}\;\;s\to+\infty.$
(3.8)

Moreover,

  * $\bullet$

If $n_{0}=2$, namely $I(s)\sim l_{2}(s)$, then

$\forall|\beta|=2,\quad\left\\{\begin{array}[]{ll}|g_{\beta}(s)|\leq
C\frac{\log s}{s^{5/2}}&\text{if}\;\;\sum_{i=1}^{\ell}\beta_{i}\neq 2,\\\
\left|g_{\beta}(s)-\frac{c_{\beta}}{s^{2}}\right|\leq C\frac{\log
s}{s^{5/2}}&\text{if}\;\;\sum_{i=1}^{\ell}\beta_{i}=2.\end{array}\right.$
(3.9)

  * $\bullet$

If $n_{0}=3$, namely $I(s)\sim l_{3}(s)$, then

$I(s)\leq C_{0}e^{-\frac{s}{2}}s^{C_{0}}\quad\text{for some}\;\;C_{0}>0.$
(3.10)

###### Proof.

See Proposition 2.6, page 1196 in [FZ00] for the existence of a dominating
component, where the proof relies on $(i)$ and $(ii)$ of Lemma 3.2. If case
$(ii)$ occurs with $n_{0}=2$, we write from $(iii)$ of Lemma 3.2: for all
$\beta\in\mathbb{N}^{N}$ with $|\beta|=2$,

$\left|g^{\prime}_{\beta}(s)+\frac{g_{\beta}}{s}\sum_{i=1}^{\ell}\beta_{i}\right|\leq
C(\beta)\left(\dfrac{I(s)}{s^{\frac{3}{2}}}+\dfrac{l_{0}(s)+l_{4}(s)}{s}\right)\leq
C(\beta)\dfrac{I(s)}{s^{\frac{3}{2}}}\leq C(\beta)\frac{\log s}{s^{7/2}},$

where we used (3.8) and (3.2) from which
$l_{0}(s)+l_{4}(s)=\mathcal{O}\left(\frac{I(s)}{s}\right)$ and
$I(s)=\mathcal{O}\left(\frac{\log s}{s^{2}}\right)$. Since
$\sum\limits_{i=1}^{\ell}\beta_{i}$ is only equal to $0,1$ or $2$ if
$|\beta|=2$, estimate (3.9) follows after integration. Estimate (3.10)
immediately follows from $(i)$ of Lemma 3.2. This ends the proof of
Proposition 3.3. ∎

Let us now derive Proposition 2.1 from Proposition 3.3. Indeed, we see from
Proposition 3.3 that if case $i)$ occurs, we already have exponential decay
for $I(s)$. If case $(ii)$ occurs with $n_{0}\geq 3$, we write from part $(i)$
of Lemma 3.2,

$\left|l^{\prime}_{n_{0}}(s)+\left(\frac{n_{0}}{2}-1\right)l_{n_{0}}\right|\leq\frac{C}{s}l_{n_{0}}.$

Since $l_{n_{0}}\neq 0$ in a neighbourhood of infinity, this gives

$l_{n_{0}}(s)\leq C_{0}s^{C_{0}}e^{\left(1-\frac{n_{0}}{2}\right)s}\leq
C_{0}s^{C_{0}}e^{-\frac{s}{2}},$

which yields (2.8). If case $(ii)$ occurs with $n_{0}=2$, by definition of
$P_{2}$, we derive from (3.9) that there is a symmetric, real
$(\ell\times\ell)$-matrix $\mathcal{B}$ such that

$P_{2}g(y,s)=\frac{1}{s^{2}}\left(\frac{1}{2}\bar{y}^{T}\mathcal{B}\bar{y}-tr(\mathcal{B})\right)+o\left(\frac{1}{s^{2}}\right),$

which is (2.7). This concludes the proof of Proposition 2.1. $\square$

###  3.2. $\mathcal{C}^{1,\frac{1}{2}-\eta}$-regularity of the blow-up set.

We give the proof of Proposition 2.8 in this section. The proof uses the
argument given in [Zaa02b] treated for the case $\ell=1$. Here we shall
exploit the refined estimate (2.12) to obtain a geometric constraint on the
blow-up set. Without loss of generality, we assume $\hat{a}=0$ and
$Q_{\hat{a}}=Id$. Under the hypotheses of Proposition 2.8, we know that
$\gamma\in\mathcal{C}^{1}((-\delta_{1},\delta_{1})^{N-\ell},\mathbb{R}^{\ell})$
with $\ell\in\\{1,\cdots,N-1\\}$. If we introduce

$\Gamma(\tilde{x})=(\gamma_{1}(\tilde{x}),\cdots,\gamma_{\ell}(\tilde{x}),\tilde{x}),\quad\tilde{x}=(x_{\ell+1},\cdots,x_{N}),$

then

$\text{Im}\,\Gamma\cap B(0,2\delta)=\text{graph}(\gamma)\cap
B(0,2\delta)=S_{\delta}.$

Consider $\tilde{x}$ and $\tilde{h}$ in $\mathbb{R}^{N-\ell}$ such that
$\tilde{x}$ as well as $\tilde{x}+\tilde{h}$ are in $B(0,\delta_{1})$ and
$\Gamma(\tilde{x})$ as well as $\Gamma(\tilde{x}+\tilde{h})$ are in
$S_{\delta}$. For all $t\in[T-e^{-s_{0}-1},T)$ such that
$|\Gamma(\tilde{x})-\Gamma(\tilde{x}+\tilde{h})|\leq\sqrt{(T-t)|\log(T-t)|}$,
we use (2.12) with $x=a=\Gamma(\tilde{x}+\tilde{h})$, then with
$x=\Gamma(\tilde{x}+\tilde{h})$ and $a=\Gamma(\tilde{x})$) to find that

$\left\\{\begin{array}[]{cl}\left|(T-t)^{\frac{1}{p-1}}u(\Gamma(\tilde{x}+\tilde{h}),t)-w_{\mathcal{B}(\Gamma(\tilde{x}+\tilde{h}))}(0,s)\right|&\leq
Ce^{-\frac{s}{2}}s^{\frac{3}{2}+C_{0}},\\\
\left|(T-t)^{\frac{1}{p-1}}u(\Gamma(\tilde{x}+\tilde{h}),t)-w_{\mathcal{B}(\Gamma(\tilde{x}))}(\bar{y}_{\Gamma(\tilde{x}),\Gamma(\tilde{x}+\tilde{h}),s},s)\right|&\leq
Ce^{-\frac{s}{2}}s^{\frac{3}{2}+C_{0}},\end{array}\right.$ (3.11)

where $\bar{y}_{\Gamma(\tilde{x}),\Gamma(\tilde{x}+\tilde{h}),s}$ is defined
as

$\bar{y}_{a_{1},a_{2},s}=e^{\frac{s}{2}}\big{(}(a_{1}-a_{2})\cdot
Q_{a_{1}}e_{1},\cdots,(a_{1}-a_{2})\cdot Q_{a_{1}}e_{\ell}\big{)}.$ (3.12)

Since $\Gamma$ is $\mathcal{C}^{1}$, we have

$|\Gamma(\tilde{x}+\tilde{h})-\Gamma(\tilde{x})|\leq C|\tilde{h}|.$

Let us fix $t=\tilde{t}(\tilde{x},\tilde{h})$ such that

$|\Gamma(\tilde{x}+\tilde{h})-\Gamma(\tilde{x})|=\sqrt{(T-\tilde{t})|\log(T-\tilde{t})|},$
(3.13)

and take $\tilde{h}\in B_{N-\ell}(0,h_{1}(s_{0}))$ for some $h_{1}(s_{0})>0$,
then we have $\tilde{t}\geq T-e^{-s_{0}-1}$. Hence, if
$\tilde{s}=-\log(T-\tilde{t})$, we have by (3.11),

$\left|w_{\mathcal{B}(\Gamma(\tilde{x}+\tilde{h}))}(0,\tilde{s})-w_{\mathcal{B}(\Gamma(\tilde{x}))}(\bar{y}_{\Gamma(\tilde{x}),\Gamma(\tilde{x}+\tilde{h}),\tilde{s}},\tilde{s})\right|\leq
Ce^{-\frac{\tilde{s}}{2}}\tilde{s}^{\frac{3}{2}+C_{0}}.$ (3.14)

Similarly, by changing the roles of $\tilde{x}$ and $\tilde{x}+\tilde{h}$, we
get

$\left|w_{\mathcal{B}(\Gamma(\tilde{x}))}(0,\tilde{s})-w_{\mathcal{B}(\Gamma(\tilde{x}+\tilde{h}))}(\bar{y}_{\Gamma(\tilde{x}+\tilde{h}),\Gamma(\tilde{x}),\tilde{s}},\tilde{s})\right|\leq
Ce^{-\frac{\tilde{s}}{2}}\tilde{s}^{\frac{3}{2}+C_{0}},$ (3.15)

where $\bar{y}_{\Gamma(\tilde{x}+\tilde{h}),\Gamma(\tilde{x}),\tilde{s}}$ is
defined as in (3.12).

From a Taylor expansion for $w_{\mathcal{B}}(\bar{y},\tilde{s})$ near
$\bar{y}=0$, we write

$w_{\mathcal{B}}(\bar{y},\tilde{s})=w_{\mathcal{B}}(0,\tilde{s})+\bar{y}\cdot\nabla
w_{\mathcal{B}}(0,\tilde{s})+\frac{1}{2}\bar{y}^{T}\nabla^{2}w_{\mathcal{B}}(0,\tilde{s})\bar{y}+\mathcal{O}\left(|\bar{y}|^{3}|\nabla^{3}w_{\mathcal{B}}(z,\tilde{s})|\right),$
(3.16)

for some $z$ between $0$ and $\bar{y}$.

Since (2.3) and (2.9) also hold in $\mathcal{C}_{loc}^{k}$ by parabolic
regularity, we deduce that

$|\nabla
w_{\mathcal{B}}(0,\tilde{s})|=\mathcal{O}\left(\frac{\log\tilde{s}}{\tilde{s}^{2}}\right),\quad\nabla^{2}w_{\mathcal{B}}(0,\tilde{s})=-\frac{\kappa}{4p\tilde{s}}I_{\ell\times\ell}+\mathcal{O}\left(\frac{\log\tilde{s}}{\tilde{s}^{2}}\right).$

From [MZ98] (see Theorem 1), we know that
$\|\nabla^{3}w_{\mathcal{B}}(\tilde{s})\|_{L^{\infty}}\leq\frac{C_{3}}{\tilde{s}^{\frac{3}{2}}}$.
Substituting all these above estimates into (3.16) yields

$w_{\mathcal{B}}(\bar{y},\tilde{s})\leq
w_{\mathcal{B}}(0,\tilde{s})-\frac{\kappa}{8p\tilde{s}}|\bar{y}|^{2}+\frac{C_{3}|\bar{y}|^{3}}{6\tilde{s}^{\frac{3}{2}}}+\frac{C\log\tilde{s}}{\tilde{s}^{2}}.$

Therefore, we have

$\forall|\bar{y}|\leq\frac{3\kappa}{8C_{3}p}\sqrt{\tilde{s}},\quad
w_{\mathcal{B}}(\bar{y},\tilde{s})\leq
w_{\mathcal{B}}(0,\tilde{s})-\frac{\kappa}{16p\tilde{s}}|\bar{y}|^{2}.$ (3.17)

We claim from (3.14), (3.15) and (3.17) the following:

$\left|w_{\mathcal{B}(\Gamma(\tilde{x}))}(0,\tilde{s})-w_{\mathcal{B}(\Gamma(\tilde{x}+\tilde{h}))}(0,\tilde{s})\right|\leq
Ce^{-\frac{\tilde{s}}{2}}\tilde{s}^{\frac{3}{2}+C_{0}},$ (3.18)

Indeed, if
$w_{\mathcal{B}(\Gamma(\tilde{x}))}(0,\tilde{s})-w_{\mathcal{B}(\Gamma(\tilde{x}+\tilde{h}))}(0,\tilde{s})\geq
0$, then we have by (3.17) and (3.15),

$\displaystyle 0$ $\displaystyle\leq
w_{\mathcal{B}(\Gamma(\tilde{x}))}(0,\tilde{s})-w_{\mathcal{B}(\Gamma(\tilde{x}+\tilde{h}))}(0,\tilde{s})$
$\displaystyle\leq
w_{\mathcal{B}(\Gamma(\tilde{x}))}(0,\tilde{s})-w_{\mathcal{B}(\Gamma(\tilde{x}+\tilde{h}))}(\bar{y}_{\Gamma(\tilde{x}+\tilde{h}),\Gamma(\tilde{x}),\tilde{s}},\tilde{s})\leq
Ce^{-\frac{\tilde{s}}{2}}\tilde{s}^{\frac{3}{2}+C_{0}}.$

If
$w_{\mathcal{B}(\Gamma(\tilde{x}))}(0,\tilde{s})-w_{\mathcal{B}(\Gamma(\tilde{x}+\tilde{h}))}(0,\tilde{s})\leq
0$, then we do as above and use (3.14) instead of (3.15) to obtain (3.18).
From (3.18), (3.14) and (3.17), we get

$\frac{\kappa}{16p\tilde{s}}|\bar{y}_{\Gamma(\tilde{x}),\Gamma(\tilde{x}+\tilde{h}),\tilde{s}}|^{2}\leq
w_{\mathcal{B}(\Gamma(\tilde{x}))}(0,\tilde{s})-w_{\mathcal{B}(\Gamma(\tilde{x}))}(\bar{y}_{\Gamma(\tilde{x}),\Gamma(\tilde{x}+\tilde{h}),\tilde{s}},\tilde{s})\leq
Ce^{-\frac{\tilde{s}}{2}}\tilde{s}^{\frac{3}{2}+C_{0}}.$

Hence, we obtain

$|\bar{y}_{\Gamma(\tilde{x}),\Gamma(\tilde{x}+\tilde{h}),\tilde{s}}|^{2}\leq
Ce^{-\frac{\tilde{s}}{2}}\tilde{s}^{\frac{5}{2}+C_{0}}.$ (3.19)

From the definition (3.12), we have

$|\bar{y}_{\Gamma(\tilde{x}),\Gamma(\tilde{x}+\tilde{h}),\tilde{s}}|=e^{\frac{\tilde{s}}{2}}d(\Gamma(\tilde{x}),\pi_{\Gamma(\tilde{x}+\tilde{h})}),$
(3.20)

where we recall $\pi_{\Gamma(\tilde{x}+\tilde{h})}$ is the tangent plan of $S$
at $\Gamma(\tilde{x}+\tilde{h})$. In the other hand, we claim that

$d(\Gamma(\tilde{x}),T_{\Gamma(\tilde{x}+\tilde{h})})\geq\frac{|\gamma_{i}(\tilde{x}+\tilde{h})-\gamma_{i}(\tilde{x})-\tilde{h}\cdot\nabla\gamma_{i}(\tilde{x})|}{\sqrt{1+|\nabla\gamma_{i}(\tilde{x})|^{2}}},$
(3.21)

where $S_{i}$ is the surface of equation $x_{i}=\gamma_{i}(\tilde{x})$,
$T_{i,\Gamma(\tilde{x}+\tilde{h})}$ is the tangent plan of $S_{i}$ at
$\Gamma(\tilde{x}+\tilde{h})$. Indeed, we note that

$d(\Gamma(\tilde{x}),T_{i,\Gamma(\tilde{x}+\tilde{h})})=\frac{|\gamma_{i}(\tilde{x}+\tilde{h})-\gamma_{i}(\tilde{x})-\tilde{h}\cdot\nabla\gamma_{i}(\tilde{x})|}{\sqrt{1+|\nabla\gamma_{i}(\tilde{x})|^{2}}},$

and $Im\,\Gamma\subset S_{i}$, hence, (3.21) follows from
$d(\Gamma(\tilde{x}),T_{\Gamma(\tilde{x}+\tilde{h})})\geq
d(\Gamma(\tilde{x}),T_{i,\Gamma(\tilde{x}+\tilde{h})})$.

Combining (3.19), (3.20), (3.21) together with the relation
$\tilde{s}=-\log(T-\tilde{t})$ yields

$|\gamma_{i}(\tilde{x}+\tilde{h})-\gamma_{i}(\tilde{x})-\tilde{h}\cdot\nabla\gamma_{i}(\tilde{x})|^{2}\leq
C(T-\tilde{t})^{\frac{3}{2}}|\log(T-\tilde{t})|^{\frac{5}{2}+C_{0}}.$

If we denote $A=|\Gamma(\tilde{x}+\tilde{h})-\Gamma(\tilde{x})|\leq
C|\tilde{h}|$, then we have by the relation (3.13),

$|\log(T-\tilde{t})|\sim 2|\log A|,\quad T-\tilde{t}\sim\frac{A^{2}}{2|\log
A|}\quad\text{as}\quad A\to 0.$

Hence,

$|\gamma_{i}(\tilde{x}+\tilde{h})-\gamma_{i}(\tilde{x})-\tilde{h}\cdot\nabla\gamma_{i}(\tilde{x})|^{2}\leq
CA^{3}|\log A|^{1+C_{0}}\leq C|\tilde{h}|^{3}|\log|\tilde{h}||^{1+C_{0}},$

which yields (2.25). This concludes the proof of Proposition 2.8. $\square$

###  3.3. A geometric constraint linking the blow-up behavior of the solution
to the regularity of the blow-up set.

This section is devoted to the proof of Proposition 2.7. The proof follows
ideas given in [Zaa06]. Recall from the hypothesis that
$\gamma_{a}\in\mathcal{C}^{1,\alpha^{*}}((-\epsilon_{a},\epsilon_{a})^{N-\ell},\mathbb{R}^{\ell})$
for some $\alpha^{*}\in(0,\frac{1}{2})$ and $\epsilon_{a}>0$, and that
$\gamma_{a,i}(0)=\nabla\gamma_{a,i}(0)=0$, we have for all
$|\tilde{\xi}|<\epsilon_{a}$,

$|\gamma_{a,i}(\tilde{\xi})|\leq
C|\tilde{\xi}|^{1+\alpha^{*}}\quad\text{and}\quad|\nabla\gamma_{a,i}(\tilde{\xi})|\leq
C|\tilde{\xi}|^{\alpha^{*}}.$ (3.22)

In what follows, $k\in\\{\ell+1,\cdots,N\\}$ is fixed, and we use indexes $i$
and $m$ for the range $1,\cdots,\ell$, index $j$ for the range
$\ell+1,\cdots,N$.

We now use (3.22) to approximate all the terms appearing in (2.22).

$(a)$ Term $\tau_{k}(a)\cdot\eta_{i}(b)$. From the local coordinates (2.23),
we have

$\eta_{i}(b)=\frac{1}{\sqrt{1+|\nabla\gamma_{a,i}(e^{-\frac{s}{2}}\tilde{y})|^{2}}}\left(\eta_{i}(a)-\sum_{j=\ell+1}^{N}\frac{\partial\gamma_{a,i}}{\partial\xi_{j}}(e^{-\frac{s}{2}}\tilde{y})\tau_{j}(a)\right).$

Using (3.22) and the fact that $\tau_{k}(a)\cdot\eta_{i}(a)=0$ and
$\tau_{k}(a)\cdot\tau_{j}(a)=\delta_{k,j}$, we obtain

$\displaystyle\left|\tau_{k}(a)\cdot\eta_{i}(b)+\frac{\partial\gamma_{a,i}}{\partial\xi_{k}}(e^{-\frac{s}{2}}\tilde{y})\right|$
$\displaystyle=\left|\left(1-\frac{1}{\sqrt{1+|\nabla\gamma_{a,i}(e^{-\frac{s}{2}}\tilde{y})|^{2}}}\right)\frac{\partial\gamma_{a,i}}{\partial\xi_{k}}(e^{-\frac{s}{2}}\tilde{y})\right|$
$\displaystyle\leq\left|\frac{\partial\gamma_{a,i}}{\partial\xi_{k}}(e^{-\frac{s}{2}}\tilde{y})\right||\nabla\gamma_{a,i}(e^{-\frac{s}{2}}\tilde{y})|^{2}$
$\displaystyle\leq\left|\frac{\partial\gamma_{a,i}}{\partial\xi_{k}}(e^{-\frac{s}{2}}\tilde{y})\right|e^{-\alpha^{*}s}.$
(3.23)

$(b)$ Term $\tau_{k}(a)\cdot\tau_{j}(b)$. From (2.23) and (3.22), we have

$|b-a|\leq\left|\sum_{i=1}^{\ell}\gamma_{a,i}(e^{-\frac{s}{2}}\tilde{y})\right|+e^{-\frac{s}{2}}|\tilde{y}|\leq
Ce^{-\frac{s}{2}}.$

Since $\eta_{i}$ and $\tau_{j}$ are $\mathcal{C}^{\alpha^{*}}$, it holds that

$|\eta_{i}(a)-\eta_{i}(b)|+|\tau_{j}(a)-\tau_{j}(b)|\leq
C|a-b|^{\alpha^{*}}\leq Ce^{-\alpha^{*}\frac{s}{2}}.$

This follows that

$\begin{array}[]{c}|\eta_{i}(a)\cdot\eta_{m}(b)-\delta_{i,m}|+|\tau_{k}(a)\cdot\tau_{j}(b)-\delta_{k,j}|\leq
Ce^{-\alpha^{*}\frac{s}{2}},\\\
|\eta_{i}(a)\cdot\tau_{j}(b)|+|\eta_{i}(b)\cdot\tau_{j}(a)|\leq
Ce^{-\alpha^{*}\frac{s}{2}}.\end{array}$ (3.24)

$(c)$ The point $Y(a,y,s)$. Using (2.18), (2.21) and (2.23), we write

$\displaystyle Y_{m}$ $\displaystyle=Y\cdot
e_{m}=\big{(}Q_{a}y+e^{\frac{s}{2}}(a-b)\big{)}\cdot Q_{b}e_{m}$
$\displaystyle=\left\\{\sum_{i=1}^{\ell}y_{i}\eta_{i}(a)+\sum_{j=\ell+1}^{N}y_{j}\tau_{j}(a)-e^{\frac{s}{2}}\left[\sum_{i=1}^{\ell}\gamma_{a,i}(e^{-\frac{s}{2}}\tilde{y})\eta_{i}(a)+\sum_{j=\ell+1}^{N}e^{-\frac{s}{2}}y_{j}\tau_{j}(a)\right]\right\\}\cdot
Q_{b}e_{m}$
$\displaystyle=\left\\{\sum_{i=1}^{\ell}\big{[}y_{i}-e^{\frac{s}{2}}\gamma_{a,i}(e^{-\frac{s}{2}}\tilde{y})\big{]}\eta_{i}(a)\right\\}\cdot
Q_{b}e_{m}.$

From (2.18), we write for $m\in\\{1,\cdots,\ell\\}$,

$\displaystyle Y_{m}-y_{m}$
$\displaystyle=\left\\{\left(y_{m}-e^{\frac{s}{2}}\gamma_{a,m}(e^{-\frac{s}{2}}\tilde{y})\right)\eta_{m}(a)\cdot\eta_{m}(b)-y_{m}\eta_{m}(a)\cdot\eta_{m}(a)\right\\}$
$\displaystyle+\sum_{i=1,i\neq
m}^{\ell}\left(y_{i}-e^{\frac{s}{2}}\gamma_{a,i}(e^{-\frac{s}{2}}\tilde{y})\right)\eta_{i}(a)\cdot\eta_{m}(b),$

and for $n\in\\{\ell+1,\cdots,N\\}$,

$Y_{n}=\sum_{i=1}^{\ell}\left(y_{i}-e^{\frac{s}{2}}\gamma_{a,i}(e^{-\frac{s}{2}}\tilde{y})\right)\eta_{i}(a)\cdot\tau_{n}(b).$

Using (3.24) yields

$|Y_{m}-y_{m}|\leq Ce^{-\alpha^{*}\frac{s}{2}}\quad\text{and}\quad|Y_{k}|\leq
Ce^{-\alpha^{*}\frac{s}{2}}.$

Hence, if we write

$\bar{Y}=(Y_{1},\cdots,Y_{\ell})\quad\text{and}\quad\tilde{Y}=(Y_{\ell+1},\cdots,Y_{N}),$

then

$|\bar{y}-\bar{Y}|\leq
Ce^{-\alpha^{*}\frac{s}{2}}\quad\text{and}\quad|\tilde{Y}|\leq
Ce^{-\alpha^{*}\frac{s}{2}}.$ (3.25)

$(d)$ Term $\frac{\partial w_{b}}{\partial y_{i}}(Y,s)$. From Proposition 2.5
and the parabolic regularity, we have that

$\sup_{s\geq
s^{\prime}}\left\|w_{b}(y,s)-w_{\mathcal{B}(b)}(\bar{y},s)\right\|_{W_{loc}^{2,\infty}(|\bar{y}|<2)}\leq
Ce^{-\frac{s}{2}}s^{C_{0}}.$ (3.26)

This implies

$\displaystyle\left|\frac{\partial w_{b}}{\partial y_{i}}(Y,s)-\frac{\partial
w_{\mathcal{B}(b)}}{\partial
y_{i}}(\bar{y},s)\right|+\sum_{m=\ell+1}^{N}\left|\frac{\partial
w_{b}}{\partial y_{m}}(Y,s)\right|$
$\displaystyle+\sup_{|z|<2,(m,n)\neq(i,i),i\geq\ell+1}\left|\frac{\partial^{2}w_{b}}{\partial
y_{m}\partial y_{n}}(z,s)\right|\leq Ce^{-\frac{s}{2}}s^{C_{0}}.$ (3.27)

Similarly, from (2.2) and (2.20),

$\sup_{s\geq-\log
T}\left\|w_{a}(y,s)-\left\\{\kappa+\frac{\kappa}{2ps}\left(\ell-\frac{|\bar{y}|^{2}}{2}\right)\right\\}\right\|_{W_{loc}^{2,\infty}(|\bar{y}|<2)}\leq
C\frac{\log s}{s^{2}}.$ (3.28)

From (3.26) and (3.28), we deduce that

$\sup_{s\geq
s^{\prime\prime}}\left\|w_{\mathcal{B}(a)}(y,s)-\left\\{\kappa+\frac{\kappa}{2ps}\left(\ell-\frac{|\bar{y}|^{2}}{2}\right)\right\\}\right\|_{W_{loc}^{2,\infty}(|\bar{y}|<2)}\leq
C\frac{\log s}{s^{2}}.$ (3.29)

Using (3.29), we have for $|z|\leq 2$,

$\left|\frac{\partial^{2}w_{\mathcal{B}(b)}}{\partial
y_{i}^{2}}(z,s)+\frac{\kappa}{2ps}\right|\leq C\frac{\log
s}{s^{2}}\quad\text{and}\quad\left|\frac{\partial^{2}w_{\mathcal{B}(b)}}{\partial
y_{i}\partial y_{m}}(z,s)\right|\leq C\frac{\log s}{s^{2}},\;\;m\neq i.$

Note that $\frac{\partial w_{\mathcal{B}(b)}}{\partial y_{i}}(0,s)=0$, we then
take the Taylor expansion of $\frac{\partial w_{\mathcal{B}(b)}}{\partial
y_{i}}(\bar{y},s)$ near $\bar{y}=0$ up to the first order to get

$\left|\frac{\partial w_{\mathcal{B}(b)}}{\partial
y_{i}}(\bar{y},s)+Y_{i}\frac{\kappa}{2ps}\right|\leq C|\bar{y}|\frac{\log
s}{s^{2}}.$

Using (3.27) and (3.25) yields

$\left|\frac{\partial w_{b}}{\partial
y_{i}}(Y,s)+y_{i}\frac{\kappa}{2ps}\right|\leq
Ce^{-\frac{s}{2}}s^{C_{0}}+C|\bar{y}|\frac{\log
s}{s^{2}}+\frac{C}{s}e^{-\alpha^{*}\frac{s}{2}}.$ (3.30)

$(e)$ Term $\frac{\partial w_{b}}{\partial y_{j}}(Y,s)$. We just use (3.27)
and (3.25) to get

$\left|\frac{\partial w_{b}}{\partial y_{j}}(Y,s)-\frac{\partial
w_{b}}{\partial y_{j}}(\bar{y},0,\cdots,0,s)\right|\leq
Ce^{-(1+\alpha^{*})\frac{s}{2}}.$ (3.31)

Estimate (2.24) then follows by substituting (3.31), (3.30), (3.27), (3.23)
and (3.24) into (2.22). This concludes the proof of Proposition 2.7. $\square$

###  3.4. Further refined asymptotic behavior.

We prove Proposition 2.10 in this subsection. We first refine estimate (2.10)
and find following terms in the expansion which is of order
$e^{-\frac{s}{2}}$. Using the geometric constraint, we show that all terms of
order $e^{-\frac{s}{2}}$ must be identically zero, which gives a better
estimate for
$\|W_{a}(Q_{a}y,s)-w_{\mathcal{B}(a)}(\bar{y},s)\|_{L^{2}_{\rho}}$. We then
repeat the process and use again Proposition 2.7 in order to get the term of
order $\frac{1}{s}e^{-\frac{s}{2}}$ and conclude the proof of Proposition
2.10.

Let us define

$g_{a}(y,s)=W_{a}(Q_{a}y,s)-w_{\mathcal{B}(a)}(\bar{y},s),$ (3.32)

and denote by

$I_{a}(s)^{2}=\|g_{a}(s)\|_{L^{2}_{\rho}}^{2},\quad
l_{a,n}(s)=\|P_{n}g_{a}(s)\|_{L^{2}_{\rho}},\quad r_{a,k}(s)=\|\sum_{n\geq
k}P_{n}g_{a}(s)\|_{L^{2}_{\rho}}.$

From (2.10), we have

$I_{a}(s)=\mathcal{O}\left(e^{-\frac{s}{2}}s^{\mu}\right)\;\;\text{for
some}\;\;\mu>0.$ (3.33)

Note that Lemma 3.2 also holds with $W_{1}=W_{a}$ and $W_{2}=w_{\mathcal{B}}$.
We claim the following:

######  Lemma 3.4.

Assume that $I_{a}(s)=\mathcal{O}\left(e^{-\frac{s}{2}}s^{\mu_{0}}\right)$ for
some $\mu_{0}\in\mathbb{R}$. There exists $s_{4}>0$ such that for all $s\geq
s_{4}$,

$\sum_{n=0}^{2}l_{a,n}(s)+r_{a,4}(s)\leq Ce^{-\frac{s}{2}}s^{\mu_{0}-1},$
(3.34)

and

$\forall\beta\in\mathbb{N}^{N},\;|\beta|=3,\quad\left|\frac{d}{ds}\left(g_{a,\beta}(s)e^{\frac{s}{2}}s^{|\bar{\beta}|}\right)\right|\leq
Cs^{|\bar{\beta}|+\mu_{0}-\frac{3}{2}},$ (3.35)

where $\bar{\beta}=(\beta_{1},\cdots,\beta_{\ell})$,
$|\bar{\beta}|=\sum\limits_{i=1}^{\ell}\beta_{i}$.

###### Proof.

From $(i)$ and $(iv)$ of Lemma 3.2, we write for all $s\geq s_{3}$,

$n=0,1,2,\;\;\left|\frac{d}{ds}\left(l_{a,n}(s)e^{(n/2-1)s}\right)\right|\leq
Ce^{(n/2-\frac{3}{2})s}s^{\mu_{0}-1},$

and

$\left|\frac{d}{ds}\left(r_{a,4}(s)e^{s}\right)\right|\leq
Ce^{\frac{s}{2}}s^{\mu_{0}-1}.$

The estimate (3.34) then follows after integration of the above inequalities .
As for (3.35), we just use part $(iii)$ of Lemma 3.2 and (3.34) (note that
$l_{a,5}\leq r_{a,4}$ by definition (3.6)). This ends the proof of Lemma 3.4.
∎

Using (3.33) and applying Lemma 3.4 a finite number of steps, we obtain the
following:

######  Lemma 3.5.

There exist $s_{5}>0$ and continuous functions $a\to\lambda_{\beta}(a)$ for
all $\beta\in\mathbb{N}^{N}$ with $|\beta|=3$ and
$|\bar{\beta}|=\sum\limits_{i=1}^{\ell}\beta_{i}=0$ such that for all $a\in
S_{\delta}$ and $s\geq s_{5}$ ,

$\left\|g_{a}(y,s)-e^{-\frac{s}{2}}\sum_{|\beta|=3,|\bar{\beta}|=0}\lambda_{\beta}(a)h_{\beta}(y)\right\|_{L^{2}_{\rho}}\leq
Ce^{-\frac{s}{2}}s^{d-\frac{1}{2}},$

for some $d\in\left(0,\frac{1}{2}\right)$, where $h_{\beta}$ is defined by
(3.4).

###### Proof.

We first show that there is $s_{5}>0$ such that

$\forall s\geq s_{5},\quad I_{a}(s)\leq Ce^{-\frac{s}{2}}s^{d}\quad\text{for
some }\;\;d\in(0,\frac{1}{2}).$ (3.36)

From (3.33), if $\mu\in(0,\frac{1}{2})$, we are done. If $\mu\geq\frac{1}{2}$,
we apply Lemma 3.4 with $\mu_{0}=\mu$ to get

$\sum_{n=0}^{2}l_{a,n}(s)+r_{a,4}(s)\leq Ce^{-\frac{s}{2}}s^{\mu-1},$

and

$\forall|\beta|=3,\quad|g_{a,\beta}(s)|\leq
Ce^{-\frac{s}{2}}s^{\mu-\frac{1}{2}}.$

Hence,

$I_{a}(s)\leq Ce^{-\frac{s}{2}}s^{\mu-\frac{1}{2}}.$

Estimate (3.36) then follows by repeating this process a finite number of
steps.

Now using (3.36) and Lemma 3.4 with $\mu_{0}=d$,
\- if $|\beta|=3$ and $|\bar{\beta}|\geq 1$, we integrate (3.35) on
$[s,+\infty)$ to derive

$\forall|\beta|=3,\,|\bar{\beta}|\geq 1,\quad|g_{a,\beta}(s)|\leq
Ce^{-\frac{s}{2}}s^{d-\frac{1}{2}},$

\- if $|\beta|=3$ and $|\bar{\beta}|=0$, by integrating (3.35) on $[s_{5},s]$,
we deduce that there exists continuous functions $a\to\lambda_{\beta}(a)$ such
that

$\forall|\beta|=3,\,|\bar{\beta}|=0,\quad|g_{a,\beta}(s)-\lambda_{\beta}(a)e^{-\frac{s}{2}}|\leq
Ce^{-\frac{s}{2}}s^{d-\frac{1}{2}}.$

This concludes the proof of Lemma 3.5. ∎

Now we shall use the geometric constraint on the asymptotic behavior of the
solution given in Proposition 2.7 to show that all the coefficients
$\lambda_{\beta}(a)$ with $|\beta|=3$ and $\bar{\beta}=0$ in Lemma 3.5 have to
be identically zero. In particular, we claim the following:

######  Lemma 3.6.

There exists $s_{6}>0$ such that for all $s\geq s_{6}$,

$\forall a\in S_{\delta},\quad\|g_{a}(s)\|_{L^{2}_{\rho}}\leq
Ce^{-\frac{s}{2}}s^{d-\frac{1}{2}}\quad\text{for some
}\;\;d\in(0,\frac{1}{2}).$

###### Proof.

Consider $a\in S_{\delta}$, we aim at proving that

$\forall\beta\in\mathbb{N}^{N},\,|\beta|=3,\,|\bar{\beta}|=0,\quad\lambda_{\beta}(a)=0,$

where $\lambda_{\beta}(a)$ is introduced in Lemma 3.5 and
$|\bar{\beta}|=\sum_{i=1}^{\ell}\beta_{i}$.

From (2.20), (3.32) and the fact that the estimate given in Lemma 3.5 also
holds in $W^{2,\infty}(|y|<2)$ by parabolic regularity, we write for all
$k\geq\ell+1$ and $s\geq s_{5}+1$,

$\sup_{a\in S_{\delta},|y|<2}\left|\frac{\partial w_{a}}{\partial
y_{k}}(y,s)-e^{-\frac{s}{2}}\sum_{|\beta|=3,\bar{\beta}=0}\lambda_{\beta}(a)\frac{\partial
h_{\beta}}{\partial y_{k}}(y)\right|\leq Ce^{-\frac{s}{2}}s^{d-\frac{1}{2}}.$
(3.37)

Take $y=(\bar{y},\tilde{y})$, where
$\bar{y}=(y_{1},\cdots,y_{\ell})=(0,\cdots,0)$ and $\tilde{y}\in
B_{N-\ell}(0,1)$, then use Proposition 2.9 and (2.24), we obtain

$\left|\frac{\partial w_{a}}{\partial y_{k}}(y,s)-\frac{\partial
w_{b}}{\partial y_{k}}(0,s)\right|\leq
Ce^{-(1+\alpha^{*})\frac{s}{2}}s^{C_{0}}+Ce^{-s}s^{C_{0}+1},$ (3.38)

for some $\alpha^{*}\in(0,\frac{1}{2})$.

From (3.37) and (3.38), we get

$\left|\sum_{|\beta|=3,|\bar{\beta}|=0}\lambda_{\beta}(a)\frac{\partial
h_{\beta}}{\partial
y_{k}}(y)-\sum_{|\beta|=3,|\bar{\beta}|=0}\lambda_{\beta}(b)\frac{\partial
h_{\beta}}{\partial y_{k}}(0)\right|\leq Cs^{d-\frac{1}{2}}.$ (3.39)

From (2.23) and Proposition 2.9, we see that $b\to a$ as $s\to+\infty$. Since
$a\to\lambda_{\beta}(a)$ is continuous, $d\in(0,\frac{1}{2})$,
$h_{\beta_{1}}(0)=\cdots=h_{\beta_{\ell}}(0)=h_{0}(0)=1$ from definition
(3.4), and

$\frac{\partial h_{\beta}}{\partial
y_{k}}(y)=\beta_{k}h_{\beta_{k}-1}(y_{k})\prod_{j=1,j\neq
k}^{N}h_{\beta_{j}}(y_{j}),$

we derive by passing to the limit in (3.39),

$\displaystyle\sum_{|\beta|=3,|\bar{\beta}|=0}\lambda_{\beta}(a)\beta_{k}h_{\beta_{k}-1}(y_{k})\prod_{j=\ell+1,j\neq
k}^{N}h_{\beta_{j}}(y_{j})$
$\displaystyle\qquad=\sum_{|\beta|=3,|\bar{\beta}|=0}\lambda_{\beta}(a)\beta_{k}h_{\beta_{k}-1}(0)\prod_{j=\ell+1,j\neq
k}^{N}h_{\beta_{j}}(0).$

By the orthogonality of the polynomials $h_{i}$, this yields

$\beta_{k}\lambda_{\beta}(a)=0,\quad\forall
k\geq\ell+1,\;\forall|\beta|=3\;\text{with}\;|\bar{\beta}|=0.$

Take $\beta$ arbitrary with $|\beta|=3$ and $|\bar{\beta}|=0$, then there
exists $k\geq\ell+1$ such that $\beta_{k}\geq 1$, which implies that
$\lambda_{\beta}(a)=0$. This ends the proof of Lemma 3.6. ∎

Let us now give the proof of Proposition 2.10 from Lemmas 3.6 and 3.4.

###### Proof of Proposition 2.10.

From Lemmas 3.6 and 3.4, we see that for all $s\geq
s_{7}=\max\\{s_{4},s_{5},s_{6}\\}$,

$\sum_{n=0}^{2}l_{a,n}(s)+r_{a,4}(s)\leq Cs^{-\frac{s}{2}}s^{d-\frac{3}{2}},$

and

$\forall|\beta|=3,\quad\left|\frac{d}{ds}\left(g_{a,\beta}(s)s^{\frac{s}{2}}s^{|\bar{\beta}|}\right)\right|\leq
Ce^{|\bar{\beta}|+d-2},$

for some $d\in(0,\frac{1}{2})$. Integrating this equation between $s$ and
$+\infty$ if $|\bar{\beta}|=0$ and between $s_{7}$ and $s$ if
$|\bar{\beta}|\geq 1$, we get

$\forall|\beta|=3,\quad|g_{a,\beta}(s)|\leq Ce^{-\frac{s}{2}}s^{d-1}.$

Hence,

$\forall s\geq s_{7},\quad I_{a}(s)=\|g_{a}(s)\|_{L^{2}_{\rho}}\leq
Ce^{-\frac{s}{2}}s^{d-1}.$

With this new estimate, we use again Lemma 3.4 with $\mu_{0}=d-1$ to show that
there exists $s_{8}>0$ such that for all $s\geq s_{8}$,

$\sum_{n=0}^{2}l_{a,n}(s)+r_{a,4}(s)\leq Ce^{-\frac{s}{2}}s^{d-2},$

and

$\forall|\beta|=3,\quad\left|\frac{d}{ds}\left(g_{a,\beta}(s)e^{\frac{s}{2}}s^{|\bar{\beta}|}\right)\right|\leq
Cs^{|\bar{\beta}|+d-\frac{5}{2}}.$

This new equation implies that for all $|\beta|=3$ and $s\geq s_{8}$,
\- if $|\bar{\beta}|=0$ or $|\bar{\beta}|\geq 2$, we have
$|g_{a,\beta}(s)|\leq Ce^{-\frac{s}{2}}s^{d-\frac{3}{2}}$,
\- if $|\bar{\beta}|=1$, we obtain the existence of continuous functions
$a\to\lambda_{\beta}(a)$ such that

$\left|g_{a,\beta}(s)-\frac{e^{-\frac{s}{2}}}{s}\lambda_{\beta}(a)\right|\leq
Ce^{-\frac{s}{2}}s^{d-\frac{3}{2}}.$ (3.40)

This concludes the proof of Proposition 2.10. ∎

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