# Profile for a simultaneously blowing up solution to a complex valued
semilinear heat equation

Nejla Nouaili,
<EMAIL_ADDRESS>
CEREMADE, Université Paris Dauphine, Paris Sciences et Lettres.
Hatem Zaag111This author is supported by the ERC Advanced Grant no. 291214,
BLOWDISOL and by ANR Project ANAÉ ref. ANR-13-BS01-0010-03.,
<EMAIL_ADDRESS>
Université Paris 13, Sorbonne Paris Cité,
LAGA, CNRS (UMR 7539), F-93430, Villetaneuse, France

###### Abstract

We construct a solution to a complex nonlinear heat equation which blows up in
finite time $T$ only at one blow-up point. We also give a sharp description of
its blow-up profile. The proof relies on the reduction of the problem to a
finite dimensional one and the use of index theory to conclude. We note that
the real and imaginary parts of the constructed solution blow up
simultaneously, with the imaginary part dominated by the real.

Mathematical Subject classification: 35K57, 35K40, 35B44.
Keywords: Simultaneous blow-up, Complex heat equation.

##  1 Introduction

This paper is concerned with blow-up solutions of the complex heat equation

$\partial_{t}u=\Delta u+u^{2},$ (1)

where $u(t):x\in\mathbb{R}^{N}\to\mathbb{C}$ and $\Delta$ denotes the
Laplacian.
If we write $u(x,t)=v(x,t)+i\tilde{v}(x,t)$, where $v$ and
$\tilde{v}\in\mathbb{R}$, we obtain the following reaction-diffusion system:

$\begin{array}[]{lll}\partial_{t}v&=&\Delta v+v^{2}-\tilde{v}^{2},\\\
\partial_{t}\tilde{v}&=&\Delta\tilde{v}+2v\tilde{v},\end{array}$ (2)

where $(x,t)\in\mathbb{R}^{N}\times(0,T)\mbox{, }v(0,x)=v_{0}(x)$ and
$\tilde{v}(0,x)=\tilde{v}_{0}(x)$.
Equation (1) has a strong relation with the viscous Constantin-Lax-Majda
equation, which is a one dimensional model for the vorticity equation. For
more details see Okamoto, Sakajo and Wunsch [OSW08], Sakajo [Sak03a] and
[Sak03b] and Guo, Ninomiya, Shimojo and Yanagida in [GNSY13].

The Cauchy problem for system (2) can be solved in
$(L^{\infty}(\mathbb{R}^{N}))^{2}$, locally in time. We say that
$u(t)=v(t)+i\tilde{v}(t)$ blows up in finite time $T<\infty$, if $u(t)$ exists
for all $t\in[0,T)$ and $\lim_{t\to
T}\|v(t)\|_{L^{\infty}}+\|\tilde{v}(t)\|_{L^{\infty}}=+\infty.$ In that case,
$T$ is called the blow-up time of the solution. A point
$x_{0}\in\mathbb{R}^{N}$ is said to be a blow-up point if there is a sequence
$\\{(x_{j},t_{j})\\}$, such that $x_{j}\to x_{0}$, $t_{j}\to T$ and
$|v(x_{j},t_{j})|+|\tilde{v}(x_{j},t_{j})|\to\infty$ as $j\to\infty$. The set
of all blow-up points is called the blow-up set.

When $u$ is real (i.e., $\tilde{v}\equiv 0$), then system (2) is reduced to
the scalar equation

$\partial_{t}u=\Delta u+u^{p}\mbox{, where }p=2.$ (3)

The blow-up question for equation (3), with $p>1$, has been studied
intensively by many authors and no list can be exhaustive.
When it comes to deriving the blow-up profile, the situation is completely
understood in one space dimension (however, less is understood in higher
dimensions, see Velázquez [Vel92, Vel93a, Vel93b] and Zaag [Zaa02a, Zaa02b,
Zaa02c] for partial results). In one space dimension, given $a$ a blow-up
point, this is the situation:

  * •

either

$\sup_{|x-a|\leq
K\sqrt{(T-t)\log(T-t)}}\left|(T-t)u(x,t)-f\left(\frac{x-a}{\sqrt{(T-t)\log(T-t)}}\right)\right|\to
0,$ (4)

  * •

or for some $m\in\mathbb{N}$, $m\geq 2$, and $C_{m}>0$

$\sup_{|x-a|<K(T-t)^{1/2m}}\left|(T-t)u(x,t)-f_{m}\left(\frac{C_{m}(x-a)}{(T-t)^{1/2m}}\right)\right|\to
0,$ (5)

as $t\to T$, for any $K>0$, where

$f(z)=\frac{8}{8+|z|^{2}}\mbox{ and }f_{m}(z)=\frac{1}{1+|z|^{2m}}\mbox{, for
all }z\in\mathbb{R}^{N}.$ (6)

From Bricmont and Kupiainen [BK94] and Herrero and Velázquez [HV93], we have
examples of initial data leading to each of the aboved-mentioned scenarios.
Note that (4) corresponds to the fundamental mode of the harmonic oscillator
in the leading order, whereas (5) corresponds to higher modes. Moreover,
Herrero and Velázquez proved the genericity of the behavior (4) in one space
dimension in [HV92] and [HV94], and only announced the result in the higher
dimensional case (the result has never been published). Note also that the
stability of such a profile with respect to initial data has been proved by
Fermanian Kammerer, Merle and Zaag in [FKZ00] and [FKMZ00]. For more results
on equation (3), see [Bal77], [GK85], [GK87], [GK89], [HV93], [HV94], [MM04],
[MM09], [MZ98], [MZ00], [Miz07] and [QS07].

As we inferred above, equation (1) appears as a complex generalization of the
real valued equation (3). Note that there is another complex generalization of
(3). Indeed, Filippas and Merle consider in [FM95] the following equation:

$\partial_{t}u=\Delta u+|u|^{p-1}u\mbox{ with }u\in\mathbb{C}\mbox{ and }p>1,$
(7)

and generalize to this equation the results first proved in the real case by
Giga and Kohn [GK85, GK87, GK89]. Our equation (1) appears then as a “twin“ of
equation (7). However there is a fundamental difference between the two.
Indeed, equation (7) has a variational structure, which allows to use various
energy techniques, unlike equation (1), where such techniques certainly fail.
Therefore, considering equation (1) appears as a highly challenging task.

Considering equation (1) with non-real solutions, we have the following blow-
up results from [GNSY13]:

(A) A non-simultaneous blow-up criterion, see Theorem 1.5 in [GNSY13]:
Assume that

$v_{0},\;\tilde{v}_{0}\in C^{1}(\mathbb{R}^{N}),\;0\leq v_{0}\leq
M,\;v_{0}\not\equiv M,\;0<\tilde{v}_{0}\leq L,$ (8)

$\lim_{|x|\to\infty}v_{0}(x)=M,\;\;\lim_{|x|\to\infty}\tilde{v}_{0}(x)=0,$ (9)

for some constants $L>0$ and $M>0$. Then, the solution of (2) blows up at time
$t=T(M)$ with $\tilde{v}\not\equiv 0$. Moreover, the component $v$ blows up
only at space infinity and $\tilde{v}$ remains bounded.

(B) A Fourier-based blow-up criterion, see Theorem 1.2 in [GNSY13]:
If the Fourier transform of initial data of (1) is real and positive, then the
solution blows up.

(C) A simultaneous blow-up criterion, see Theorem 1.3 in [GNSY13]:
If $v_{0}$ is even, $\tilde{v}_{0}$ is odd with $\tilde{v}_{0}(x)>0$ for
$x>0$, then the fact that the blow-up set is compact implies that $v$ and
$\tilde{v}$ blow up simultaneously.

Following the description in (4) and (5) together with the work of [GNSY13],
we see that the blow-up profile derivation remains open, in the non-real case.
Indeed, either we have the description (4) or (5), with a zero imaginary part,
or we have blow-up solutions from [GNSY13], with a non-trivial imaginary part,
and no profile description.

In this paper, we give the first example of a complex-valued blow-up solution
of equation (1), with a non-trivial imaginary part, and a full description of
its blow-up profile, obeying behavior (4) (note that our method extends with
no difficulty to the construction of an analogous solution obeying the
behavior (5); however, the proof should be even more technical). Let us note
that the blow-up behavior we give here is not predicted by [GNSY13] (see
details in the remarks following our result). More precisely, this is our
result:

###### Theorem 1

(Existence of a blow-up solution for equation (1) with the description of its
profile) There exists $T>0$ such that equation (1) has a solution
$u(x,t)=v(x,t)+i\tilde{v}(x,t)$ in $\mathbb{R}^{N}\times[0,T)$ such that:
(i) the solution $u$ blows up in finite time $T$ only at the origin;
(ii) there holds that

$\left\|(T-t)u(.,t)-f\left(\frac{.}{\sqrt{(T-t)|\log(T-t)|}}\right)\right\|_{L^{\infty}}\leq\frac{C}{\sqrt{|\log(T-t)|}},$
(10)

where $f$ is defined by (6).
(iii) For all $R>0$,

$\sup_{|x|\leq
R\sqrt{T-t}}\left|(T-t)\tilde{v}(x,t)-\frac{\sum_{i=1}^{N}C_{i}}{|\log(T-t)|^{2}}\left(\frac{x_{i}^{2}}{T-t}-2\right)\right|\leq\frac{C}{|\log(T-t)|^{\alpha}},$
(11)

where $(C_{1},C_{2},..,C_{N})\neq(0,0,..,0)$ and $2<\alpha\leq 2+\eta$ for
some small $\eta>0$.
(iv) For all $x\neq 0$, $u(x,t)\to u^{*}(x)$ uniformly on compacts sets of
$\mathbb{R}^{N}\backslash\\{0\\}$, and

$u^{*}(x)\sim\frac{16|\log|x||}{|x|^{2}}\mbox{ as }x\to 0.$ (12)

Remarks:

1) Note that the real and imaginary parts of $u$ blow up simultaneously at
$x=0$. However the real part dominates the imaginary part in the sense that

$v(0,t)\sim\frac{1}{T-t}>>\frac{2\left|\sum_{i=1}^{N}C_{i}\right|}{(T-t)|\log(T-t)|^{2}}\sim|\tilde{v}(0,t)|\mbox{
as }t\to\infty.$

2) As announced right before the statement of our theorem, the solution we
construct is new and doesn’t obey the criteria given in [GNSY13]: this is
clear from (24) below. The proof relies on the reduction of the problem to a
$2(N+1)-$dimensional problem (a $4-$dimensional one if $N=1$; see Lemma 3.4
below). In the real case treated by Merle and Zaag in [MZ97], the problem was
of dimension $N+1$. Since that number is equal to the dimension of the blow-up
parameters ($1$ for the blow-up time and $N$ for the blow-up point), the
authors of [MZ97] were able to show the stability of the behavior (10) with
respect to initial data, of course in the real case. Here, in the complex
case, since the dimension of our problem $2(N+1)$ exceeds that of the blow-up
parameters $(N+1)$, we suspect our solution to be unstable with respect to
perturbations in initial data.

Our proof uses some ideas developed by Bricmont and Kupiainen [BK94] and Merle
and Zaag [MZ97] to construct a blow-up solution for the semilinear heat
equation (3) obeying the behavior (4). In [EZ11], Ebde and Zaag use the same
ideas to show the persistence of the profile (4) under perturbations of
equation (1) in the real case by lower order terms involving $u$ and $\nabla
u$. In [MZ08], Masmoudi and Zaag adapted that method to the case of the
following complex Ginzburg-Landau equation, where no gradient structure
exists:

$\partial_{t}u=(1+i\beta)\Delta u+(1+i\delta)|u|^{p-1}u\mbox{, with $\beta$
and $\delta$ are reals},$

(note that the case $\beta=0$ and $\delta$ small was first studied by Zaag in
[Zaa98]).
More precisely, the proof relies on the understanding of the dynamics of the
selfsimilar version of (2) (see system (17) below) around the profile (4).
Moreover, we proceed in two steps:

  * •

First, we reduce the question to a finite-dimensional problem: we show that it
is enough to control a $(N+1)$-dimensional variable in order to control the
solution (which is infinite dimensional) near the profile.

  * •

Second, we proceed by contradiction to solve the finite-dimensional problem
and conclude using index theory.

Surprisingly enough, we would like to mention that this kind of methods has
proved to be successful in various situations including hyperbolic and
parabolic PDE, in particular with energy-critical exponents. This was the case
for the construction of multi-solitons for the semilinear wave equation in one
space dimension by Côte and Zaag [CZ13], the wave maps by Raphaël and
Rodnianski [RR12], the Schrödinger maps by Merle, Raphaël and Rodnianski
[MRR11], the critical harmonic heat flow by Schweyer [Sch12] and the two-
dimensional Keller-Segel equation by Raphaël and Schweyer [RS13].

We proceed in 3 sections to prove Theorem 1. We first give in Section 2 an
equivalent formulation of the problem in the scale of the well-known
similarity variables. Section 3 is devoted to the proof of the similary
variables formulation (this is a central part in our argument). Finally, we
conclude the proof of Theorem 1 in Section 4.
Acknowledgement. We would like to thank the referee for his valuable remarks
which undoubtedly improved the presentation of our paper.

##  2 Formulation of the problem

For simplicity, we give the proof in one dimension. The adaptation to higher
dimensions is straightforward. We would like to find initial data
$u_{0}=v_{0}+i\tilde{v}_{0}$ such that the solution $u=v+i\tilde{v}$ of
equation (2) blows up in time $T$ with

$\displaystyle\lim_{t\to
T}\left\|(T-t)u(x,t)-f\left(\frac{x}{\sqrt{(T-t)|\log(T-t)|}}\right)\right\|_{L^{\infty}}=0,$
(13)

where $f$ is defined in (6).

This is the main estimate and the other results of Theorem 1 will appear as
by-products of the proof (see Section 4 for the proof of all the estimates of
Theorem 1).
Introducing the following self-similar transformation of problem (2):

$\begin{array}[]{c}w(y,s)=(T-t)v(x,t)\mbox{,
}\tilde{w}(y,s)=(T-t)\tilde{v}(x,t),\\\ y=\frac{x-a}{\sqrt{T-t}}\mbox{,
}s=-\log(T-t),\end{array}$ (14)

we see that (13) is equivalent to finding $s_{0}>0$ and initial data at
$s_{0}$, $W_{0}(y,s_{0})=w_{0}(y,s_{0})+i\tilde{w}_{0}(y,s_{0})$, such that
$W(y,s)=w(y,s)+i\tilde{w}(y,s)$ satisfies

$\displaystyle\lim_{s\to\infty}\left\|W(y,s)-f\left(\frac{y}{\sqrt{s}}\right)\right\|_{L^{\infty}}=0.$
(15)

Introducing

$w=\varphi+q\mbox{ and }\tilde{w}=\tilde{q}\mbox{ where
}\varphi(y,s)=f\left(\frac{y}{\sqrt{s}}\right)+\frac{1}{4s},$ (16)

the problem is then reduced to constructing a function $Q=q+i\tilde{q}$ such
that

$\lim_{s\to\infty}\|Q(y,s)\|_{L^{\infty}}=0,$

and $(q,\tilde{q})$ is a solution of the following equation for all
$(y,s)\in\mathbb{R}\times[s_{0},\infty)$,

$\begin{array}[]{lll}\displaystyle\partial_{s}q&=&\displaystyle({\cal{L}}+V)q+b(y,s)+R(y,s),\\\
\displaystyle\partial_{s}\tilde{q}&=&\displaystyle({\cal{L}}+V)\tilde{q}+\tilde{b}(y,s),\end{array}$
(17)

where

$\displaystyle{\cal{L}}=\partial_{y}^{2}-\frac{1}{2}y\partial_{y}+1\mbox{,
}V(y,s)=2\left(\varphi(y,s)-1\right),$ (18) $b(y,s)=q^{2}-\tilde{q}^{2}\mbox{,
}\tilde{b}(y,s)=2q\tilde{q},$ (19)

and

$\displaystyle
R(y,s)=\partial_{y}^{2}\varphi-\frac{1}{2}y\partial_{y}\varphi-\varphi+\varphi^{2}-\partial_{s}\varphi.$
(20)

The control of $q$ and $\tilde{q}$ near $0$ obeys two facts:

  * •

Localization: the fact that our profile $\varphi(y,s)$ dramatically changes
its value from $1+\frac{1}{4s}$ in the region near $0$ to $\frac{1}{4s}$ in
the region near infinity, according to a free boundary moving at the speed
$\sqrt{s}$. This will require different treatments in the regions
$|y|<2K_{0}\sqrt{s}$ and $|y|>2K_{0}\sqrt{s}$ for some $K_{0}$ to be chosen.

  * •

Spectral information: the fact that the operator ${\cal L}$ is selfadjoint,
$b$ and $\tilde{b}$ are quadratic in $(q,\tilde{q})$ and that

$\|R(s)\|_{L^{\infty}}+\|V(s)\|_{L^{2}_{\rho}}\to 0\mbox{ as }s\to\infty$

from (16) and (18), which shows that the dynamics of equation (17) near $0$
are driven by the spectral properties of ${\cal L}$. This will require a
decomposition of the solution according to the spectrum of ${\cal L}$. Note
that the operator ${\cal L}$ is self-adjoint in the Hilbert space

$L^{2}_{\rho}=\\{g\in L^{2}_{loc}(\mathbb{R},\mathbb{C})\mbox{,
}\|g\|_{L^{2}_{\rho}}^{2}\equiv\int_{\mathbb{R}}|g|^{2}e^{-\frac{|y|^{2}}{4}}dy<+\infty\\}\mbox{
where }\rho(y)=\displaystyle\frac{e^{-\frac{|y|^{2}}{4}}}{(4\pi)^{1/2}}.$

The spectrum of ${\cal L}$ is explicitly given by

$spec({\cal L})=\\{1-\frac{m}{2}\mbox{, }m\in\mathbb{N}\\}.$

All the eigenvalues are simple, the eigenfunctions are dilations of Hermite’s
polynomial and given by

$h_{m}(y)=\sum_{n=0}^{[\frac{m}{2}]}\frac{m!}{n!(m-2n)!}(-1)^{n}y^{m-2n}.$
(21)

Note that ${\cal L}$ has two positive (or expanding) directions ($\lambda=1$
and $\lambda=\frac{1}{2}$), and a zero direction ($\lambda=0$). Complying with
the localization and spectral information facts, we will decompose $q$ and
$\tilde{q}$ accordingly as stated above:

  * •

first, we consider a non-increasing cut-off function $\chi_{0}\in
C^{\infty}_{0}(\mathbb{R}^{+},[0,1])$ such that $supp(\chi_{0})\subset[0,2]$,
$\chi_{0}(\xi)=1$ for $\xi<1$ and $\chi_{0}(\xi)=0$ for $\xi>2$, then
introduce

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

where $K_{0}\geq 1$ will be chosen large enough so that various technical
estimates hold. Then, we write $q=q_{b}+q_{e}$ and
$\tilde{q}=\tilde{q}_{b}+\tilde{q}_{e}$, where the inner parts and the outer
parts are given by

$q_{b}=q\chi\mbox{, }\tilde{q}_{b}=\tilde{q}\chi\mbox{, }q_{e}=q(1-\chi)\mbox{
and }\tilde{q}_{e}=\tilde{q}(1-\chi).$

Let us remark that

$supp(q_{b}(s))\subset B(0,2K_{0}\sqrt{s})\mbox{ and
}supp(q_{e}(s))\subset\mathbb{R}\setminus B(0,K_{0}\sqrt{s}),$

and the same holds for $\tilde{q}_{b}$ and $\tilde{q}_{e}$.

  * •

Second, we study $q_{b}$ and $\tilde{q}_{b}$ using the structure of ${\cal
L}$, isolating the nonnegative directions. More precisely we decompose $q_{b}$
and $\tilde{q}_{b}$ as follows

$\begin{array}[]{lll}q_{b}(y,s)&=&\sum_{0}^{2}q_{m}(s)h_{m}(y)+q_{-}(y,s),\\\
\tilde{q}_{b}(y,s)&=&\sum_{0}^{2}\tilde{q}_{m}(s)h_{m}(y)+\tilde{q}_{-}(y,s),\\\
\end{array}$ (22)

where $q_{m}$ (respectively $\tilde{q}_{m}$) is the projection of $q_{b}$
(respectively $\tilde{q}_{b}$) on $h_{m}$, $q_{-}(y,s)=P_{-}(q_{b})$
(respectively $\tilde{q}_{-}(y,s)=P_{-}(\tilde{q}_{b})$) and $P_{-}$ is the
projection on $\\{h_{i},\;i\geq 3\\}$ the negative subspace of the ${\cal L}$.

In summary, we can decompose $q$ (respectively $\tilde{q}$) in 5 components as
follows:

$\begin{array}[]{lll}q(y,s)&=&\sum_{m=0}^{2}q_{m}(s)h_{m}(y)+q_{-}(y,s)+q_{e}(y,s),\\\
\tilde{q}(y,s)&=&\sum_{m=0}^{2}\tilde{q}_{m}(s)h_{m}(y)+\tilde{q}_{-}(y,s)+\tilde{q}_{e}(y,s).\end{array}$
(23)

Here and throughout the paper, we call $q_{-}(y,s)$ (respectively
$\tilde{q}_{-}$) the negative part of $q$ (respectively $\tilde{q}$), $q_{2}$
(respectively $\tilde{q}_{2}$), the null mode of $q$ (respectively
$\tilde{q}$).

##  3 The construction method in selfsimilar variables

This section is devoted to the proof of the existence of a solution
$(q,\tilde{q})$ of system (17) satisfying
$\|q(s)\|_{L^{\infty}}+\|\tilde{q}(s)\|_{L^{\infty}}\to 0$. This is a central
argument in our proof. In Section 4, we use this solution and give the proof
of Theorem 1. Though we refer to the earlier work by Merle and Zaag [MZ97] for
purely technical details, we insist on the fact that we can completely split
from that paper as long as ideas and arguments are considered. We hope that
the explanation of the strategy we give in this section will be more reader
friendly.

We proceed in 3 subsections:
\- In the first subsection, we give all the arguments of the proof without the
details, which are left for the following subsection (readers not interested
in technical details may stop here).
\- In the second subsection, we give various estimates concerning initial
data.
\- In the third subsection, we give the dynamics of system (17) near the zero
solution, in accordance with the decomposition (23).

###  3.1 The proof without technical details

Given $s_{0}>0$, we consider initial data for equation (17) of the following
form:

$\begin{array}[]{lll}\displaystyle
q_{d_{0},d_{1}}(y,s_{0})&=&\displaystyle\frac{A}{s_{0}^{2}}(d_{0}+d_{1}y)\chi(2y,s_{0}),\\\
\displaystyle\tilde{q}_{\tilde{d}_{0},\tilde{d}_{1},\tilde{d}_{2}}(y,s_{0})&=&\displaystyle\left[\frac{\tilde{A}}{s_{0}^{\alpha}}(\tilde{d}_{0}+\tilde{d}_{1}y)+\frac{\tilde{B}}{s_{0}^{2}}h_{2}(y)\right]\chi(2y,s_{0}),\end{array}$
(24)

for some constants $A$, $\tilde{A}$ and $\tilde{B}$ will be fixed later and
the parameters $(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in[-2,2]^{4}$. The
solution of equation (17) with initial data (24) will be denoted by
$(q,\tilde{q})(s_{0},A,\tilde{A},\tilde{B},(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1}),y,s)$,
or, when there is no ambiguity by
$(q,\tilde{q})(s_{0},d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1},y,s)$ or even
$(q,\tilde{q})(y,s)$. We will show that given $\tilde{B}\in\mathbb{R}$, if $A$
and $\tilde{A}$ are fixed large enough, then $s_{0}$ is fixed large enough
depending on $A$, $\tilde{A}$ and $\tilde{B}$, we can also fixe the parameters
$(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in[-2,2]^{4}$, so that the solution
$(q,\tilde{q})\left(s_{0},A\,\tilde{A},\tilde{B},d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1},y,s\right)$
will converge to $0$ as $s\to\infty$. Thanks to the decomposition given in
(23), in order to control $(q,\tilde{q})(s)$ near $(0,0)$, it is enough to
control it in some shrinking set defined as follows:

###### Definition 3.1

(Definition of a shrinking set for the components of $(q,\tilde{q})$) For all
$A\geq 1$, $\tilde{A}\geq 1$, $0<\eta<\frac{1}{2}$, $2<\alpha<2+\eta$ and
$s\geq e$, we define $V_{A}(s)$ (respectively $\tilde{V}_{\tilde{A}}(s)$) as
the set of all functions $r$ (respectively $\tilde{r}$) in $L^{\infty}$ such
that:

$|r_{m}(s)|\leq As^{-2}\;m=0,1,\;\;|r_{2}(s)|\leq A^{2}(\log s)s^{-2},$
$\forall y\in\mathbb{R},\;|r_{-}(y,s)|\leq
A(1+|y|^{3})s^{-2},\;\|r_{e}(s)\|_{L^{\infty}}\leq A^{2}s^{-\frac{1}{2}},$

(respectively

$|\tilde{r}_{m}(s)|\leq\tilde{A}s^{-\alpha}\;m=0,1,\;\;|\tilde{r}_{2}(s)|\leq\tilde{A}^{2}s^{-2+\eta},$
$\forall
y\in\mathbb{R},\;|\tilde{r}_{-}(y,s)|\leq\tilde{A}(1+|y|^{3})s^{-\alpha},\;\|\tilde{r}_{e}(s)\|_{L^{\infty}}\leq\tilde{A}^{2}s^{-\alpha+3/2}),$

where $r_{-}$, $r_{e}$ and $r_{m}$ (respectively $\tilde{r}_{-}$,
$\tilde{r}_{e}$ and $\tilde{r}_{m}$) are defined in Section 2.

As a matter of fact, if $s\geq e$ and $(r,\tilde{r})\in
V_{A}(s)\times\tilde{V}_{\tilde{A}}(s)$, then one easily derives that

$\|r(s)\|_{L^{\infty}}\leq C\frac{A^{2}}{\sqrt{s}}\mbox{ and
}\|\tilde{r}(s)\|_{L^{\infty}}\leq C\frac{\tilde{A}^{2}}{s^{\alpha-3/2}},$
(25)

(see Proposition 3.7 page 157 in [MZ97] for details). Thus, our aim become the
following:

###### Proposition 3.2

(Existence of a solution of (17) trapped in
$V_{A}(s)\times\tilde{V}_{\tilde{A}}(s)$) There exists ${A_{1}}$ such that for
all $A\geq{A_{1}}$ and $\tilde{A}\geq{A_{1}}$, $0<\eta<\frac{1}{10}$ and
$2<\alpha<2+\eta$, there exists ${s_{0,1}}(A,\tilde{A})$ such that for all
$s_{0}\geq{s_{0,1}}$ and $|\tilde{B}|\leq 1$, there exists
$(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})$, such that,
if $(q,\tilde{q})$ is a solution of (17) with initial data at $s_{0}$ given by
(24), then

$\forall s\geq s_{0},\;\;q(s)\in V_{A}(s)\mbox{ and
}\tilde{q}(s)\in\tilde{V}_{\tilde{A}}(s).$

The aim of this section is to prove this proposition.

In the following lemma, we find a set
$D_{A,\tilde{A},\tilde{B},\eta,\alpha,s_{0}}=D_{s_{0}}$ such that
$(q,\tilde{q})(s_{0})\in V_{A}(s_{0})\times\tilde{V}_{\tilde{A}}(s_{0})$,
whenever $(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in D_{s_{0}}$. More
precisely, we claim the following:

###### Lemma 3.3

(Choice of parameters $d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1}$ to have
initial data in $V_{A}(s)\times\tilde{V}_{\tilde{A}}(s)$ at $s=s_{0}$) For
each $|\tilde{B}|\leq 1$, $A\geq 1$, $\tilde{A}\geq 1$, $0<\eta<\frac{1}{10}$
and $2<\alpha<2+\eta$, there exists $s_{0,2}(A,\tilde{A},\tilde{B})\geq e$
such that for all $s_{0}\geq s_{0,2}$:
If initial data for equation (17) are given by (24): then, there exists a
cuboid

$D_{A,\tilde{A},\tilde{B},\eta,\alpha,s_{0}}=D_{s_{0}}\subset[-2,2]^{4},$ (26)

such that, for all $(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in D_{s_{0}}$,
we have

$(q,\tilde{q})(s_{0},A,\tilde{A},\tilde{B},d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in
V_{A}(s_{0})\times\tilde{V}_{\tilde{A}}(s_{0}).$

Proof: The proof is purely technical and follows as the analogous step in
[MZ97], for that reason we refer the reader to Lemma 3.5 page 156 and Lemma
3.9 page 160 in [MZ97]. $\blacksquare$

Let us consider $|\tilde{B}|\leq 1$, $A\geq 1$, $\tilde{A}\geq 1$,
$0<\eta<1/10$, $2<\alpha<2+\eta$,
$(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in D_{s_{0}}$ and $s_{0}\geq
s_{0,1}$ defined in Lemma 3.3. From the local Cauchy theory, we define a
maximal solution $(q,\tilde{q})$ to equation (17) with initial data (24), and
a maximal time
$s_{*}(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in[s_{0},+\infty]$ such that,
for all $s\in[s_{0},s_{*})$, $(q,\tilde{q})(s)\in
V_{A}(s)\times\tilde{V}_{\tilde{A}}(s)$ and:

  * •

either $s_{*}=\infty$,

  * •

or $s_{*}<\infty$ and from continuity,
$(q,\tilde{q})(s_{*})\in\partial\left(V_{A}(s_{*})\times\tilde{V}_{\tilde{A}}(s_{*})\right)$,
in the sense that when $s=s^{*}$, one ’$\leq$’ symbol in the definition of
$V_{A}(s_{*})$ and $\tilde{V}_{\tilde{A}}(s_{*})$ is replaced by the symbol
’$=$’.

Our aim is to show that for all $|\tilde{B}|\leq 1$, for $A$, $\tilde{A}$ and
$s_{0}$ large enough, one can find a parameter
$(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})$ in $D_{s_{0}}$ such that

$s_{*}(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})=\infty.$ (27)

We argue by contradiction, and assume that for all
$(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in D_{s_{0}}$,
$s_{*}(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})<+\infty$. As we have just
stated, one of the symbols ’$\leq$’ in the definition of $V_{A}(s)$ and
$\tilde{V}_{\tilde{A}}(s)$ should be replaced by ’$=$’ symbols when $s=s_{*}$.
In fact, this is possible only with the ’$\leq$’ symbols concerning the
components $q_{0}$, $q_{1}$, $\tilde{q}_{0}$ or $\tilde{q}_{1}$, as one sees
in the following:

###### Lemma 3.4

(Reduction to a finite dimensional problem) There exists $A_{3}>0$ such that
for each $A\geq A_{3}$ and $\tilde{A}\geq A_{3}$ there exists
$s_{0,3}(A,\tilde{A})\geq s_{0,2}(A,\tilde{A})$ such that if $s_{0}\geq
s_{0,3}$, then
$(q_{0}(s_{*}),q_{1}(s_{*}),\tilde{q}_{0}(s_{*}),\tilde{q}_{1}(s_{*}))\in\partial\left(\left[-\frac{A}{s_{*}^{2}},\frac{A}{s_{*}^{2}}\right]^{2}\times\left[-\frac{\tilde{A}}{s_{*}^{\alpha}},\frac{\tilde{A}}{s_{*}^{\alpha}}\right]^{2}\right)$.

Proof: This is a direct consequence of the dynamics of equation (17), as we
will show in Subsection 3.3 below.
Just to give a flavor of the argument, we invite the reader to look at
Proposition 3.8 below, where we project system (17) on the different
components of $q$ and $\tilde{q}$ introduced in (23). There, one can see that
the components $q_{2}$, $q_{-}$ and $q_{e}$ (respectively $\tilde{q}_{2}$,
$\tilde{q}_{-}$ and $\tilde{q}_{e}$) correspond to decreasing directions of
the flow and since they are “small“ at $s=s_{0}$ (see Lemma 3.7 below), they
remain small up for $s\in[s_{0},s_{*}]$, and can not touch their bounds. Thus,
only $q_{0}$, $q_{1}$, $\tilde{q}_{0}$ or $\tilde{q}_{1}$ may touch their
boundary at $s=s_{*}$.
For more details on the arguments, see Subsection 3.3 below. This ends the
proof of Lemma 3.4.$\blacksquare$

From Lemma 3.4, we may define the rescaled flow $\Phi$ at $s=s_{*}$ for the
four expanding directions, namely $q_{0}$, $q_{1}$, $\tilde{q}_{0}$ and
$\tilde{q}_{1}$, as follows:

$\begin{array}[]{lccl}\Phi:&D_{s_{0}}&\to&\partial([-1,1]^{4})\\\
&(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})&\to&\displaystyle\left(\frac{s_{*}^{2}q_{0}}{A},\frac{s_{*}^{2}q_{1}}{A},\frac{s_{*}^{\alpha}\tilde{q}_{0}}{\tilde{A}},\frac{s_{*}^{\alpha}\tilde{q}_{1}}{\tilde{A}}\right)_{d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1}}(s_{*}).\end{array}$
(28)

In particular,

$\mbox{either }\omega q_{m}(s_{*})=\frac{A}{s_{*}^{2}}\mbox{ or
}\tilde{\omega}q_{\tilde{m}}(s_{*})=\frac{\tilde{A}}{s_{*}^{\alpha}},$ (29)

for some $m$, $\tilde{m}\in\\{0,1\\}$ and $\omega$,
$\tilde{\omega}\in\\{-1,1\\}$, both depending on
$(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})$. In the following lemma, we show
that $q_{m}$ (or $\tilde{q}_{m}$) actually crosses its boundary at $s=s_{*}$,
resulting in the continuity of $s_{*}$ and $\Phi$. More precisely, we have the
following:

######  Lemma 3.5 (Transverse crossing)

There exists $A_{4}>0$ such that for all $A\geq A_{4}$ and $\tilde{A}\geq
A_{4}$, there exists $s_{0,4}(A,\tilde{A})\geq s_{0,3}(A,\tilde{A})$ such that
if $s_{0}\geq s_{0,4}$ and (29) holds for some $s_{*}\geq s_{0,4}$, then

$\mbox{either }\omega\frac{dq}{ds}(s_{*})>0\mbox{ or
}\tilde{\omega}\frac{d\tilde{q}_{\tilde{m}}}{ds}(s_{*})>0.$ (30)

Clearly, from the transverse crossing, we see that

$(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\mapsto
s_{*}(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\mbox{ is continuous,}$

hence by definition (28), $\Phi$ is continuous. In order to find a
contradiction and conclude, we crucially use the particular form we choose for
initial data in (24). More precisely, we have the following:

######  Lemma 3.6 (Degree $1$ on the boundary)

There exists $A_{5}>0$ such that for each $A\geq A_{5}$ and $\tilde{A}\geq
A_{5}$, there exists $s_{0,5}(A,\tilde{A})\geq s_{0,4}(A,\tilde{A})$ such that
if $s_{0}\geq s_{0,4}$, then the mapping
$(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\to(q_{0}(s_{0}),q_{1}(s_{0}),\tilde{q}_{0}(s_{0}),\tilde{q}_{1}(s_{0}))$
maps $\partial D_{s_{0}}$ into
$\partial\left([-\frac{A}{s_{0}^{2}},\frac{A}{s_{0}^{2}}]^{2}\times[-\frac{\tilde{A}}{s_{0}^{\alpha}},\frac{\tilde{A}}{s_{0}^{\alpha}}]^{2}\right)$,
and has degree one on the boundary.

Indeed, from this lemma and the transverse crossing property of Lemma 3.5, we
see that if $(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in\partial D_{s_{0}}$,
then $s_{*}(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})=s_{0}$,
$\Phi(s_{*}(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1}),d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})=\left(\frac{s^{2}_{*}q_{0}}{A},\frac{s^{2}_{*}q_{1}}{A},\frac{s^{\alpha}_{*}\tilde{q}_{0}}{A},\frac{s^{\alpha}_{*}\tilde{q}_{1}}{A}\right)(s_{0})$
and $\Phi$ defined in (28) is a continuous function from the cuboid
$D_{s_{0}}\subset\mathbb{R}^{4}$ to $\partial[-1,1]^{4}$, whose restriction to
$\partial D_{s_{0}}$ is of degree 1. This is a contradiction. Thus,
Proposition 3.2 is proved, and from identity (25), we have constructed a
solution $(q,\tilde{q})$ to system (17), such that

$\|q(s)\|_{L^{\infty}}+\|\tilde{q}(s)\|_{L^{\infty}}\to 0\mbox{ as
}s\to\infty.$

In the following subsections, we give the proofs of the technical steps of the
current subsection (namely Lemmas 3.3, 3.4, 3.5 and 3.6), referring to earlier
work when the proof is the same.

###  3.2 Preparation of initial data

In this subsection, we study initial data given by (24). More precisely, we
state a lemma which directly implies Lemmas 3.3 and 3.6. It also shows the
(relative) smallness of the components $q_{2}$, $q_{-}$, $q_{e}$,
$\tilde{q}_{2}$, $\tilde{q}_{-}$ and $\tilde{q}_{e}$, an information which
will be useful for the next subsection, dedicated to the dynamics of equation
(17), crucial for the proofs of the reduction to a finite dimensional problem
(Lemma 3.4) and the transverse crossing property (Lemma 3.5). More precisely,
we claim the following:

###### Lemma 3.7

(Decomposition of initial data in different components) For each
$|\tilde{B}|\leq 1$, $A\geq 1$, $\tilde{A}\geq 1$, $0<\eta<\frac{1}{10}$ and
$2<\alpha<2+\eta$, there exists $s_{0,6}(A,\tilde{A},\tilde{B})\geq e$ such
that for all $s_{0}\geq s_{0,6}$:
(i) there exists a cuboid

$D_{s_{0}}\subset[-2,2]^{4},$ (31)

such that the mapping
$(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\to(q_{0}(s_{0}),q_{1}(s_{0}),\tilde{q}_{0}(s_{0}),\tilde{q}_{1}(s_{0}))$
is linear and one to one from $D_{s_{0}}$ onto
$[-\frac{A}{s_{0}^{2-\eta}},\frac{A}{s_{0}^{2-\eta}}]^{2}\times[-\frac{\tilde{A}}{s_{0}^{\alpha}},\frac{\tilde{A}}{s_{0}^{\alpha}}]^{2}$
and maps the boundary $\partial D_{s_{0}}$ into the boundary
$\partial\left([-\frac{A}{s_{0}^{2}},\frac{A}{s_{0}^{2}}]^{2}\times[-\frac{\tilde{A}}{s_{0}^{\alpha}},\frac{\tilde{A}}{s_{0}^{\alpha}}]^{2}\right)$.
Moreover, it is of degree one on the boundary.
(ii) For all $(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in D_{s_{0}}$, we have

$\begin{array}[]{l}|q_{2}(s_{0})|\leq CAe^{-\gamma s_{0}}\mbox{, for some
}\gamma>0,\;\;|q_{-}(y,s_{0})|\leq\frac{c}{s_{0}^{2}}(1+|y|^{3})\mbox{ and
}q_{e}(y,s_{0})=0,\\\ |d_{0}|+|d_{1}|\leq 1,\end{array}$ (32)

and

$\begin{array}[]{l}|\tilde{q}_{2}(s_{0})-\frac{\tilde{B}}{s_{0}^{2}}|\leq
C\tilde{A}e^{-\gamma s_{0}}\mbox{, for some
}\gamma>0,\;\;|\tilde{q}_{-}(y,s_{0})|\leq\frac{c}{s_{0}^{\alpha}}(1+|y|^{3})\mbox{
and }\tilde{q}_{e}(y,s_{0})=0,\\\ |\tilde{d}_{0}|+|\tilde{d}_{1}|\leq
1.\end{array}$ (33)

Proof: Since we have almost the same definition of the set $V_{A}$, and almost
the same expression of initial data as in [MZ97], we refer the reader to Lemma
3.5 page 156 and Lemma 3.9 page 160 from [MZ97]. $\blacksquare$

###  3.3 Reduction to a finite-dimensional problem

This subsection is dedicated to the proof of Lemmas 3.4 and 3.5. They both
follow from the understanding of the flow of equation (17) in the set
$V_{A}(s)\times\tilde{V}_{\tilde{A}}(s)$. Accordingly, this crucially relies
on the projection of equation (17) with respect to the decomposition given in
(23). More precisely, we claim the following:

######  Proposition 3.8 (Dynamics of equation (17))

There exists $A_{7}\geq 1$ such that for all $A\geq A_{7}$, $\tilde{A}\geq
A_{7}$, $0<\eta<\frac{1}{10}$, $2<\alpha<2+\eta$ and $\theta\geq 0$, there
exists $s_{0,7}(A,\tilde{A},\theta)$ such that the following holds for all
$s_{0}\geq s_{0,7}$:
Assume that for some $\tau\geq s_{0}$ and for all $s\in[\tau,\tau+\theta]$,

$(q(s),\tilde{q}(s))\in V_{A}(s)\times\tilde{V}_{A}(s).$

Then, the following holds for all $s\in[\tau,\tau+\theta]$:
(i)(Differential inequalities satisfied by the expanding and null modes) For
$m=0$ and $1$, we have:

$\left|q^{\prime}_{m}(s)-(1-\frac{m}{2})q_{m}(s)\right|\leq\frac{C}{s^{2}},$
$\left|\tilde{q}^{\prime}_{m}(s)-(1-\frac{m}{2})\tilde{q}_{m}(s)\right|\leq\frac{C\tilde{A}^{2}}{s^{3-\eta}},$
$\left|\tilde{q}^{{}^{\prime}}_{2}(s)+\frac{2}{s}\tilde{q}_{2}(s)\right|\leq\frac{C\tilde{A}}{s^{\alpha+1}}.$

(ii)(Control of the null and negative modes) Moreover, we have:

$\displaystyle|q_{2}(s)|$
$\displaystyle\leq\frac{\tau^{2}}{s^{2}}|q_{2}(\tau)|+\frac{CA(s-\tau)}{s^{3}},$
$\displaystyle|\tilde{q}_{2}(s)|$
$\displaystyle\leq\frac{\tau^{2}}{s^{2}}|\tilde{q}_{2}(\tau)|+\frac{C\tilde{A}(s-\tau)}{s^{\alpha+1}},$
$\displaystyle\left\|\frac{q_{-}(s)}{1+|y|^{3}}\right\|_{L^{\infty}}$
$\displaystyle\leq
Ce^{-\frac{(s-\tau)}{2}}\left\|\frac{q_{-}(\tau)}{1+|y|^{3}}\right\|_{L^{\infty}}+C\frac{e^{-(s-\tau)^{2}}\|q_{e}(\tau)\|_{L^{\infty}}}{s^{3/2}}+\frac{C(1+s-\tau)}{s^{2}},$
$\displaystyle\left\|\frac{\tilde{q}_{-}(s)}{1+|y|^{3}}\right\|_{L^{\infty}}$
$\displaystyle\leq
Ce^{-\frac{(s-\tau)}{2}}\left\|\frac{\tilde{q}_{-}(\tau)}{1+|y|^{3}}\right\|_{L^{\infty}}+C\frac{e^{-(s-\tau)^{2}}\|\tilde{q}_{e}(\tau)\|_{L^{\infty}}}{s^{3/2}}+\frac{C(1+s-\tau)}{s^{\alpha}},$
$\displaystyle\|q_{e}(s)\|_{L^{\infty}}$ $\displaystyle\leq
Ce^{-\frac{(s-\tau)}{2}}\|q_{e}(\tau)\|_{L^{\infty}}+C\frac{e^{s-\tau}}{s^{3/2}}\left\|\frac{q_{-}(\tau)}{1+|y|^{3}}\right\|_{L^{\infty}}+\frac{C(1+s-\tau)}{s^{1/2}},$
$\displaystyle\|\tilde{q}_{e}(s)\|_{L^{\infty}}$ $\displaystyle\leq
Ce^{-\frac{(s-\tau)}{2}}\|\tilde{q}_{e}(\tau)\|_{L^{\infty}}+C\frac{e^{s-\tau}}{s^{3/2}}\left\|\frac{\tilde{q}_{-}(\tau)}{1+|y|^{3}}\right\|_{L^{\infty}}+\frac{C(1+s-\tau)}{s^{\alpha-3/2}}.$

Let us first insist on the fact that the derivation of Lemmas 3.4 and 3.5
follows from Proposition 3.8, exactly as in the real case treated in [MZ97]
(see pages 163 to 166 and 158 to 159 in [MZ97] ). For that reason, we only
focus in the following on the proof of Proposition 3.8.

Proof of Proposition 3.8:

The proof of Proposition 3.8 consists in the projection of the two equations
of system (17) on the different components of $q$ and $\tilde{q}$ defined in
(23).
When $\tilde{q}\equiv 0$, the proof is already available from Lemma 3.13 pages
167 and Lemma 3.8 page 158 from [MZ97].
When $\tilde{q}\not\equiv 0$, since the equation satisfied by $\tilde{q}$ in
(17) shares the same linear part as the equation in $q$, the proof is similar
to the argument in [MZ97]. For that reason, we only give the ideas here, and
kindly ask the interested reader to look at Lemma 3.13 page 167 and Lemma 3.8
page 158 in [MZ97] for the technical details.

(i) Multiplying the two equations in (17) by $\chi(y,s)k_{m}(y)\rho(y)$, for
$m=0,1,2$ and integrating in $y\in\mathbb{R}$, we proceed as in pages 158-159
from [MZ97] and we get the differential inequalities given in (i) with no
difficulties.

(ii) For convenience, we separate the contribution of $q$ and $\tilde{q}$ in
the quadratic term $b(y,s)$ defined in (19) by writing $b(y,s)=B(y,s)-N(y,s)$
with

$B(y,s)=q^{2}\mbox{, and }N(y,s)=\tilde{q}^{2}.$ (34)

Let us first recall equations of $(q,\tilde{q})$ in their Duhamel formulation,

$\begin{array}[]{lll}q(s)&=&K(s,\tau)q(\tau)+\int_{\tau}^{s}d\sigma
K(s,\sigma)B(q(\sigma))+\int_{\tau}^{s}d\sigma
K(s,\sigma)R(\sigma)-\int_{\tau}^{s}d\sigma K(s,\sigma)N(\sigma),\\\
\tilde{q}(s)&=&K(s,\tau)\tilde{q}(\tau)+\int_{\tau}^{s}d\sigma
K(s,\sigma)\tilde{b}(\sigma),\end{array}$ (35)

where $K$ is the fundamental solution of the operator ${\cal L}+V$. We write
$q=\alpha+\beta+\gamma+\delta$ and $\tilde{q}=\tilde{\alpha}+\tilde{\beta}$,
where

$\begin{array}[]{l}\alpha(s)=K(s,\tau)q(\tau),\;\;\beta(s)=\int_{\tau}^{s}d\sigma
K(s,\sigma)B(q(\sigma)),\\\ \gamma(s)=\int_{\tau}^{s}d\sigma
K(s,\sigma)R(\sigma),\;\;\delta(s)=-\int_{\tau}^{s}d\sigma
K(s,\sigma)N(\sigma).\end{array}$ (36)
$\tilde{\alpha}(s)=K(s,\tau)\tilde{q}(\tau),\;\;\tilde{\beta}(s)=\int_{\tau}^{s}d\sigma
K(s,\sigma)\tilde{b}(\sigma).\\\ $ (37)

We assume that $(q(s),\tilde{q}(s))\in V_{A}(s)\times\tilde{V}_{A}(s)$ for
each $s\in[\tau,\tau+\theta]$. Clearly, proceeding as the derivation of Lemma
3.13 page 167 in [MZ97], (ii) of Proposition 3.8 follows from the following:

######  Lemma 3.9 (Projection of the Duhamel formulation)

There exists $A_{8}\geq 1$ such that for all $A\geq A_{8}$, $\tilde{A}\geq
A_{8}$ and $\theta>0$ there exists $s_{0,8}(A,\tilde{A},\theta)\geq
s_{0,7}(A)$, such that for all $s_{0}\geq s_{0,8}(A,\tilde{A},\theta)$, if we
assume that for some $\tau\geq s_{0}$ and for all $s\in[\tau,\tau+\theta]$,
$q(s)\in V_{A}(s)$ and $\tilde{q}(s)\in\tilde{V}_{\tilde{A}}(s)$, then
(i) (Linear terms)

$\begin{array}[]{lll}|\alpha_{2}(s)|&\leq&\frac{\tau^{2}}{s^{2}}|q_{2}(\tau)|+\frac{CA(s-\tau)}{s^{3}},\\\
\left\|\frac{\alpha_{-}(s)}{1+|y|^{3}}\right\|_{L^{\infty}}&\leq&Ce^{-\frac{(s-\tau)}{2}}\left\|\frac{q_{-}(\tau)}{1+|y|^{3}}\right\|_{L^{\infty}}+C\frac{e^{-(s-\tau)^{2}}\|q_{e}(\tau)\|_{L^{\infty}}}{s^{3/2}}+\frac{C}{s^{2}},\\\
\|\alpha_{e}(s)\|_{L^{\infty}}&\leq&Ce^{-\frac{(s-\tau)}{2}}\|q_{e}(\tau)\|_{L^{\infty}}+Ce^{s-\tau}s^{3/2}\left\|\frac{q_{-}(\tau)}{1+|y|^{3}}\right\|_{L^{\infty}}+\frac{C}{\sqrt{s}},\end{array}$
(38)

and

$\begin{array}[]{lll}|\tilde{\alpha}_{2}(s)|&\leq&\frac{\tau^{2}}{s^{2}}|\tilde{q}_{2}(\tau)|+\frac{CA(s-\tau)}{s^{\alpha+1}},\\\
\left\|\frac{\tilde{\alpha}_{-}(s)}{1+|y|^{3}}\right\|_{L^{\infty}}&\leq&Ce^{-\frac{(s-\tau)}{2}}\left\|\frac{\tilde{q}_{-}(\tau)}{1+|y|^{3}}\right\|_{L^{\infty}}+C\frac{e^{-(s-\tau)^{2}}\|\tilde{q}_{e}(\tau)\|_{L^{\infty}}}{s^{3/2}}+\frac{C}{s^{\alpha}},\\\
\|\tilde{\alpha}_{e}(s)\|_{L^{\infty}}&\leq&Ce^{-\frac{(s-\tau)}{2}}\|\tilde{q}_{e}(\tau)\|_{L^{\infty}}+Ce^{s-\tau}s^{3/2}\left\|\frac{\tilde{q}_{-}(\tau)}{1+|y|^{3}}\right\|_{L^{\infty}}+\frac{C}{s^{\alpha-3/2}}.\end{array}$
(39)

(ii) (Nonlinear terms)

$\displaystyle|\beta_{2}(s)|$ $\displaystyle\leq\frac{(s-\tau)}{s^{3}},$
$\displaystyle|\beta_{-}(y,s)|$
$\displaystyle\leq\frac{(s-\tau)}{s^{2}}(1+|y|^{3}),$
$\displaystyle\|\beta_{e}(s)\|_{L^{\infty}}$
$\displaystyle\leq\frac{(s-\tau)}{\sqrt{s}},$ $\displaystyle|\delta_{2}(s)|$
$\displaystyle\leq C\frac{(s-\tau)}{s^{3}},$ $\displaystyle|\delta_{-}(y,s)|$
$\displaystyle\leq C\frac{(s-\tau)}{s^{2}}(1+|y|^{3}),$
$\displaystyle\|\delta_{e}(s)\|_{L^{\infty}}$ $\displaystyle\leq
C\frac{(s-\tau)}{\sqrt{s}},$ $\displaystyle|\tilde{\beta}_{2}(s)|$
$\displaystyle\leq\frac{(s-\tau)}{s^{\alpha+1}},$
$\displaystyle|\tilde{\beta}_{-}(y,s)|$
$\displaystyle\leq\frac{(s-\tau)}{s^{\alpha}}(1+|y|^{3}),$
$\displaystyle\|\tilde{\beta}_{e}(s)\|_{L^{\infty}}$
$\displaystyle\leq\frac{(s-\tau)}{s^{\alpha-3/2}}.$

(iii) (Source term)

$|\gamma_{2}(s)|\leq C(s-\tau)s^{-3},\;|\gamma_{-}(y,s)|\leq
C(s-\tau)(1+|y|^{3})s^{-2},\;\|\gamma_{e}(s)\|_{L^{\infty}}\leq(s-\tau)s^{-1/2}.$

Proof: We consider, $A\geq 1$, $\tilde{A}\geq 1$, $\theta>0$, and
$s_{0}\geq\theta$. The terms $\alpha$, $\beta$ and $\gamma$ are already
present in the case of the real-valued semilinear heat equation, so we refer
to Lemma 3.13 page 167 in [MZ97] for the estimates involving them. As for
$\tilde{\alpha}$, since the definition of $\tilde{V}_{\tilde{A}}(s)$ is
different from the definition of $V_{A}(s)$, the reader will have absolutely
no difficulty to adapt Lemma 3.13 of [MZ97] to the new situation. Thus, we
only focus on the new terms $\delta(y,s)$ and $\tilde{\beta}(y,s)$. Note that
since $s_{0}\geq\theta$, if we take $\tau\geq s_{0}$, then $\tau+\theta\leq
2\tau$ and if $\tau\leq\sigma\leq s\leq\tau+\theta$, then

$\frac{1}{2\tau}\leq\frac{1}{s}\leq\frac{1}{\sigma}\leq\frac{1}{\tau}.$

Let us first derive the following bounds when $(q(s),\tilde{q}(s))\in
V_{A}(s)\times\tilde{V}_{\tilde{A}}(s)$:

######  Proposition 3.10 (Bounds for $(q(s),\tilde{q}(s))\in
V_{A}(s)\times\tilde{V}_{\tilde{A}}(s)$)

For all $s\geq e$, we consider $r\in V_{A}(s)$ and
$\tilde{r}\in\tilde{V}_{\tilde{A}}(s)$, where the shrinking sets $V_{A}(s)$
and $\tilde{V}_{\tilde{A}}(s)$ are given in Definition 3.1. Then, we have:

$\begin{array}[]{l}(i)\;for\;all\;y\in\mathbb{R},\;\;|r(y,s)|\leq
CA^{2}\frac{\log s}{s^{2}}(1+|y|^{3}),\\\ (ii)\;\|r(s)\|_{L^{\infty}}\leq
C\frac{A^{2}}{\sqrt{s}},\\\
(iii)\;for\;all\;y\in\mathbb{R},\;\;|\tilde{r}(y,s)|\leq
C\frac{\tilde{A}^{2}}{s^{2-\eta}}(1+|y|^{3}),\;\;|\tilde{r}_{b}(y,s)|\leq\frac{C\tilde{A}}{s^{\alpha-3/2}},\\\
(iv)\;\|\tilde{r}(s)\|_{L^{\infty}}\leq
C\frac{\tilde{A}^{2}}{s^{\alpha-3/2}}.\end{array}$

Proof: The proof is omitted since it is the same as the corresponding part in
[MZ97]. See Proposition 3.7 page 157 in [MZ97] for details. $\blacksquare$

Then, we recall from Bricmont and Kupiainen [BK94] the following estimates on
$K(s,\sigma)$, the semigroup generated by ${\cal L}+V$:

###### Lemma 3.11

(Properties of $K(s,\sigma)$):
(i) For all $s\geq\sigma>1$ and $y$, $x\in\mathbb{R}$, we have

$|K(s,\sigma,y,x)|\leq Ce^{(s-\sigma){\cal L}}(y,x),$

where $e^{\psi{\cal L}}$ is given by

$e^{\psi{\cal L}}(y,x)=\frac{e^{\psi}}{\sqrt{4\pi(1-e^{-\psi})}}{\rm
exp}\left[-\frac{(ye^{-\psi/2}-x)^{2}}{4(1-e^{-\psi})}\right].$

(ii)We have for all $s\geq\tau\geq 1$, with $s\leq 2\tau$,

$\displaystyle\left|\int K(s,\tau,y,x)(1+|x|^{m})dx\right|\leq C\int
e^{(s-\tau){\cal L}}(y,x)(1+|x|^{m})dx\leq e^{s-\tau}(1+|y|^{m}).$ (40)

Proof:
(i) See page 181 in [MZ97]
(ii) See Corollary 3.14 page 168 in [MZ97]. $\blacksquare$

Estimates on $\delta$ defined in (36):
Consider $s\in[\tau,\tau+\theta]$. Since $\tilde{q}(s)\in\tilde{V}_{A}(s)$ by
assumption, using (iii) and (iv) of Lemma 3.10, we see that

$\forall
y\in\mathbb{R},\;|\tilde{q}(y,s)|\leq\min\left(\frac{C\tilde{A}^{2}}{s^{2-\eta}}(1+|y|^{3}),\frac{C\tilde{A}^{2}}{s^{\alpha-3/2}}\right),$
(41)

hence by definition (34) of $N$, we obtain

$\forall y\in\mathbb{R},\;|N(y,s)|\leq
C\tilde{A}^{4}\min\left(\frac{(1+|y|^{3})}{s^{\alpha+\frac{1}{2}-\eta}},\frac{1}{s^{2\alpha-3}},\frac{1+|y|^{6}}{s^{4-2\eta}}\right).$
(42)

Using Lemma 3.11 and the definition (36) of $\delta$, we write

$\begin{array}[]{lll}|\delta(y,s)|&\leq&\displaystyle\int_{\tau}^{s}d\sigma\int_{\mathbb{R}}\left|K(s,\sigma,y,x)N(x,\sigma)\right|dx\\\
&\leq&\displaystyle\int_{\tau}^{s}d\sigma\int_{\mathbb{R}}e^{(s-\sigma){\cal
L}}(y,x)\frac{C\tilde{A}^{4}(1+|x|^{3})}{s^{\alpha+1/2-\eta}}dx\\\
&\leq&\displaystyle\frac{C\tilde{A}^{4}(s-\tau)}{s^{\alpha+1/2-\eta}}e^{s-\tau}(1+|y|^{3})\leq\frac{(s-\tau)}{s^{2}}(1+|y|^{3}),\\\
\end{array}$ (43)

for $s_{0}$ large enough, since $\eta<1/2$.
Using the following bounds in (42) and proceeding similarly, we see that

$\forall
y\in\mathbb{R},\;|\delta(y,s)|\leq(s-\tau)\min\left(\frac{1+|y|^{3}}{s^{2}},\frac{1}{\sqrt{s}},\frac{1+|y|^{6}}{s^{3}}\right),$

since $\alpha>2$, $\eta<1/2$, and provided that $s_{0}$ is large enough.
By definition of $q_{m}$, $q_{-}$ and $q_{e}$ for $m\leq 2$, we write

$\begin{array}[]{lll}|\delta_{m}(s)|&\leq&\left|\int_{\mathbb{R}}\chi(y,s)\delta(y,s)k_{m}(y)\rho(y)dy\right|\leq
C\int_{\mathbb{R}}|\delta(y,s)|(1+|y|^{2})\rho(y)dy\leq\frac{C(s-\tau)}{s^{3}},\\\
|\delta_{-}(y,s)|&=&\left|\chi(y,s)\delta(y,s)-\sum_{i=0}^{2}\delta_{i}(s)k_{i}(y)\right|\leq(s-\tau)(1+|y|^{3})\frac{C}{s^{2}}.\\\
|\delta_{e}(y,s)|&=&\left|(1-\chi(y,s))\delta(y,s)\right\|\leq(s-\tau)\frac{C}{\sqrt{s}}.\end{array}$
(44)

Estimates on $\tilde{\beta}$ defined in (37):
Consider $s\in[\tau,\tau+\theta]$. Since $q(s)\in V_{A}(s)$ by assumption,
using (i) and (ii) of Lemma 3.10, we see that

$\forall y\in\mathbb{R},\;|q(y,s)|\leq CA^{2}\min\left(\frac{\log
s}{s^{2}}(1+|y|^{3}),\frac{1}{\sqrt{s}}\right).$

Using (41) and the definition (19) of $\tilde{b}$, we see that

$\forall y\in\mathbb{R},\;|\tilde{b}(y,s)|\leq
CA^{2}\tilde{A}^{2}\min\left(\frac{\log
s}{s^{\alpha+1/2}}(1+|y|^{3}),\frac{1}{s^{\alpha-1}},\frac{1+|y|^{6}}{s^{4-\eta}}\right).$

Using the definition (37) of $\tilde{\beta}$ and arguing as for estimate
(LABEL:esdelta), we see that

$\forall
y\in\mathbb{R},\;|\tilde{\beta}(y,s)|\leq(s-\tau)\min\left(\frac{1+|y|^{3}}{s^{\alpha}},\frac{1}{s^{\alpha-3/2}},\frac{1+|y|^{6}}{s^{\alpha+1}}\right),$

provided that $s_{0}$ is large enough, since $\eta<\frac{1}{2}$ and
$\alpha<2+\eta<3-\eta$. Arguing as for (44), we get the desired estimates.
This concludes the proof of Lemma 3.9. $\blacksquare$

Since item (ii) of Proposition 3.8 follows from Lemma 3.9, exactly as for
Lemma 3.13 page 167 in [MZ97], this also ends the proof of Proposition 3.8 .
$\blacksquare$

Since Lemmas 3.4 and 3.5 follow from Proposition 3.8 exactly as in [MZ97] (see
pages 163 to 166 and 158 to 159 in that paper), this is also the conclusion of
the proof of Lemmas 3.4 and 3.5. Recalling that we have already justified that
Lemmas 3.3 and 3.6 hold (see Lemma 3.7 above), and given that Proposition 3.2
is the consequence of Lemmas 3.3, 3.4, 3.5 and 3.6, this is also the
conclusion of Proposition 3.2. $\blacksquare$

##  4 Asymptotic behavior of $u(t)$

We prove Theorem 1 in this section. We will first derive (ii) and (iii) from
Section 3, then we will prove (i) and (iv).
Consider $0<|\tilde{B}|\leq 1$. Using Proposition 3.2, Lemma 3.7 and
Proposition 3.8, we see that if $A=\tilde{A}=\max(1,{A_{1}},A_{7})$,
$0<\eta<\frac{1}{10}$, $2<\alpha<2+\eta$ and $T\leq T_{9}(\tilde{B})$ for some
$T_{9}(\tilde{B})\leq\min(T_{1},T_{6},T_{7})$ where $T_{i}=-\log s_{0,i}$,
then there exists a parameter $(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})$ such
that if $(q(s_{0}),\tilde{q}(s_{0}))$ is given by (24), where $s_{0}=-\log T$,
then

$\begin{array}[]{ll}\forall s\geq-\log T,\;q(s)\in
V_{A}(s),\;\tilde{q}(s)\in\tilde{V}_{\tilde{A}}(s),&\;\left|\tilde{q}_{2}^{{}^{\prime}}(s)+\frac{2}{s}\tilde{q}_{2}(s)\right|\leq
C\frac{\tilde{A}}{s^{\alpha+1}}\leq\frac{\mu_{0}}{s^{\alpha+1}},\\\ \mbox{with
}\mu_{0}=\frac{\alpha-2}{4}|\tilde{B}|s_{0}^{\alpha-2},&\end{array}$ (45)

and

$\left|\tilde{q}_{2}(s_{0})-\frac{\tilde{B}}{s_{0}^{2}}\right|\leq
C\tilde{A}e^{-\gamma s_{0}}\leq\frac{|\tilde{B}|}{4s_{0}^{2}}.$

As announced earlier, we use this property to derive (ii) and (iii) of Theorem
1, then we will prove (i) and (iv).

(ii) This directly follows from (45), (25) and the selfsimilar transformation
(14).
(iii) From (45), we see that

$\forall s\geq-\log
T,\;|\left(s^{2}\tilde{q}_{2}\right)^{{}^{\prime}}|\leq\frac{\mu_{0}}{s^{\alpha-1}},$
(46)

which means that $s^{2}\tilde{q}_{2}(s)$ has some limit $l$ as $s\to\infty$.
Integrating this inequality between $s$ and $+\infty$, we obtain

$|s^{2}\tilde{q}_{2}(s)-l|\leq\frac{\mu_{0}}{(2-\alpha)s^{\alpha-2}}.$ (47)

Putting $s=s_{0}$ in this identity, then using (45), we see that

$|s_{0}^{2}\tilde{q}_{2}(s_{0})-l|\leq\frac{|\tilde{B}|}{4}\mbox{ and
}|s_{0}^{2}\tilde{q}_{2}(s_{0})-\tilde{B}|\leq\frac{|\tilde{B}|}{4},$

Thus, it follows that

$|l-\tilde{B}|\leq\frac{|\tilde{B}|}{2}\mbox{, hence
}|l|\geq\frac{|\tilde{B}|}{2}>0\mbox{ and }l\not\equiv 0.$

We then write from the decomposition (23) that for all $s\geq-\log T$, $R>0$
and $|y|\leq R$, $\tilde{q}_{e}(y,s)=0$, hence,

$\tilde{q}(y,s)-\frac{l}{s^{2}}h_{2}(y)=\sum_{i=0}^{1}\tilde{q}_{i}(s)h_{i}(y)+(\tilde{q}_{2}(s)-\frac{l}{s^{2}})h_{2}(y)+\tilde{q}_{-}(y,s).$

Using the fact that for all $s\geq-\log T$,
$\tilde{q}(s)\in\tilde{V}_{\tilde{A}}(s)$ (see (45) above), Definition 3.1 for
$\tilde{V}_{\tilde{A}}(s)$, together with (47), we see that for all
$s\geq-\log T,\;R>0\mbox{ and }|y|\leq R$

$\left|\tilde{q}(y,s)-\frac{l}{s^{2}}h_{2}(y)\right|\leq\frac{C(\tilde{A},R)}{s^{\alpha}}.$

Using the definition (16) of $\tilde{q}$ and (14) of $\tilde{w}$, we get the
desired conclusion.

(i) If $x_{0}=0$, then we see from (10) and (11) that $|v(0,t)|\sim(T-t)^{-1}$
as $t\to T$. Hence $u$ blows up at time $T$ at $x_{0}=0$. It remains to prove
that any $a\neq 0$ is not a blow-up point. The following result from Giga and
Kohn [GK89] allows us to conclude:

###### Proposition 4.1

(Giga and Kohn - No blow-up under the ODE threshold) For all $C_{0}>0$, there
is $\eta_{0}>0$ such that if $v(\xi,\tau)$ solves

$\left|v_{t}-\Delta v\right|\leq C_{0}(1+|v|^{p})$

and satisfies

$|v(\xi,\tau)|\leq\eta_{0}(T-t)^{-1}$

for all $(\xi,\tau)\in B(a,r)\times[T-r^{2},T)$ for some $a\in\mathbb{R}$ and
$r>0$, then $v$ does not blow up at (a,T).

Proof:  See Theorem 2.1 page 850 in [GK89]. Note that the proof of Giga and
Kohn is valid also when $u$ is complex valued. $\blacksquare$
Indeed, since we see from (10) that

$\sup_{|x-x_{0}|\leq|x_{0}|/2}(T-t)^{-1}|u(x,t)|\leq\left|\varphi\left(\frac{|x_{0}|/2}{\sqrt{(T-t)|\log(T-t)|}}\right)\right|+\frac{C}{\sqrt{|\log(T-t)|}}\to
0$

as $t\to T$, $x_{0}$ is not a blow-up point of $u$ from Proposition 4.1. This
concludes the proof of (i) of Theorem 1.
(iv) Arguing as Merle did in [Mer92], we derive the existence of a blow-up
profile $u^{*}\in C^{2}(\mathbb{R}^{*})$ such that $u(x,t)\to u^{*}(x)$ as
$t\to T$, uniformly on compact sets of $\mathbb{R}^{*}$. The profile
$u^{*}(x)$ is not defined at the origin. In the following, we would like to
find its equivalent as $x\to 0$ and show that it is in 444 singular at the
origin. We argue as in Masmoudi and Zaag [MZ08]. Consider $K_{0}>0$ to be
fixed large enough later. If $x_{0}\neq 0$ is small enough, we introduce for
all $(\xi,\tau)\in\mathbb{R}\times[-\frac{t_{0}(x_{0})}{T-t_{0}(x_{0})},1)$,

$\displaystyle V(x_{0},\xi,\tau)$ $\displaystyle=(T-t_{0}(x_{0}))v(x,t),$ (48)
$\displaystyle\tilde{V}(x_{0},\xi,\tau)$
$\displaystyle=(T-t_{0}(x_{0}))\tilde{v}(x,t),$ (49)
$\displaystyle\mbox{where, }x$
$\displaystyle=x_{0}+\xi\sqrt{T-t_{0}(x_{0})},\;t=t_{0}(x_{0})+\tau(T-t_{0}(x_{0})),$
(50)

and $t_{0}(x_{0})$ is uniquely determined by

$|x_{0}|=K_{0}\sqrt{(T-t_{0}(x_{0}))|\log(T-t_{0}(x_{0}))|}.$ (51)

From the invariance of problem (2) under dilation,
$(V(x_{0},\xi,\tau),\tilde{V}(x_{0},\xi,\tau))$ is also a solution of (2) on
its domain. From (50), (51), (11) and (10), we have

$\sup_{|\xi|<2|\log(T-t_{0}(x_{0}))|^{1/4}}\left|V(x_{0},\xi,0)-f(K_{0})\right|\leq\frac{C}{|\log(T-t_{0}(x_{0}))|^{1/4}}\to
0\mbox{ as }x_{0}\to 0$

and

$\sup_{|\xi|<2|\log(T-t_{0}(x_{0}))|^{1/4}}\left|\tilde{V}(x_{0},\xi,0)\right|\leq\frac{C}{|\log(T-t_{0}(x_{0}))|^{1/4}}\to
0\mbox{ as }x_{0}\to 0.$

Using the continuity with respect to initial data for problem (2) associated
to a space-localization in the ball $B(0,|\xi|<|\log(T-t_{0}(x_{0}))|^{1/4})$,
we show as in Section 4 of [Zaa98] that

$\begin{array}[]{l}\sup_{|\xi|\leq|\log(T-t_{0}(x_{0}))|^{1/4},\;0\leq\tau<1}|V(x_{0},\xi,\tau)-U_{K_{0}}(\tau)|\leq\epsilon(x_{0})\mbox{
as }x_{0}\to 0\\\
\sup_{|\xi|\leq|\log(T-t_{0}(x_{0}))|^{1/4},\;0\leq\tau<1}|\tilde{V}(x_{0},\xi,\tau)|\leq\epsilon(x_{0})\mbox{
as }x_{0}\to 0,\end{array}$

where $U_{K_{0}}(\tau)=((1-\tau)+\frac{K_{0}^{2}}{8})^{-1}$ is the solution of
the PDE (2) with constant initial data $\varphi(K_{0})$. Making $\tau\to 1$
and using (50), we see that

$\begin{array}[]{lll}v^{*}(x_{0})=\lim_{t\to
T}v(x,t)&=&(T-t_{0}(x_{0}))^{-1}\lim_{\tau\to 1}V(x_{0},0,\tau)\\\
&\sim&(T-t_{0}(x_{0}))^{-1}U_{K_{0}}(1)\end{array}$

as $x_{0}\to 0$. We note also that

$|\tilde{v}^{*}(x_{0})|\leq\epsilon(x_{0})(T-t_{0}(x_{0}))^{-1}.$

Since we have from (51)

$\log(T-t_{0}(x_{0}))\sim 2\log|x_{0}|\mbox{ and
}T-t_{0}(x_{0})\sim\frac{|x_{0}|^{2}}{2K_{0}^{2}|\log|x_{0}||},$

as $x_{0}\to 0$, this yields (iv) of Theorem 1 and concludes the proof of
Theorem 1. $\blacksquare$

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