# Construction of type I blowup solutions for a higher order semilinear
parabolic equation

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###### Abstract.

We consider the higher-order semilinear parabolic equation

$\partial_{t}u=-(-\Delta)^{m}u+u|u|^{p-1},$

in the whole space $\mathbb{R}^{N}$, where $p>1$ and $m\geq 1$ is an odd
integer. We exhibit type I non self-similar blowup solutions for this equation
and obtain a sharp description of its asymptotic behavior. The method of
construction relies on the spectral analysis of a non self-adjoint linearized
operator in an appropriate scaled variables setting. In view of known spectral
and sectorial properties of the linearized operator obtained by Galaktionov
[Galaktionov(2001)], we revisit the technique developed by Merle-Zaag [Merle
and Zaag(1997)] for the classical case $m=1$, which consists in two steps: the
reduction of the problem to a finite dimensional one, then solving the finite
dimensional problem by a classical topological argument based on the index
theory. Our analysis provides a rigorous justification of a formal result in
[Galaktionov(2001)].

###### Key words and phrases:

Higher order parabolic equation, Blowup solution, Blowup profile, Stability

###### 1991 Mathematics Subject Classification:

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

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 semilinear parabolic equation

$\left\\{\begin{array}[]{rl}\partial_{t}u&=\mathbf{A}_{m}u+u|u|^{p-1},\\\
u(0)&=u_{0}\end{array}\right.\quad\mathbf{A}_{m}\equiv-(-\Delta)^{m},$ (1.1)

where $u(t):\mathbb{R}^{N}\to\mathbb{R}$ with $N\geq 1$, $\Delta$ stands for
the standard Laplace operator in $\mathbb{R}^{N}$, and the exponents $p$ and
$m$ are fixed,

$p>1\quad\textup{and}\quad m\in\mathbb{N},\;m\geq 1\;\;\textup{odd}.$

The higher-order semilinear parabolic equation (1.1) is a natural generation
of the classical semilinear heat equation $(m=1)$. It arises in many physical
applications such as theory of thin film, lubrication, convection-explosion,
phase translation, or applications to structural mechanics (see the Petetier-
Troy book [Peletier and Troy(2001)] and references therein).

By standard results the local Cauchy problem for equation (1.1) can be solved
in $L^{1}\cap L^{\infty}$ thanks to the integral representation

$u(t)=\mathcal{K}_{m}(t)\ast u_{0}+\int_{0}^{t}\mathcal{K}_{m}(t-s)\ast
u(s)|u(s)|^{p-1}ds,$ (1.2)

where $\mathcal{K}_{m}(t)$ is the fundamental solution of the linear parabolic
$\partial_{t}\mathcal{K}_{m}=\mathbf{A}_{m}\mathcal{K}_{m}$, defined via the
inverse Fourier transform

$\mathcal{K}_{m}(x,t)=\frac{1}{(2\pi)^{N}}\int_{\mathbb{R}^{N}}e^{-|\xi|^{2m}t-\imath(\xi\cdot
x)}d\xi,\quad\mathcal{K}_{m}(x,0)=\delta(x).$

From Fujita [Fujita(1966)] ($m=1$) and Galaktionov-Pohozaev [Galaktionov and
Pohozaev(2002)] ($m>1$), we know that

$p_{F}\triangleq 1+\frac{2m}{N},$

is the critical Fujita exponent for the problem in the following sense. If
$p>p_{F}$, for any sufficient small initial data $u_{0}\in
L^{1}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$ the Cauchy problem (1.1)
admits a global solution satisfying $u(t)\to 0$ as $t\to+\infty$ uniformly in
$\mathbb{R}^{N}$. If $p\in(1,p_{F}]$ and the initial data $u_{0}\not\equiv
0,\int_{\mathbb{R}^{N}}u_{0}dx\geq 0$ for $m>1$ or $u_{0}\geq 0$ for $m=1$,
then the corresponding solution to problem (1.1) blows up in some finite time
$T>0$, namely that

$\lim_{t\to T}\|u(t)\|_{L^{\infty}(\mathbb{R}^{N})}=+\infty.$

Here $T$ is called the blowup time, and a point $a\in\mathbb{R}^{N}$ is called
a blowup point if and only if there exists a sequence $(a_{n},t_{n})\to(a,T)$
such that $|u(a_{n},t_{n})|\to+\infty$ as $n\to+\infty$. A solution of (1.1)
is called Type I blowup if it satisfies

$c(T-t)^{-\frac{1}{p-1}}\leq\|u(t)\|_{L^{\infty}}\leq
C(T-t)^{-\frac{1}{p-1}},$ (1.3)

otherwise, it is of Type II blowup. In addition, we call a blowup solution
self-similar if it is of the form

$u(x,t)=(T-t)^{-\frac{1}{p-1}}\Phi(y),\quad y=\frac{x}{(T-t)^{\frac{1}{2m}}},$
(1.4)

where $\Phi$ is not identically constant. Obviously, the self-similar blowup
solution is of Type I.

When $m=1$, problem (1.1) reduces to the classical semilinear heat equation

$\partial_{t}u=\Delta u+u|u|^{p-1},$ (1.5)

which has been extensively studied in the last four decades, and no rewiew can
be exhaustive. Given our interest in the construction of solutions with a
prescribed blowup behavior, we only mention previous work in this direction.
The first conctructive result was given by Bricmont-Kupiainen [Bricmont and
Kupiainen(1994)] who showed the existence of type I blowup solution to
equation (1.5) according to the asymptotic dynamic

$\sup_{x\in\mathbb{R}^{N}}\left|(T-t)^{\frac{1}{p-1}}u(x,t)-\kappa\left(1+\frac{(p-1)}{4p}\frac{|x|^{2}}{(T-t)|\log(T-t)|}\right)^{-\frac{1}{p-1}}\right|\to
0\quad\textup{as}\;\;t\to T,$ (1.6)

for some universal positive constant $\kappa=\kappa(p)$. Note that the authors
of [Bricmont and Kupiainen(1994)] also exhibited finite time blowup solutions
that verify other asymptotic behaviors which are expected to be unstable. Note
also that Bressan [Bressan(1990), Bressan(1992)] made a similar construction
in the case of an exponential nonlinearity. Later, Merle-Zaag [Merle and
Zaag(1997)] suggested a modification of the argument of [Bricmont and
Kupiainen(1994)] and obtained the stability of the constructed solution
verifying (1.6) under small perturbations of initial data. The stability of
the asymptotic behavior (1.6) had been observed numerically by Berger-Kohn
[Berger and Kohn(1988)] (see also Nguyen [Nguyen(2017)] for other numerical
analysis). In particular, Herrero-Velázquez [Herrero and Velázquez(1992)]
proved that the blowup dynamic (1.6) is generic in one dimensional case, and
they announced the same for higher dimensional case (but never published it).

The method of [Bricmont and Kupiainen(1994)] and [Merle and Zaag(1997)] relies
on the understanding of the spectral property of the linearized operator
around an expected profile in the similarity variables setting. Roughly
speaking, the linearized operator possesses a finite number of positive
eigenvalues, a null eigenvalue and a negative spectrum; then they proceed in
two steps:

  * •

Reduction of an infinite dimensional problem to a finite dimensional one in
the sense that the control of the error reduces to the control of the
components corresponding to the positive eigenvalues.

  * •

Solving the finite dimensional problem thanks to a classical topological
argument based on the index theory.

This general two-step procedure has been extended to various situations such
as the case of the complex Ginzgburg-Landau equation by Masmoudi-Zaag
[Masmoudi and Zaag(2008)], Nouaili-Zaag [Nouaili and Zaag(2018)] (see also
Zaag [Zaag(1998)] for an earlier work); the complex semilinear heat equation
with no variational structure by Duong [Duong()], Nouaili-Zaag [Nouaili and
Zaag(2015)]; non-scaling invariant semilinear heat equations by Ebde-Zaag
[Ebde and Zaag(2011)], Nguyen-Zaag [Nguyen and Zaag(2016)], Duong-Nguyen-Zaag
[Duong et al.(2018)Duong, Nguyen, and Zaag]. We also mention the work of
Tayachi-Zaag [Tayachi and Zaag(2015), Tayachi and Zaag(2016)] and Ghoul-
Nguyen-Zaag [Ghoul et al.(2017)Ghoul, Nguyen, and Zaag] dealing with a
nonlinear heat equation with a double source depending on the solution and its
gradient in some critical setting. In [Ghoul et al.(2018a)Ghoul, Nguyen, and
Zaag, Ghoul et al.(2018b)Ghoul, Nguyen, and Zaag], we successfully adapted the
method to construct a stable blowup solution for a non variational semilinear
parabolic system.

As for the present paper, we aim at extending the above mentioned method to
construct for problem (1.1) finite time blowup solutions satisfying some
prescribed asymptotic behavior. Although the general idea is the same as for
the classical case (1.5), we would like to emphasis that the above mentioned
strategy is heavy and its implementation never being straightforward, the
context and difficulties are different for each specific problem. As a step
forward to better understanding the blowup dynamics for (1.1), we obtain the
following result.

######  Theorem 1.1 (Type I blowup solutions for (1.1) with a prescibed
behavior).

Let $m\geq 1$ be an odd integer. There exists initial data $u_{0}\in
L^{\infty}(\mathbb{R}^{N})$ such that the corresponding solution $u(t)$ to
equation (1.1) satisfies
$(i)$ $u(t)$ blows up in a finite time $T=T(u_{0})$ only at the origin.
$(ii)$ (Asymptotic behavior)

$\sup_{x\in\mathbb{R}^{N}}\left|(T-t)^{\frac{1}{p-1}}u(x,t)-\Phi\left(\frac{x}{\big{[}(T-t)|\log(T-t)|\big{]}^{\frac{1}{2m}}}\right)\right|\leq\frac{C}{|\log(T-t)|^{\frac{1}{2m}}},$
(1.7)

where

$\Phi(\xi)=\kappa\Big{(}1+B_{m,p}|\xi|^{2m}\Big{)}^{-\frac{1}{p-1}},$ (1.8)

with

$\kappa=(p-1)^{-\frac{1}{p-1}},\quad
B_{m,p}=(-1)^{m+1}\frac{2(p-1)(2m)!}{p((4m)!-2[(2m)!]^{2})}.$ (1.9)

$(iii)$ (Final blowup profile) There exists
$u^{*}\in\mathcal{C}(\mathbb{R}^{N}\backslash\\{0\\},\mathbb{R})$ such that
$u(x,t)\to u^{*}(x)$ as $t\to T$ uniformly on compact subsets of
$\mathbb{R}^{N}\backslash\\{0\\}$, where

$u^{*}(x)\sim\kappa\left(\frac{B_{m,p}|x|^{2m}}{2m|\log|x||}\right)^{-\frac{1}{p-1}}\quad\textup{as}\quad|x|\to
0.$ (1.10)

######  Remark 1.2.

We believe that such a blowup profile (1.8) exists for all
$m\in\mathbb{N}^{*}$. We note that the constant $B_{m,p}<0$ when $m$ is even
(see (1.9)), so the profile $\Phi$ blows up on the finite interface
$|\xi|=\xi_{*}=\big{(}-B_{m,p}\big{)}^{-\frac{1}{2m}}$. This says that the
case when $m$ is even would lead to type II blowup solutions in the sense of
(1.3). Although main ideas for a full justification of such a blowup behavior
with $m$ even remains the same, the proof would be very delicate and will be
addressed in a separate work.

######  Remark 1.3.

The blowup solution described in Theorem 1.1 is not self-similar in the sense
of (1.4). Note that in contrast to blowup solutions of the classical second
order semilinear heat equation (1.5), Budd-Galaktionov-Williams [Budd et
al.(2004)Budd, Galaktionov, and Williams] through numerical and asymptotic
calculations conjectured that there are at least $2\lfloor\frac{m}{2}\rfloor$
nontrivial self-similar blowup solutions to (1.1), and that profiles having a
single maximum correspond to stable (generic) self-similar blowup solutions.

######  Remark 1.4.

The proof of Theorem 1.1 (in dimension $N=1$ for simplicity) involves a
detailed description of the set of initial data leading to the asymptotic
dynamic (1.7). In particular, our initial data is roughly of the form (see
formula (3.13) below)

$u_{0}(x)=e^{-s_{0}}\left[\Phi\left(xe^{s_{0}/2m}\right)+\frac{A}{s_{0}^{2}}\chi\left(2xe^{s_{0}/2m},s_{0}\right)\sum_{k=1}^{2m-1}d_{k}\psi_{k}(xe^{s_{0}/2m})\right],$

where $A$ and $s_{0}$ are large fixed constants, $\Phi$ is the profile defined
by (1.8), $\chi$ is some smooth cut-off function,
$\big{(}d_{0},\cdots,d_{2m-1}\big{)}\in\mathbb{R}^{2m}$ are free parameters,
$\psi_{k},0\leq k\leq 2m-1$ are the eigenfunctions of the linearized operator
(see Proposition 2.1 for a precise definition) corresponding to the positive
eigenvalue $\lambda_{k}=1-\frac{k}{2m}$. Through a topological argument, we
show that there exists a suitable choice of parameters
$\big{(}d_{0},\cdots,d_{2m-1}\big{)}$ such that the solution to equation (1.1)
with the initial datum $u_{0}$ satisfies the conclusion of Theorem 1.1. In
some sense, our constructed solution is $2(m-1)$-codimension stable in the
following sense. The $2m$ components of the linearized solution corresponding
to $\lambda_{k}=1-\frac{k}{2m}$ have the exponential growth
$e^{\lambda_{k}s}$. However, the first two modes corresponding to
$\lambda_{0}$ and $\lambda_{1}$ can be eliminated by means of the time and
space translation invariance of the problem. Hence, by fixing $2(m-1)$
directions $\psi_{2},\cdots,\psi_{2m-1}$ and perturbing the remaining
components (in $L^{\infty}$), we still obtain the same asymptotic dynamic
(1.7) of the perturbed solution. The proof of $2(m-1)$-codimenison stability
would require some Lipschitz regularity of the considered initial data set and
it would be addressed separately in another work.

######  Remark 1.5.

According to our construction, the asymptotic dynamic (1.7) lies on the center
manifold generated by eigenfunctions corresponding to the null eigenvalue
$\lambda_{2m}=0$. Our analysis can be extended to construct for equation (1.1)
a finite time blowup solution having a different asymptotic dynamic from
(1.7). Such solutions particularly have asymptotic dynamics laying on the
stable manifold generated by eigenfunctions corresponding to the negative
eigenvalue $\lambda_{k}=1-\frac{k}{2m}<0$ with $k\geq 2m+1$. As explained in
Remark 1.4, the corresponding initial data leading to such solutions would
involve $k$ parameters with $k\geq 2m+1$ (consider $N=1$), so that a
topological argument is assigned in order to control the first $k$ components
corresponding to the eigenvectors $\psi_{j}$ for $0\leq j\leq k-1$. Although
the constructive method are similar for all cases, we decide to only deal with
the case of the center manifold, since the proof is the most delicate in the
sense that it requires a more refined analysis in the blowup region leading to
some logarithmic correction of the blowup variable as shown in (1.7).

#### Strategy of the proof.

Let us sketch the main ideas of the proof of Theorem 1.1.

\- Similarity variables and linearized problem. According to the scaling
invariance of the problem (1.1), we introduce the similarity variables

$u(x,t)=(T-t)^{-\frac{1}{p-1}}w(y,s),\quad
y=\frac{x}{(T-t)^{\frac{1}{2m}}},\quad s=-\ln(T-t),$ (1.11)

which leads to the new equation

$\displaystyle\partial_{s}w$
$\displaystyle=\mathbf{A}_{m}w-\frac{1}{2m}y\cdot\nabla
w-\frac{1}{p-1}w+w|w|^{p-1}$
$\displaystyle\equiv\mathscr{L}_{m}w-\frac{p}{p-1}w+w|w|^{p-1},$ (1.12)

where the linear operator $\mathscr{L}_{m}$ is given by

$\mathscr{L}_{m}=\mathbf{A}_{m}-\frac{1}{2m}y\cdot\nabla+\textup{Id}.$ (1.13)

Note that the change of variables (1.11) allows us to reduce the finite time
blowup problem to a long time behavior one at the cost of an extra scaling
term in the new equation (1.12). In the setting (1.11), proving the asymptotic
dynamic (1.7) is equivalent to proving that

$\sup_{y\in\mathbb{R}^{N}}\left|w(y,s)-\Phi\left(ys^{-\frac{1}{2m}}\right)\right|\to
0\quad\textup{as}\;\;s\to+\infty.$ (1.14)

It is then natural to linearize equation (1.12) around the expected profile
$\Phi$ by introducing

$q=w-\Phi.$

which leads to the equation of the form

$\partial_{s}q=\big{(}\mathscr{L}_{m}+V(y,s)\big{)}q+B(q)+R(y,s),$ (1.15)

where $B(q)$ is built to be quadratic, $R$ measures the error generated by
$\Phi$ and is uniformly bounded by $\mathcal{O}(s^{-1})$, and $V$ is the
potential defined as

$V(y,s)=p\left(\Phi^{p-1}-\kappa^{p-1}\right).$

Our goal is to construct for equation (1.15) a solution $q$ defined for all
$(y,s)\in\mathbb{R}^{N}\times[s_{0},+\infty)$ such that

$\sup_{y\in\mathbb{R}^{N}}|q(y,s)|\to 0\quad\textup{as}\;\;s\to+\infty.$

\-  Properties of the linearized operator. In view of equation (1.15), we see
that the nonlinear quadratic and the error term are small and can be
negligible in comparison with the linear term. Roughly speaking, the linear
part will play an important role in the dynamic of the solution. It is
essential to determine the spectrum and corresponding eigenfunctions of both
$\mathscr{L}_{m}$ and its adjoint $\mathscr{L}_{m}^{*}$. According to
[Galaktionov(2001)], the spectrum of the linear operator $\mathscr{L}_{m}$
comprises real simple eigenvalues only,

$\textup{spec}(\mathscr{L}_{m})=\left\\{\lambda_{\beta}=1-\frac{|\beta|}{2m},\;\beta=(\beta_{1},\cdots,\beta_{n})\in\mathbb{N}^{n},\;|\beta|=\beta_{1}+\cdots+\beta_{n}\right\\},$

and the corresponding eigenfunction $\psi_{\beta}$ with $|\beta|=n$ is
polynomial of order $n$ (see Proposition 2.1 below). Moreover, the family of
the eigenfunctions $\\{\psi_{\beta}\\}_{\beta\in\mathbb{N}^{n}}$ forms a
complete subset in $L^{2}_{\rho}(\mathbb{R}^{N})$ where $\rho$ is some
exponentially decaying weight function.

Depending on the asymptotic behavior of the potential $V$, we observe that

  * •

Inside the blowup region, $|y|\leq Ks^{\frac{1}{2m}}$ for some $K$ large, the
effect of $V$ is regarded as a perturbation of $\mathscr{L}_{m}$.

  * •

Outside the blowup region, $|y|\geq Ks^{\frac{1}{2m}}$, the full linear part
$\mathscr{L}_{m}+V$ behaves like $\mathscr{L}_{m}-\frac{p}{p-1}$, which has a
purely negative spectrum. Hence, the control of the solution in this region is
simple.

\-  Decomposition of the solution and reduction to a finite dimensional
problem. According to the spectrum of $\mathscr{L}_{m}$, we decompose

$q(y,s)=\sum_{|\beta|=0}^{2m}q_{\beta}(s)\psi_{\beta}(y)+q_{-}(y,s),$

where $q_{-}$ is the projection of $q$ on the subspace of $\mathscr{L}_{m}$
corresponding to strictly negative eigenvalues. Since the spectrum of the
linear part of the equation satisfied by $q_{-}$ is negative, it is
controllable to zero. We would like to notice that we do not use the Feymann-
Kac representation111In [Bricmont and Kupiainen(1994)] and [Merle and
Zaag(1997)], the kernel $\mathcal{K}$ of the heat semigroup associated to the
linear operator $\mathscr{L}_{1}+V$ is defined through the Feymann-Kac
representation
$\mathcal{K}(s,\sigma,y,x)=e^{(s-\sigma)\mathscr{L}_{1}}(y,x)\int
d\mu_{y,x}^{s-\sigma}(\omega)e^{\int_{0}^{s-\sigma}V(\omega(\tau),\sigma+\tau)d\tau},$
where $e^{t\mathscr{L}_{1}}(y,x)$ is given by Mehler’s formula and
$d\mu_{y,x}^{t}$ is the oscillator measure on the continuous path:
$\omega:[0,t]\to\mathbb{R}^{N}$ with $\omega(0)=x$ and $\omega(t)=y$. as for
the case $m=1$ treated in [Merle and Zaag(1997)], because of its complicated
implementation for higher order cases $m\geq 2$. To avoid such a formula, we
further decompose

$q_{-}(y,s)=\sum_{|\beta|=2m+1}^{M}q_{\beta}(s)\psi_{\beta}+q_{{}_{M,\bot}}(y,s),$

for some $M$ large enough (typically $M\geq 4\|V\|_{L^{\infty}_{y,s}}$). A
direct projection yields

$q_{\beta}^{\prime}=\left(1-\frac{|\beta|}{2m}\right)q_{\beta}+\mathcal{O}\left(s^{-\frac{|\beta|+2}{2m}}\right),\quad|\beta|=2m+1,\cdots,M,$

from which we obtain the rough bound
$|\theta_{\beta}(s)|=\mathcal{O}(s^{-\frac{|\beta|+2}{2m}})$ for
$|\beta|=2m+1,\cdots,M$. For the infinite part $q_{{}_{M,\bot}}$, we explore
the properties of the semigroup $e^{s\mathscr{L}_{m}}$ and a standard Gronwall
inequality to close the estimate for this part.

The control of the null mode $q_{2m}$ is delicate since the potential has in
some sense a critical size in our analysis. In particular, we need a careful
refinement of the asymptotic behavior of $V$ to derive the sharp ODE

$q_{\beta}^{\prime}=-\frac{2}{s}q_{\beta}+\mathcal{O}\left(\frac{1}{s^{3}}\right)\quad\textup{for}\;\;|\beta|=2m,$

which shows a negative spectrum after changing the variable $\tau=\ln s$,
hence, the rough bound $|q_{\beta}(s)|=\mathcal{O}\left(\frac{\log
s}{s^{2}}\right)$ for $|\beta|=2m$. Here the precise value of $B_{m,p}$ given
in (1.9) is crucial for many algebraic identities to derive this sharp ODE. At
this stage, we reduce the infinite dimensional problem to a finite dimensional
one in the sense that it remains to control a finite number of positive modes
$q_{\beta}$ for $|\beta|\leq 2m-1$. This is done through a classical
topological argument based on the index theory.

The rest of the paper is organized as follows: In Section 2 we recall basic
spectral properties of the linearized operator $\mathscr{L}$ and its adjoint
$\mathscr{L}^{*}$, then we perform a formal spectral analysis to derive an
approximate blowup profile served for our analysis later. In Section 3 we give
all arguments for the proof of Theorem 1.1 without going to technical details
(the reader who is not interested in the technical details can stop at this
section). Section 4 is the hearth of our analysis: it is devoted to the study
of the dynamic of the linearized problem from which we reduce the problem to a
finite dimensional one.

##  2\. A formal approach via spectral analysis.

In this section we first recall basic spectral properties of the linearized
operator, then present a formal approach based on a spectral analysis to
derive the blowup profile given in Theorem 1.1.

###  2.1. Spectral properties of the linearized operator.

In this subsection, we recall from [Galaktionov(2001)] the basic spectral
properties of the linear operator $\mathscr{L}_{m}$ and its adjoint
$\mathscr{L}_{m}^{*}$. The case $m=1$ is well known, since we can rewrite

$\mathscr{L}_{1}=\frac{1}{\rho_{1}}\nabla.(\rho_{1}\nabla)+\textup{Id}\quad\textup{with}\quad\rho_{1}(y)=(4\pi)^{-\frac{N}{2}}e^{-\frac{|y|^{2}}{4}},$

which is a self-adjoint operator in the weighted Hilbert space
$L^{2}(e^{-|y|^{2}/4}dy)$ with the domain
$\mathcal{D}(\mathscr{L})=H^{2}(e^{-|y|^{2}/4}dy)$. It has a real discrete
spectrum and the corresponding eigenfunctions are derived from Hermite
polynomials.

For $m\geq 2$, the operator $\mathscr{L}_{m}$ is not symmetric and does not
admit a self-adjoint extension. Then we denote $\mathscr{L}^{*}_{m}$ the
formal adjoint of $\mathscr{L}_{m}$ as

$\mathscr{L}^{*}_{m}f=\mathbf{A}_{m}f+\frac{1}{2m}\nabla.(yf)+f,$ (2.1)

in the sense that

$\big{<}\mathscr{L}_{m}f,g\big{>}=\big{<}f,\mathscr{L}^{*}_{m}g\big{>}\quad\textup{for}\quad(f,g)\in\mathcal{D}(\mathscr{L}_{m})\times\mathcal{D}(\mathscr{L}^{*}_{m}).$
(2.2)

From [Èĭ del′man(1969)] and [Fedorjuk(1977)], we know that the following
elliptic equation

$\mathbf{A}_{m}F+\frac{1}{2m}\nabla.(yF)=0\quad\textup{in}\;\;\mathbb{R}^{N}\quad\textup{with}\;\;\int_{\mathbb{R}^{N}}F(y)dy=1.$
(2.3)

has a unique radial solution given by the explicit formula

$F(y)=(2\pi)^{-\frac{N}{2}}\int_{0}^{\infty}e^{-s^{2m}}s^{\frac{N}{2}}J_{\frac{N-2}{2}}(s|y|)ds,$
(2.4)

with $J_{\nu}$ being the Bessel function. In particular, $F$ satisfies the
estimate

$|F(y)|<De^{-d|y|^{\nu}}\quad\textup{in}\;\;\mathbb{R}^{N},\;\;\textup{where}\;\;\nu=\frac{2m}{2m-1},$
(2.5)

for some positive constants $D$ and $d$ depending on $m$ and $N$.
We introduce the weight functions

$\rho^{*}=\rho^{-1},\quad\rho(y)=e^{-a|y|^{\nu}},$ (2.6)

where $a=a(m,N)\in(0,d]$ is a small constant, $d$ and $\nu$ are introduced in
(2.5).

######  Proposition 2.1 (Spectral properties of $\mathscr{L}_{m}$ and
$\mathscr{L}^{*}_{m}$, [Galaktionov(2001)]).

Let $m\in\mathbb{N}^{*}$, we have
$(i)$ $\mathscr{L}_{m}:H^{2m}_{\rho}(\mathbb{R}^{N})\to
L^{2}_{\rho}(\mathbb{R}^{N})$ is a bounded linear operator with the spectrum

$\textup{spec}(\mathscr{L}_{m})=\left\\{\lambda_{\beta}=1-\frac{|\beta|}{2m},\quad\beta=(\beta_{1},\cdots,\beta_{N})\in\mathbb{N}^{N},\quad|\beta|=0,1,2,\cdots\right\\}.$
(2.7)

The set of eigenfunctions $\\{\psi_{\beta}\\}_{|\beta|\geq 0}$ is complete in
$L^{2}_{\rho}(\mathbb{R}^{N})$, where $\psi_{\beta}$ has the separable
decomposition

$\psi_{\beta}(y)=\psi_{\beta_{1}}(y_{1})\cdots\psi_{\beta_{N}}(y_{N})\quad\textup{with}\quad\psi_{k}(\xi)=\frac{1}{\sqrt{k!}}\sum_{j=0}^{\left\lfloor\frac{k}{2m}\right\rfloor}\frac{(-1)^{jm}}{j!}\partial_{\xi}^{2jm}\xi^{k}.$
(2.8)

$(ii)$ $\mathscr{L}^{*}_{m}:H^{2m}_{\rho^{*}}(\mathbb{R}^{N})\to
L^{2}_{\rho^{*}}(\mathbb{R}^{N})$ is a bounded linear operator with the
spectrum

$\textup{spec}(\mathscr{L}^{*}_{m})=\left\\{\lambda_{\beta}^{*}=1-\frac{|\beta|}{2m},\quad\beta=(\beta_{1},\cdots,\beta_{N})\in\mathbb{N}^{N},\quad|\beta|=0,1,2,\cdots\right\\}.$
(2.9)

The set of eigenfunctions $\\{\psi_{\beta}^{*}\\}_{|\beta|\geq 0}$ is complete
in $L^{2}_{\rho^{*}}(\mathbb{R}^{N})$, where $\psi_{\beta}^{*}$’s are given by

$\psi^{*}_{\beta}(y)=\frac{(-1)^{|\beta|}}{\sqrt{\beta!}}\partial^{\beta}_{y}F(y),$
(2.10)

where $\beta!=\beta_{1}!\cdots\beta_{N}!$,
$\;\partial^{\beta}_{y}=\frac{\partial^{\beta_{1}}}{\partial
y_{1}}\cdots\frac{\partial^{\beta_{N}}}{\partial y_{N}}$ and the function $F$
is defined by (2.4).

$(iii)$ (Orthogonality)

$\left<\psi_{\beta},\psi_{\gamma}^{*}\right>=\delta_{\beta,\gamma},$ (2.11)

where $\delta_{\beta,\gamma}$ is the Kronecker delta.

######  Remark 2.2.

We note from the orthogonality (2.11) and the definition of $\psi_{k}$ that
for all polynomials $P_{n}(y)$ of degree $n<|\gamma|$, we have
$\big{<}P_{n},\psi^{*}_{\gamma}\big{>}=0$.

For $N=1$, we have

$\displaystyle\psi_{k}(y)$ $\displaystyle=a_{k}y^{k}\quad\textup{for}\;\;0\leq
k<2m,\quad a_{k}=\frac{1}{\sqrt{k!}},$ $\displaystyle\psi_{2mj}(y)$
$\displaystyle=\sum_{i=0}^{j}c_{2mi}y^{2mi}\;\;\textup{and}\quad
y^{2mj}=\sum_{i=0}^{j}c^{\prime}_{2mi}\psi_{2mi}(y)\quad\textup{for}\quad
j\in\mathbb{N}.$

We end this subsection by recalling basic properties of the semigroup
$e^{s\mathscr{L}_{m}}$ for $s>0$.

######  Lemma 2.3 (Properties of the semigroup $e^{s\mathscr{L}_{m}}$).

The kernel of the semigroup $e^{s\mathscr{L}_{m}}$ is given by

$e^{s\mathscr{L}_{m}}(y,x)=\frac{e^{s}}{\big{[}(1-e^{-s})\big{]}^{\frac{N}{2m}}}F\left(\frac{ye^{-\frac{s}{2m}}-x}{\big{[}2m(1-e^{-s})\big{]}^{\frac{1}{2m}}}\right),\quad\forall
s>0,$ (2.12)

where $F$ is defined as in (2.4). The action of $e^{s\mathscr{L}_{m}}$ is
defined by

$e^{s\mathscr{L}_{m}}g(y)=\int_{\mathbb{R}^{N}}e^{s\mathscr{L}_{m}}(y,x)g(x)dx.$
(2.13)

We also have the following estimates:
$(i)\;$ $\|e^{s\mathscr{L}_{m}}g\|_{L^{\infty}}\leq Ce^{s}\|g\|_{L^{\infty}}$
for all $g\in L^{\infty}(\mathbb{R}^{N})$.
$(ii)\,$ $\|e^{s\mathscr{L}_{m}}\textup{div}(g)\|_{L^{\infty}}\leq
Ce^{s}(1-e^{-s})^{-\frac{1}{2m}}\|g\|_{L^{\infty}}$ for all $g\in
L^{\infty}(\mathbb{R}^{N})$.
$(iii)\,$ If $|f(x)|\leq\eta(1+|x|^{M+1})$ for all $x\in\mathbb{R}^{N}$, then

$\left|e^{s\mathscr{L}_{m}}\Pi_{{}_{M,\bot}}f(y)\right|\leq C\eta
e^{-\frac{Ms}{2m}}(1+|y|^{M+1}),\quad\forall y\in\mathbb{R}^{N},$ (2.14)

where $\Pi_{{}_{M,\bot}}$ is defined as in (3.10).

###### Proof.

The formula (2.12) can be verified by a direct computation thanks to equation
(2.3). The estimates $(i)$-$(iii)$ are straightforward from the definitions
(2.12) and (2.13). ∎

###  2.2. Approximate blowup profile.

In this subsection we recall the formal approach of [Galaktionov(2009)] (see
also [Budd et al.(2004)Budd, Galaktionov, and Williams]) to figure out an
appropriate blowup profile for our analysis later. This approach had been used
in several problems involving the second order Laplacian, see for example
[Bricmont and Kupiainen(1994)], [Tayachi and Zaag(2016)], [Ghoul et
al.(2017)Ghoul, Nguyen, and Zaag, Ghoul et al.(2018a)Ghoul, Nguyen, and Zaag,
Ghoul et al.(2018b)Ghoul, Nguyen, and Zaag]. The argument relies on the basis
of the known spectral properties of the rescaled operator $\mathscr{L}_{m}$
and its adjoint $\mathscr{L}^{*}_{m}$ given in the previous subsection. For
simplicity, we consider the one dimensional case and symmetric positive
solutions.

Let us introduce

$\bar{w}=w-\kappa,$

where $\kappa=(p-1)^{-\frac{1}{p-1}}$ is the constant equilibrium to equation
(1.12). This yields the following perturbed equation

$\partial_{s}\bar{w}=\mathscr{L}_{m}\bar{w}+\frac{p}{2\kappa}\bar{w}^{2}+R(\bar{w}),$
(2.15)

where $|R(\bar{w})|\leq C|\bar{w}|^{3}$ for $\bar{w}\ll 1$.
From Proposition 2.1, we know that $\psi_{n}$ with $n\geq 2m+1$ correspond to
negative eigenvalues of $\mathscr{L}_{m}$. Therefore, we may consider

$\bar{w}(y,s)=\sum_{i=0}^{m}\bar{w}_{2i}(s)\psi_{2i}(y),$ (2.16)

where $\sum_{i=0}^{m}|\bar{w}_{2i}(s)|\to 0$ as $s\to+\infty$. Plugging this
ansatz to equation (2.15), taking the scalar product with $\psi_{2j}^{*}$ and
using Proposition 2.1, we find that for $0\leq j\leq m$,

$\bar{w}^{\prime}_{2j}(s)=\left(1-\frac{j}{m}\right)w_{2j}(s)+\frac{p}{2\kappa}\left<\left(\sum_{j=0}^{m}\bar{w}_{2i}(s)\psi_{2i}(y)\right)^{2},\psi^{*}_{2j}\right>+\mathcal{O}(\sum_{i=0}^{m}|\bar{w}_{2i}(s)|^{3}).$
(2.17)

Assuming that $\bar{w}_{2m}$ is dominant, i.e.

$\textup{for}\;\;0\leq j\leq
m-1,\quad|\bar{w}_{2j}(s)|\ll|\bar{w}_{2m}(s)|\quad\textup{as}\;\;s\to+\infty,$
(2.18)

then system (2.17) reduces to

$\displaystyle\bar{w}^{\prime}_{2j}(s)$
$\displaystyle=\left(1-\frac{j}{m}\right)w_{2j}(s)+\mathcal{O}(|\bar{w}_{2m}(s)|^{2})\quad\textup{for}\quad
0\leq j\leq m-1,$ $\displaystyle\bar{w}^{\prime}_{2m}(s)$
$\displaystyle=\frac{p}{2\kappa}\bar{\mu}_{2m}\bar{w}_{2m}^{2}+o(|\bar{w}_{2m}(s)|^{2}),\quad\textup{where}\quad\bar{\mu}_{2m}=\left<\psi_{2m}^{2},\psi_{2m}^{*}\right>.$

Solving this system yields

$\bar{w}_{2m}(s)=-\frac{2\kappa}{p\bar{\mu}_{2m}}\frac{1}{s}+\mathcal{O}\left(\frac{\ln^{2}s}{s}\right),\quad\sum_{i=0}^{m-1}|\bar{w}_{2i}(s)|=\mathcal{O}\left(\frac{1}{s^{2}}\right)\quad\textup{as}\;\;s\to+\infty,$

which is in agreement with the assumption (2.18).
Hence, from (2.16) and the definition of $\psi_{2m}$, we derive the following
asymptotic behavior

$w(y,s)=\kappa-\frac{2\kappa}{p\bar{\mu}_{2m}s}\left(\frac{y^{2m}+(-1)^{m}(2m)!}{\sqrt{(2m)!}}\right)+\mathcal{O}\left(\frac{\ln
s}{s^{2}}\right),$ (2.19)

where the convergence takes place in $L^{2}_{\rho}(\mathbb{R})$ as well as
uniformly in compact sets by standard parabolic regularity.

The expansion (2.19) provides a relevant variable for blowup, namely
$z=ys^{-\frac{1}{2m}}$, that governs the behavior in the intermediate region.
In particular, we try to search formally a solution $w$ of equation (1.12) of
the form

$w(y,s)=\Phi(z)+\frac{(-1)^{m+1}2\kappa\sqrt{(2m)!}}{p\bar{\mu}_{2m}s}+\mathcal{O}\left(\frac{1}{s^{1+\epsilon}}\right),$
(2.20)

for some $\epsilon>0$, with the boundary condition $\Phi(0)=\kappa$.
Plugging this ansatz to equation (1.12) and comparing the leading order terms,
we arrive at

$-\frac{z}{2m}\Phi^{\prime}-\frac{1}{p-1}\Phi+\Phi^{p}=0,\quad\Phi(0)=\kappa.$
(2.21)

Solving this ODE yields

$\Phi(z)=\kappa\Big{(}1+B_{m,p}z^{2m}\Big{)}^{-\frac{1}{p-1}}\quad\textup{for
some $B_{m,p}\in\mathbb{R}$}.$

By matching expansions (2.19) and (2.20), we find that

$B_{m,p}=\frac{2(p-1)}{p\bar{\mu}_{2m}\sqrt{(2m)!}},\quad\bar{\mu}_{2m}=\left<\psi_{2m}^{2},\psi^{*}_{2m}\right>.$

Let us compute $\bar{\mu}_{2m}$ in the following. Using (2.8), we write

$\displaystyle\psi_{2m}(y)$
$\displaystyle=\frac{1}{\sqrt{(2m)!}}\big{(}y^{2m}+(-1)^{m}(2m)!\big{)},$
$\displaystyle\psi_{4m}(y)$
$\displaystyle=\frac{1}{\sqrt{(4m)!}}\left(y^{4m}+(-1)^{m}\frac{(4m)!}{(2m)!}y^{2m}+(4m)!\right),$
$\displaystyle\psi^{2}_{2m}(y)$
$\displaystyle=\frac{1}{(2m)!}\left[\sqrt{(4m)!}\psi_{4m}+(-1)^{m+1}\sqrt{(2m)!}\left(\frac{(4m)!}{(2m)!}-2(2m)!\right)\psi_{2m}+c_{0}(m)\psi_{0}\right].$

Using the orthogonality relation (2.11), the definition (2.10) of
$\psi_{2m}^{*}$ and the fact that $\int_{\mathbb{R}}F(y)dy=1$, we compute by
an integration by parts,

$\displaystyle\bar{\mu}_{2m}$
$\displaystyle=\frac{(-1)^{m+1}}{\sqrt{(2m)!}}\left(\frac{(4m)!}{(2m)!}-2(2m)!\right)\int_{\mathbb{R}}\psi_{2m}(y)\psi^{*}_{2m}(y)dy$
$\displaystyle=\frac{(-1)^{m+1}}{\sqrt{(2m)!}}\left(\frac{(4m)!}{\big{[}(2m)!\big{]}^{2}}-2\right)\int_{\mathbb{R}}(y^{2m}+c_{2m,0})\partial_{y}^{2m}F(y)dy$
$\displaystyle=(-1)^{m+1}\frac{(2m)}{\sqrt{(2m)!}}\left(\frac{(4m)!}{\big{[}(2m)!\big{]}^{2}}-2\right)\int_{\mathbb{R}}F(y)dy$
$\displaystyle=(-1)^{m+1}\sqrt{(2m)!}\left(\frac{(4m)!}{[(2m)!]^{2}}-2\right).$
(2.22)

In conclusion, we have derived the following candidate for the blowup profile
in the similarity variables:

$w(y,s)\sim\varphi(y,s):=\kappa\left(1+B_{m,p}\frac{y^{2m}}{s}\right)^{-\frac{1}{p-1}}+\frac{A_{m,p}}{s},$
(2.23)

where

$B_{m,p}=\frac{2(p-1)}{p\bar{\mu}_{2m}\sqrt{(2m)!}},\quad
A_{m,p}=\frac{(-1)^{m+1}2\kappa\sqrt{(2m)!}}{p\bar{\mu}_{2m}}.$ (2.24)

Since we want the profile $\varphi$ bounded, this requests $B_{m,p}>0$ or
$\bar{\mu}_{2m}>0$, which only happens for $m$ odd.

##  3\. Proof of Theorem 1.1 without technical details.

In this section we give all arguments of the proof of Theorem 1.1. We only
deal with the one dimensional case $N=1$ for simplicity since the analysis for
the higher dimensional cases $N\geq 2$ is exactly the same up to some
complicated calculation of the projection of (1.15) on the eigenspaces of
$\mathscr{L}_{m}$. The proof of Theorem 1.1 is completed in three parts:

  * •

In the first part we formulate the problem by linearizing the rescaled
equation (1.12) around the approximate profile $\varphi$ given by (2.23). We
also introduce a shrinking set in which the constructed solution of the
linearized equation is trapped.

  * •

In the second part we exhibit an explicit formula of the initial data and show
that the corresponding solution belongs to the shrinking set. In particular,
the reader can find how to reduce the problem to a finite dimensional one (all
technical details will be left to the next section) and the use of a
topological argument based on index theory to conclude.

  * •

The last part is devoted to the proof of items $(i)$ and $(iii)$ of Theorem
1.1. Although the argument of the proof is almost the same as for the
classical case $m=1$, we would like to sketch the main ideas for the reader
convenience.

###  3.1. Formulation of the problem.

According to the formal analysis given in Section 2.2, we introduce

$q(y,s)=w(y,s)-\varphi(y,s),$ (3.1)

and write from (1.12) the equation driving by $q$,

$\partial_{s}q=\big{(}\mathscr{L}_{m}+V(y,s)\big{)}q+B(q)+R(y,s),$ (3.2)

where $\mathscr{L}_{m}$ is the linearized operator defined by (1.13) and

$\displaystyle V(y,s)$
$\displaystyle=p\left(\varphi^{p-1}-\kappa^{p-1}\right),$ (3.3) $\displaystyle
B(q)$
$\displaystyle=(q+\varphi)|q+\varphi|^{p-1}-\varphi^{p}-p\varphi^{p-1}q,$
(3.4) $\displaystyle R(y,s)$
$\displaystyle=-\partial_{s}\varphi+\mathbf{A}_{m}\varphi-\frac{1}{2m}y\cdot\nabla\varphi-\frac{\varphi}{p-1}+\varphi^{p}.$
(3.5)

Let $\chi_{0}\in\mathcal{C}_{0}^{\infty}(\mathbb{R}_{+})$ be a cut-off
function with $\textup{supp}(\chi_{0})\subset[0,2]$ and $\chi_{0}\equiv 1$ on
$[0,1]$. We introduce

$\chi(y,s)=\chi_{0}\left(\frac{|y|}{Ks^{\frac{1}{2m}}}\right)\quad\textup{with}\;\;K\gg
1\;\;\textup{fixed},$ (3.6)

and define

$q_{e}(y,s)=q(y,s)(1-\chi(y,s)).$ (3.7)

We decompose

$q(y,s)=\sum_{k=0}^{M}q_{k}(s)\psi_{k}(y)+q_{{}_{M,\bot}}(y,s),$ (3.8)

where

  * •

$q_{k}=\Pi_{k}(q)$ is the projection of $q$ on the eigenmode corresponding to
eigenvalue $\lambda_{k}=1-\frac{k}{2m}$, defined as

$q_{k}(s)=\big{<}q,\psi_{k}^{*}\big{>}\quad\textup{with \, $\psi^{*}_{k}$ \,
being introduced in \eqref{def:psibeta*}.}$ (3.9)

  * •

$q_{{}_{M,\bot}}=\Pi_{{}_{M,\bot}}(q)$ is called the infinite dimensional part
of $q$, where $\Pi_{{}_{M,\bot}}(q)$ is the projection of $q$ on the
eigensubspace where the spectrum of $\mathscr{L}_{m}$ is lower than
$\frac{1-M}{2m}$. Note that we have the orthogonality

$\big{<}q_{{}_{M,\bot}},\psi_{k}^{*}\big{>}=0\quad\textup{for}\;k\leq M.$
(3.10)

  * •

$M$ is typically a large constant

$M=4mN\quad\textup{with}\;\;N\in\mathbb{N}^{*},\;N\geq\|V\|_{L^{\infty}_{y,s}}.$
(3.11)

which allows us to successfully apply a standard Gronwall’s inequality to the
control of the infinite dimensional part $q_{{}_{M,\bot}}$.

We aim at constructing for equation (1.15) a global in time solution $q$ such
that

$\|q(s)\|_{L^{\infty}(\mathbb{R}^{N})}\to 0\quad\textup{as}\quad s\to+\infty.$
(3.12)

According to the decomposition (3.8), it is enough to show that there exists a
solution $q$ belonging to the following set.

######  Definition 3.1 (Shrinking set to trap solutions).

For each $A>0$, for each $s>0$, we denote $\mathcal{V}_{A}(s)$ the set of all
functions $q(y,s)$ in $L^{\infty}(\mathbb{R})$ such that

$\displaystyle|q_{k}(s)|\leq\frac{A}{s^{2}}\quad\textup{for}\;\;0\leq k\leq
2m-1,\qquad\quad|q_{2m}(s)|\leq\frac{A^{2}\log s}{s^{2}},$
$\displaystyle\quad|q_{k}(s)|\leq
A^{k}s^{-\frac{k+1}{2m}}\quad\textup{for}\;\;2m+1\leq k\leq
M,\qquad\quad\|q_{e}(s)\|_{L^{\infty}(\mathbb{R})}\leq
A^{M+2}s^{-\frac{1}{2m}},$ $\displaystyle\qquad\forall
y\in\mathbb{R},\quad|q_{{}_{M,\bot}}(y,s)|\leq
A^{M+1}s^{-\frac{M+2}{2m}}(|y|^{M+1}+1),$

where $q_{k}$, $q_{{}_{M,\bot}}$ and $q_{e}$ are defined as in the
decomposition (3.8) and (3.7).

######  Remark 3.2.

By definition, we see that if $q(s)\in\mathcal{V}_{A}(s)$, then the following
bound holds

$\displaystyle|q(y,s)|$ $\displaystyle\leq
CA^{M+1}\left(\sum_{k=0}^{M+1}(|y|^{k}+1)s^{-\frac{k+1}{2m}}\right)\mathbf{1}_{\\{|y|\leq
2Ks^{\frac{1}{2m}}\\}}+\|q_{e}\|_{L^{\infty}(\mathbb{R}^{N})}$
$\displaystyle\qquad\leq CA^{M+2}s^{-\frac{1}{2m}}\quad\textup{for all}\quad
y\in\mathbb{R}.$

Hence, our goal (3.12) reduces to constructing for equation (1.15) a solution
$q(s)$ belonging to the shrinking set $\mathcal{V}_{A}(s)$ for all
$s\in[s_{0},+\infty)$.

###  3.2. Existence of solutions trapped in $\mathcal{V}_{A}$.

In this step we aim at proving that there actually exists initial data
$\phi(y)=q(y,s_{0})$ such that the corresponding solution $q(y,s)$ to (1.15)
belongs to the shrinking set $\mathcal{V}_{A}(s)$. Given $A\geq 1$ and
$s_{0}\geq e$, we consider initial data of the form

$\phi_{A,s_{0},\mathbf{d}}(y)=\frac{A}{s_{0}^{2}}\chi(2y,s_{0})\sum_{k=0}^{2m-1}d_{k}\psi_{k}(y),$
(3.13)

where $\mathbf{d}=(d_{1},\cdots,d_{2m-1})\in\mathbb{R}^{2m}$ are real
parameters to be fixed later, $\chi$ is introduced in (3.6). In particular,
the initial data (3.13) belongs to $\mathcal{V}_{A}(s_{0})$ as shown in the
following proposition.

######  Proposition 3.3 (Properties of initial data (3.13)).

For each $A\gg 1$, there exist $s_{0}=s_{0}(A)\gg 1$ and a cuboid
$\mathcal{D}_{s_{0}}\subset[-A,A]^{2m}$ such that for all
$\big{(}d_{0},\cdots,d_{2m-1}\big{)}\in\mathcal{D}_{s_{0}}$, the following
properties hold:

  * (i)

The initial data $\phi_{A,s_{0},\mathbf{d}}$ defined in (3.13) belongs to
$\mathcal{V}_{A}(s_{0})$ with strict inequalities except for the positive
modes $\phi_{k}$ with $0\leq k\leq 2m-1$, where
$\phi_{k}=\Pi_{k}(\phi_{A,s_{0},\mathbf{d}})$.

  * (ii)

The map $\Gamma:\mathcal{D}_{s_{0}}\to\mathbb{R}^{2m}$, defined as
$\Gamma(d_{0},\cdots,d_{2m-1})=\big{(}\phi_{0},\cdots,\phi_{2m-1})$, is
linear, one to one from $\mathcal{D}_{s_{0}}$ to
$\big{[}-As_{0}^{-2},As_{0}^{-2}\big{]}^{2m}$, and maps
$\partial\mathcal{D}_{s_{0}}$ into
$\partial\Big{(}\big{[}-As_{0}^{-2},As_{0}^{-2}\big{]}^{2m}\big{)}$. Moreover,
the degree of $\Gamma$ on the boundary is different from zero.

###### Proof.

The proof directly follows from the definition of the projection
$\Pi_{k}(\phi_{A,s_{0},\mathbf{d}})$ and it is straightforward, so we omit it
here. ∎

Starting with initial data $\phi_{A,s_{0},\mathbf{d}}$ belonging to
$\mathcal{V}_{A}(s_{0})$, we claim that we can fine-tune the parameters
$\big{(}d_{0},\cdots,d_{2m-1}\big{)}\in\mathcal{D}_{s_{0}}$ such that the
corresponding solution $q(s)$ to equation (1.15) stays in $\mathcal{V}_{A}(s)$
for all $s\geq s_{0}$. More precisely, we claim the following.

######  Proposition 3.4 (Existence of solutions trapped in
$\mathcal{V}_{A}(s)$).

There exist $A\gg 1$, $s_{0}=s_{0}(A)\gg 1$ and
$\big{(}d_{0},\cdots,d_{2m-1}\big{)}\in\mathcal{D}_{s_{0}}$ such that if
$q(s)$ is the solution to equation (1.15) with initial data (3.13), then
$q(s)\in\mathcal{V}_{A}(s)$ for all $s\geq s_{0}(A)$.

###### Proof.

From the local Cauchy problem of (1.1) in $L^{\infty}(\mathbb{R})$, we see
that for each initial data
$\phi_{A,s_{0},\mathbf{d}}\in\mathcal{V}_{A}(s_{0})$, equation (1.15) has a
unique solution $q(s)\in\mathcal{V}_{A}(s)$ for all $s\in[s_{0},s_{*})$ with
$s_{*}=s_{*}(\mathbf{d})$. If $s_{*}=+\infty$ for some
$\mathbf{d}=(d_{0},\cdots,d_{2m-1})\in\mathcal{D}_{s_{0}}$, we are done.
Otherwise, we proceed by contradiction and assume that
$s_{*}(\mathbf{d})<+\infty$ for all $\mathbf{d}\in\mathcal{D}_{s_{0}}$. By
continuity and the definition of $s_{*}$, we remark that
$q(s_{*})\in\partial\mathcal{V}_{A}(s_{*})$. We claim the following.

######  Proposition 3.5 (Finite dimensional reduction).

There exist $A\gg 1$, $s_{0}=s_{0}(A)\gg 1$ and
$\big{(}d_{0},\cdots,d_{2m-1}\big{)}\in\mathcal{D}_{s_{0}}$ such that the
following properties hold: If $q(s)$ is the solution to equation (1.15) with
initial data (3.13) and $q(s)\in\mathcal{V}_{A}(s)$ for all
$s\in[s_{0},s_{1}]$ and $q(s_{1})\in\partial\mathcal{V}_{A}(s_{1})$, then

  * (i)

(Reduction to a finite dimensional problem)
$\;\Big{(}q_{0},\cdots,q_{2m-1}\Big{)}(s_{1})\in\partial\left([-As_{1}^{-2},As_{1}^{-2}]^{2m}\right)$.

  * (ii)

(Transversality) There exists $\mu_{0}>0$ such that
$q(s_{1}+\mu)\not\in\mathcal{V}_{A}(s_{1}+\mu)$ for all $\mu\in(0,\mu_{0})$.

Let us postpone the proof of Proposition 3.5 to the next section and continue
our argument. From $(i)$ of Proposition 3.5, we obtain
$\big{(}q_{0},\cdots,q_{2m-1}\big{)}(s_{*})\in\partial\left([-As_{*}^{-2},As_{*}^{-2}]^{2m}\right)$,
and the following mapping is well defined

$\displaystyle\Theta:\mathcal{D}_{s_{0}}\;$
$\displaystyle\to\;\partial\left([-1,1]^{2m}\right)$
$\displaystyle(d_{0},\cdots,d_{2m-1})\;$
$\displaystyle\mapsto\;\frac{s_{*}^{2}}{A}\big{(}q_{0},\cdots,q_{2m-1}\big{)}(s_{*}).$

From the transversality given in item $(ii)$ of Proposition 3.5,
$\big{(}q_{0},\cdots,q_{2m-1}\big{)}$ actually crosses its boundary at
$s=s_{*}$, resulting in the continuity of $s_{*}$ and $\Theta$. Applying again
the transversality, we see that if
$(d_{0},\cdots,d_{2m-1})\in\partial\mathcal{D}_{s_{0}}$, then $q(s)$ leaves
$\mathcal{V}_{A}(s)$ at $s=s_{0}$, thus, $s_{*}=s_{0}$ and
$\Theta_{|\partial\mathcal{D}_{s_{0}}}=\Gamma$, the map defined in item $(ii)$
of Proposition 3.3. Using that item, we see that the degree of $\Theta$ is not
zero. Since $\Theta$ is continuous, this is a contradiction. This concludes
the proof of Proposition 3.4, assuming that Proposition 3.5 holds. ∎

###  3.3. Conclusion of Theorem 1.1.

From Proposition 3.4, there exists a solution $q$ to equation (1.15) such that
$q(s)\in\mathcal{V}_{A}(s)$ for all $s\geq s_{0}$. From Remark 3.2, we deduce
that $\|q(s)\|_{L^{\infty}(\mathbb{R})}\leq C(A)s^{-\frac{1}{2m}}$ for all
$s\geq s_{0}$. The conclusion of item $(ii)$ then follows from (1.11) and
(3.1). Item $(i)$ of Theorem 1.1 is just a direct consequence of items $(ii)$
and $(iii)$. Because the proof of item $(iii)$ is similar to the classical
case $m=1$, we only sketch the main ideas for the reader’s convenience. The
existence of the final blowup profile
$u^{*}\in\mathcal{C}(\mathbb{R}\setminus\\{0\\})$ follows from the technique
of Merle [Merle(1992)]. Here we focus on a precise description of the final
blowup profile $u^{*}$ in a neighborhood of the singularity. To do so, we
follow the technique of Herrero-Velázquez [Herrero and Velázquez(1993)] (see
also Bebernes-Bricher [Bebernes and Bricher(1992)], Zaag [Zaag(1998)] for a
similar approach) by introducing the auxiliary function

$h(\xi,\tau;x_{0})=(T-t_{0}(x_{0}))^{\frac{1}{p-1}}u(x,t),$ (3.14)

where

$\xi=\frac{x-x_{0}}{[T-t_{0}(x_{0})]^{\frac{1}{2m}}},\quad\tau=\frac{t-t_{0}(x_{0})}{T-t_{0}(x_{0})},$

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

$|x_{0}|=K\big{[}(T-t_{0}(x_{0}))|\log(T-t_{0}(x_{0}))|\big{]}^{\frac{1}{2m}},\quad
K\gg 1\;\textup{fixed}.$ (3.15)

We note that $h(\xi,\tau;x_{0})$ is also a solution to (1.1) because of the
invariance of (1.1) under dilations. From (1.7), we have

$\sup_{|\xi|\leq|\log(T-t_{0}(x_{0}))|^{\frac{1}{4m}}}\left|h(\xi,0,x_{0})-\Phi(K)\right|\leq\frac{C}{(T-t_{0}(x_{0}))^{\frac{1}{2m}}}\to
0\quad\textup{as}\;|x_{0}|\to 0.$

Let $\hat{h}_{K}(\tau)$ be the solution to (1.1) with the constant initial
datum $\Phi(K)$, defined as

$\hat{h}_{K}(\tau)=\kappa\left(1-\tau+B_{m,p}K^{2m}\right)^{-\frac{1}{p-1}},\quad\tau\in[0,1).$

By the continuity with respect to initial data for equation (1.1), one can
show that

$\sup_{|\xi|\leq|\log(T-t_{0}(x_{0}))|^{\frac{1}{4m}},0\leq\tau<1}\left|h(\xi,\tau;x_{0})-\hat{h}_{K}(\tau)\right|\leq\epsilon(x_{0})\to
0,\quad\textup{as}\;\;|x_{0}|\to 0.$

Passing to the limit $\tau\to 1$ yields

$u^{*}(x_{0})=(T-t_{0}(x_{0}))^{-\frac{1}{p-1}}\lim_{\tau\to
1}h(0,\tau;x_{0})\sim(T-t_{0}(x_{0}))^{-\frac{1}{p-1}}\hat{h}_{K}(1).$

From the definition (3.15), we compute

$\big{|}\log(T-t_{0}(x_{0}))\big{|}\sim 2m\big{|}\log|x_{0}|\big{|},\quad
T-t_{0}(x_{0})\sim\frac{|x_{0}|^{2m}}{2mK^{2m}|\log|x_{0}||}\quad\textup{as}\quad|x_{0}|\to
0,$

from which we obtain the asymptotic behavior

$u^{*}(x_{0})\sim\kappa\left(\frac{B_{m,p}|x_{0}|^{2m}}{2m|\log|x_{0}||}\right)^{-\frac{1}{p-1}}\quad\textup{as}\quad|x_{0}|\to
0.$

This concludes the proof of Theorem 1.1, assuming that Proposition 3.5 holds.

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

This section is the central part in our analysis where we give all details of
the proof of Proposition 3.5, completing hence the proof of Theorem 1.1. The
essential idea is to project equation (1.15) onto different components
according to the decomposition (3.8). In particular, we claim that Proposition
3.5 is a direct consequence of the following.

######  Proposition 4.1 (Dynamics of equation (1.15)).

For all $A\gg 1$, there exists $s_{0}=s_{0}(A)\gg 1$ such that if
$q(s)\in\mathcal{V}_{A}(s)$ for all $s\in[\tau,\tau_{1}]$ with
$\tau_{1}\geq\tau\geq s_{0}$, then the following estimates hold for all
$s\in[\tau,\tau_{1}]$:

  * (i)

(Control of the finite dimensional part)

$\displaystyle\left|q_{k}^{\prime}(s)-\left(1-\frac{k}{2m}\right)q_{k}\right|\leq\frac{C}{s^{2}}\quad\textup{for}\quad
0\leq k\leq 2m-1.$ (4.1)
$\displaystyle\quad\left|q_{2m}^{\prime}(s)+\frac{2}{s}q_{2m}\right|\leq\frac{CA^{3}}{s^{3}},$
(4.2) $\displaystyle\qquad|q_{j}(s)|\leq
Ce^{-\left(\frac{j}{2m}-1\right)(s-\tau)}|q_{j}(\tau)|+\frac{CA^{j-1}}{s^{\frac{j+1}{2m}}}\quad\textup{for}\quad
2m+1\leq j\leq M.$ (4.3)

  * (ii)

(Control of the infinite dimensional and outer part)

$\displaystyle\left\|\frac{q_{{}_{M,\bot}}(y,s)}{1+|y|^{M+1}}\right\|_{L^{\infty}(\mathbb{R})}\leq
Ce^{-\frac{M(s-\tau)}{2m}}\left\|\frac{q_{{}_{M,\bot}}(y,\tau)}{1+|y|^{M+1}}\right\|_{L^{\infty}(\mathbb{R})}+\frac{CA^{M}}{s^{\frac{M+2}{2}}}.$
(4.4) $\displaystyle\quad\left\|q_{e}(s)\right\|_{L^{\infty}(\mathbb{R})}\leq
Ce^{-\frac{(s-\tau)}{2(p-1)}}\|q_{e}(\tau)\|_{L^{\infty}(\mathbb{R})}+\frac{CA^{M+1}}{s^{\frac{1}{2m}}}(1+s-\tau).$
(4.5)

######  Remark 4.2.

Note that the sharp ODE (4.2) comes from the precise choice of the constant
$B_{m,p}$ appearing in the profile $\Phi$ defined in (1.8). Other choices
would give the error of size $\mathcal{O}\left(\frac{1}{s^{2}}\right)$ which
is too large to close the estimate for $q_{2m}$.

Let us postpone the proof of Proposition 4.1 and proceed with the proof of
Proposition 3.5.

###### Proof of Proposition 3.5 assuming Proposition 4.1.

Since the argument of the proof is similar to what was done in [Merle and
Zaag(1997)], we only sketch the proof for the reader’s convenience. We recall
from the assumption that

$q(s)\in\mathcal{V}_{A}(s)\quad\textup{for all}\quad
s\in[s_{0},s_{1}]\quad\textup{and}\quad
q(s_{1})\in\partial\mathcal{V}_{A}(s_{1}).$ (4.6)

Thus, part $(i)$ of Proposition 3.5 will be proved if we show that all the
bounds given in Definition 3.1 can be improved, except for the first $2m$
components $q_{k}$ with $0\leq k\leq 2m-1$, in the sense that for all
$s\in[s_{0},s_{1}]$:

$\displaystyle|q_{2m}(s)|<\frac{A^{2}\log
s}{s^{2}},\quad|q_{k}(s)|<\frac{A^{k}}{s^{\frac{k+1}{2m}}}\quad\textup{for}\quad
2m+1\leq k\leq M,$ (4.7)
$\displaystyle\left\|\frac{q_{{}_{M,\bot}}(y,s)}{1+|y|^{M+1}}\right\|_{L^{\infty}}<\frac{A^{M+1}}{s^{\frac{M+2}{2m}}},\quad\|q_{e}(s)\|_{L^{\infty}}<\frac{A^{M+2}}{s^{\frac{1}{2m}}}.$
(4.8)

We first deal with $q_{2m}$. We argue by contradiction by assuming that there
is $s_{*}\in[s_{0},s_{1}]$ such that for all $s\in[s_{0},s_{*})$,

$|q_{2m}(s)|<\frac{A^{2}\log s}{s^{2}},\quad|q_{2m}(s_{*})|=\frac{A^{2}\log
s_{*}}{s_{*}^{2}},$

(note that the existence of $s_{*}$ is guaranteed by item $(i)$ of Proposition
3.3). Considering $q_{2m}(s_{*})>0$ and using minimality (the case
$q_{2m}(s_{*})<0$ is similar), we have on the one hand,

$q^{\prime}_{2m}(s_{*})\geq A^{2}\frac{d}{ds}\left(\frac{\log
s_{*}}{s_{*}^{2}}\right)=\frac{A^{2}}{s_{*}^{3}}-\frac{2A^{2}\log
s_{*}}{s_{*}^{3}},$

which holds thanks to (4.6). On the other hand, we obtain from the ODE (4.2)
that

$q^{\prime}_{2m}(s_{*})\leq\frac{-2A^{2}\log
s_{*}}{s_{*}^{3}}+\frac{CA}{s_{*}^{3}},$

and the contradiction follows if $A\geq 2C+1$.
As for $q_{j}$ with $2m+1\leq j\leq M$ and $q_{{}_{M,\bot}}$ and $q_{e}$, we
argue as follows: Let $\lambda=\log A$ and assume that $s_{0}\geq\lambda$ such
that for all $\tau\geq s_{0}$ and $s\in[\tau,\tau+\lambda]$ we have

$\tau\leq s\leq\tau+\lambda\leq\tau+s_{0}\leq
2\tau,\quad\textup{hence,}\quad\frac{1}{\tau}\sim\frac{1}{s}.$

We distinguish into two cases:
\- For $s-s_{0}\leq\lambda$, we use estimates (4.3), (4.4) and (4.5) with
$\tau=s_{0}$ together with Proposition 3.3 to find that

$\displaystyle|q_{j}(s)|\leq\frac{C(1+A^{j-1})}{s^{\frac{j+1}{2m}}}<\frac{A^{j}}{s^{\frac{j+1}{2m}}}\quad\textup{for}\quad
2m+1\leq j\leq M,$
$\displaystyle\left\|\frac{q_{{}_{M,\bot}}(y,s)}{1+|y|^{M+1}}\right\|_{L^{\infty}}\leq\frac{C(1+A^{M})}{s^{\frac{M+2}{2m}}}<\frac{A^{M+1}}{s^{\frac{M+2}{2m}}},$
$\displaystyle\|q_{e}(s)\|_{L^{\infty}}\leq\frac{C(1+A^{M+1}\log
A)}{s^{\frac{1}{2m}}}<\frac{A^{M+2}}{s^{\frac{1}{2m}}},$

for $A$ large enough. Hence, estimates (4.7) and (4.8) hold for
$s-s_{0}\leq\lambda$.
\- For $s-s_{0}>\lambda$, we use estimates (4.3), (4.4) and (4.5) with
$\tau=s-\lambda>s_{0}$ together with (4.6) to write (remember that
$s\geq\lambda/2$)

$\displaystyle|q_{j}(s)|\leq
e^{-\left(\frac{j}{2m}-1\right)\lambda}\frac{A^{j}}{(s/2)^{\frac{j+1}{2m}}}+\frac{CA^{j-1}}{s^{\frac{j+1}{2m}}}<\frac{A^{j}}{s^{\frac{j+1}{2m}}}\quad\textup{for}\;\;2m+1\leq
j\leq M,$
$\displaystyle\left\|\frac{q_{{}_{M,\bot}}(y,s)}{1+|y|^{M+1}}\right\|_{L^{\infty}}\leq
Ce^{-\frac{M\lambda}{4m}}\frac{A^{M+1}}{(s/2)^{\frac{M+2}{2m}}}+\frac{CA^{M}}{s^{\frac{M+2}{2}}}<\frac{A^{M+1}}{s^{\frac{M+2}{2m}}},$
$\displaystyle\|q_{e}(s)\|_{L^{\infty}}\leq
Ce^{-\frac{\lambda}{2(p-1)}}\frac{A^{M+2}}{(s/2)^{\frac{1}{2m}}}+\frac{CA^{M+1}(1+\log
A)}{s^{\frac{1}{2m}}}<\frac{A^{M+2}}{s^{\frac{1}{2m}}}.$

Therefore, estimates (4.7) and (4.8) hold for all $s\in[s_{0},s_{1}]$, hence,
the conclusion of part $(i)$ of Proposition 3.5 follows.

Part $(ii)$ of Proposition 3.5 is a direct consequence of the dynamics of
$q_{k}$ given in (4.1) and (4.6). Indeed, from part $(i)$ of Proposition 4.1,
we know that $q_{k}(s_{1})=\epsilon\frac{A}{s_{1}^{2}}$ for some
$k\in\\{0,\cdots,2m-1\\}$ and $\epsilon=\pm 1$. Using estimate (4.1) yields

$\epsilon q^{\prime}_{k}(s_{1})\geq\left(1-\frac{k}{2m}\right)\epsilon
q_{k}(s_{1})-\frac{C}{s_{1}^{2}}\geq\left((1-\frac{k}{2m})A-C\right)\frac{1}{s_{1}^{2}}.$

Thus, for $0\leq k\leq\frac{k}{2m}$ and $A$ large enough, we have $\epsilon
q_{k}^{\prime}(s_{1})>0$. Therefore, $q_{k}$ is traversal outgoing to the
boundary curve $s\mapsto\epsilon As^{-2}$ at $s=s_{1}$. This concludes the
proof of Proposition 3.5, assuming that Proposition 4.1 holds. ∎

We now give the proof of Proposition 4.1 to complete the proof of Proposition
3.5. We divide the proof into two subsections according to the two parts of
Proposition 4.1.

###  4.1. Control of the finite dimensional part.

We prove item $(i)$ of Proposition 4.1 in this part. A direct projection of
equation (1.15) on the eigenfunction $\psi_{k}$ for $0\leq k\leq M$ yields

$q_{k}^{\prime}(s)-\left(1-\frac{k}{2m}\right)q_{k}(s)=\Pi_{k}\Big{(}Vq+B(q)+R\Big{)}.$
(4.9)

#### Estimate of $\Pi_{k}(Vq)$.

We claim the following.

######  Lemma 4.3 (Expansion of $V$).

The potential $V$ defined by (3.3) satisfies the estimate

$|V(y,s)|\leq\frac{C(1+|y|^{2m})}{s},\quad\forall y\in\mathbb{R},\;s\geq 1,$
(4.10)

and admits the following uniform expansion for all $n\in\mathbb{N}^{*}$,

$V(y,s)=\sum_{j=1}^{n}\frac{1}{s^{j}}V_{j}(y)+\mathcal{O}\left(\frac{1+|y|^{2m(n+1)}}{s^{n+1}}\right),\quad\forall|y|\leq
s^{\frac{1}{2m}},\;s>1,$ (4.11)

where $V_{j}$’s are even polynomial of degree $2mj$. More precisely, we have

$V_{1}(y)=-\frac{2}{\bar{\mu}_{2m}}\psi_{2m}\quad\textup{with}\quad\bar{\mu}_{2m}=\big{<}\psi_{2m}^{2},\psi_{2m}^{*}\big{>}.$
(4.12)

###### Proof.

Estimate (4.10) is trivial. Estimate (4.11) follows from a Taylor expansion in
the variable $z=\frac{|y|^{2m}}{s}$. Formula (4.12) comes from the definitions
(2.23) and (2.8) of $\varphi$ and $\psi_{2m}$. ∎

From Lemma 4.3, we obtain the following estimate for $\Pi_{k}(Vq)$.

######  Lemma 4.4 (Estimate of $\Pi_{k}(Vq)$).

Under the assumption of Proposition 4.1, we have

$\displaystyle\Big{|}\Pi_{k}(Vq)\Big{|}\leq\frac{C}{s^{2}}\quad\textup{for}\quad
0\leq k\leq 2m-1,$
$\displaystyle\quad\left|\Pi_{2m}(Vq)+\frac{2}{s}q_{2m}\right|\leq\frac{CA}{s^{3}},$
$\displaystyle\qquad\Big{|}\Pi_{k}(Vq)\Big{|}\leq\frac{CA^{k-2}}{s^{\frac{k+1}{2m}}}\quad\textup{for}\quad
2m+1\leq k\leq M.$

###### Proof.

By Definition 3.1, we have

$|q(y,s)|\leq\frac{CA^{M+1}}{s^{1+\frac{1}{m}}}(1+|y|^{M+1})\quad\textup{for
all}\;\;y\in\mathbb{R}.$ (4.13)

Using this, the estimate (4.10) and noting from the definition (2.10) and
(2.5) that $\psi_{k}^{*}$ is exponentially decaying, we obtain for $0\leq
k\leq 2m-1$,

$\displaystyle\left|\Pi_{k}(Vq)\right|\leq\frac{CA^{M+1}}{s^{2+\frac{1}{m}}}\int_{\mathbb{R}}(1+|y|^{M+1+2m})\psi_{k}^{*}(y)dy\leq\frac{C}{s^{2}}.$

For $2m+1\leq k\leq M$, we write from the decomposition (3.8),

$\displaystyle\Pi_{k}(Vq)=\sum_{i=0}^{M}q_{i}(s)\int_{\mathbb{R}}V(y,s)\psi_{i}(y)\psi_{k}^{*}(y)dy+\int_{\mathbb{R}}V(y,s)q_{{}_{M,\bot}}(y,s)\psi_{k}^{*}(y)dy.$

Using (4.10) and the bound of $q_{{}_{M,\bot}}$ given in Definition 3.1 yields

$\left|\int_{\mathbb{R}}V(y,s)q_{{}_{M,\bot}}(y,s)\psi_{k}^{*}(y)dy\right|\leq\frac{CA^{M+1}}{s^{1+\frac{M+1}{2m}}}\int_{\mathbb{R}}(1+|y|^{M+1+2m})\psi_{k}^{*}(y)dy\leq\frac{CA^{M+1}}{s^{1+\frac{M+1}{2m}}}.$

From the orthogonality (2.11), we note that
$\big{<}P_{n},\psi_{k}^{*}\big{>}=0$ for all polynomial $P_{n}$ of degree
$n\leq k-1$. We then use (4.11) and (4.13) to estimate

$\displaystyle\sum_{i=0}^{M}q_{i}(s)\int_{\mathbb{R}}V(y,s)\psi_{i}(y)\psi_{k}^{*}(y)dy$
$\displaystyle=\sum_{i=0}^{M}\sum_{j=1}^{n}\frac{q_{i}(s)}{s^{j}}\int_{|y|\leq
s^{\frac{1}{2m}}}V_{j}(y)\psi_{i}(y)\psi_{k}^{*}(y)dy$
$\displaystyle\quad+\mathcal{O}\left(\frac{A^{M+1}}{s^{n+1+\frac{1}{m}}}\int_{|y|\leq
s^{\frac{1}{2m}}}(1+|y|^{2m(n+1)+M+1})|\psi^{*}_{k}(y)|dy\right)$
$\displaystyle\qquad+\mathcal{O}\left(\frac{A^{M+1}}{s^{2+\frac{1}{m}}}\int_{|y|\geq
s^{\frac{1}{2m}}}(1+|y|^{2m+M+1})|\psi^{*}_{k}(y)|dy\right).$

The last term is bounded by $\mathcal{O}(e^{-cs})$ because of the exponential
decay of $\psi_{k}^{*}$. By taking $n\in\mathbb{N}^{*}$ such that
$n+1+\frac{1}{m}\geq\frac{k+2}{2m}$, the second term is bounded by
$\mathcal{O}\left(A^{M+1}s^{-\frac{k+2}{2m}}\right)$. Using the fact that
$V_{j}\psi_{i}$ is polynomial of degree $2mj+i$, we see that
$\big{<}V_{j}\psi_{i},\psi_{k}^{*}\big{>}=0$ for $2mj+i\leq k-1$. Combining
this with the bounds of $q_{i}$ given in Definition 3.1 of $\mathcal{V}_{A}$,
we obtain the rough estimate

$\displaystyle\left|\sum_{i=0}^{M}\sum_{j=1}^{n}\frac{q_{i}(s)}{s^{j}}\int_{|y|\leq
s^{\frac{1}{2m}}}V_{j}(y)\psi_{i}(y)\psi_{k}^{*}(y)dy\right|\leq\sum_{i=0}^{M}\sum_{j=1,2jm+i\geq
k}^{n}\frac{CA^{i}}{s^{j+\frac{i+1}{2m}}}$
$\displaystyle\quad\leq\frac{CA^{k-2}}{s^{\frac{k+1}{2m}}}+\sum_{i=k-1}^{M}\frac{CA^{M}}{s^{1+\frac{i+1}{2m}}}\leq\frac{CA^{k-2}}{s^{\frac{k+1}{2m}}}.$

As for $k=2m$, we need to use the precise definition (4.12) of $V_{1}$ and
process similarly as for $k\geq 2m+1$. Indeed, from decomposition (3.8) and
expansion (4.11) with $n=2$, we have

$\displaystyle\Pi_{2m}(Vq)-\frac{q_{2m}}{s}\int_{|y|\leq
s^{\frac{1}{2m}}}V_{1}(y)\psi_{2m}\psi_{2m}^{*}dy=\sum_{i=0,i\neq
2m}^{M}\frac{q_{i}(s)}{s}\int_{|y|\leq
s^{\frac{1}{2m}}}V_{1}\psi_{i}\psi_{2m}^{*}+\mathcal{O}\left(\frac{A^{M+1}}{s^{3+\frac{1}{m}}}\right).$

Using the definition (4.12) of $V_{1}$, the bound of $q_{i}$ given in
Definition 3.1 and the fact coming from the definition (2.8) of $\psi_{\beta}$
and the orthogonality (2.11) that

$\int_{\mathbb{R}}\psi_{2m}\psi_{i}\psi_{2m}^{*}dy=\int_{\mathbb{R}}\big{(}c_{0}\psi_{2m+i}+c_{1}\psi_{i}+c_{2}\psi_{i-2m}\big{)}\psi_{2m}^{*}=0\quad\textup{for}\;\;2m+1\leq
i\leq 4m-1,$

we arrive at

$\left|\Pi_{2m}(Vq)+\frac{2}{s}q_{2m}\right|\leq\sum_{i=0}^{2m-1}\frac{CA}{s^{3}}+\sum_{4m}^{M}\frac{CA^{M}}{s^{1+\frac{i+1}{2m}}}+\frac{CA^{M+1}}{s^{3+\frac{1}{m}}}\leq\frac{CA}{s^{3}}.$

This concludes the proof of Lemma 4.4. ∎

We now turn to the estimate of the main contribution of the nonlinear term
$B(q)$ under the projection $\Pi_{k}$. We begin with the following expansion.

######  Lemma 4.5 (Expansion of $B(q)$).

For all $|q|<1$, $s\geq s_{0}\gg 1$ and $|y|\leq s^{\frac{1}{2m}}$, the
function $B(q)$ defined by (3.4) admits the uniform expansion

$\left|B(q)-\sum_{j=2}^{M+1}q^{j}\sum_{l=0}^{M}\frac{1}{s^{l}}\left(B_{j,l}(ys^{-\frac{1}{2m}})+\tilde{B}_{j,l}(y,s)\right)\right|\leq
C|q|^{M+2}+\frac{C}{s^{M+1}},$ (4.14)

where $B_{j,l}(z)$’s are even polynomials of degree $l$ and

$|\tilde{B}_{j,l}(y,s)|\leq\frac{C(1+|y|^{M+1})}{s^{\frac{M+1}{2m}}}.$

Furthermore, we have the estimate for all $y\in\mathbb{R}$ and $s\geq 1$,

$|B(q)|\leq C|q|^{\bar{p}}\quad\textup{with}\;\;\bar{p}=\min\\{2,p\\}.$ (4.15)

###### Proof.

We first note from the definition (2.23) of $\varphi$ that there exist two
positive constants $c_{0},C_{0}$ such that $c_{0}\leq\varphi(y,s)\leq C_{0}$
for $|y|\leq s^{\frac{1}{2m}}$. Thus, a Taylor expansion of $B(q,\varphi)$ in
terms of $q$ yields

$\left|B(q)-\sum_{j=2}^{M+1}B_{j}(\varphi)q^{j}\right|\leq C|q|^{M+1},$

where $B_{j}(\varphi)$ admits the following expansion in terms of
$\frac{1}{s}$,

$\left|B_{j}(\varphi)-\sum_{l=0}^{M}\frac{1}{s^{l}}B_{j,l}(\Phi)\right|\leq\frac{C}{s^{M+1}}.$

Now, a Taylor expansion of $B_{j,l}(\Phi)$ in terms of the variable
$z=ys^{-\frac{1}{2m}}$ yields the desired result. This concludes the proof of
Lemma 4.5. ∎

With Lemma 4.5 at hand, we estimate the main contribution of $B(q)$ under the
projection $\Pi_{k}$. We claim the following.

######  Lemma 4.6 (Estimate for $\Pi_{k}\big{(}B(q)\big{)}$).

Under the assumption of Proposition 4.1, we have

$\displaystyle\left|\Pi_{k}(B(q))\right|\leq\frac{C}{s^{2}}\quad\textup{for}\;\;0\leq
k\leq 2m-1,\quad\left|\Pi_{2m}(B(q))\right|\leq\frac{C}{s^{3}},$
$\displaystyle\quad\left|\Pi_{k}(B(q))\right|\leq\frac{CA^{k}}{s^{\frac{k+2}{2m}}}\quad\textup{for}\;\;2m+1\leq
k\leq M.$

###### Proof.

From (4.14), the exponential decay of $\psi_{k}^{*}$ and (4.13), we have

$\displaystyle\Pi_{k}(B(q))=\sum_{j=2}^{M+1}\sum_{l=0}^{M}\frac{1}{s^{l}}\int_{|y|\leq
s^{\frac{1}{2m}}}q^{j}B_{j,l}\psi_{k}^{*}dy+\mathcal{O}\left(\frac{A^{M+1}}{s^{2(1+1/m)+\frac{M+1}{2m}}}\right)+\mathcal{O}(e^{-cs}).$

From the decomposition (3.8), we write

$q^{j}=\big{(}\sum_{i=0}^{M}q_{i}\psi_{i}+q_{{}_{M,\bot}}\big{)}^{j}\equiv\big{(}q_{<}+q_{{}_{M,\bot}}\big{)}^{j}\quad\textup{for}\quad
j\geq 2.$

From Definition 3.1 of $\mathcal{V}_{A}$ and the uniform boundedness of $q$,
we obtain the bound for $j\geq 2$,

$|q^{j}-q_{<}^{j}|\leq
C\Big{(}|q_{{}_{M,\bot}}|^{j}+|q_{<}|^{j-1}|q_{{}_{M,\bot}}|\Big{)}\leq\frac{CA^{j(M+1)}}{s^{\frac{M+4+2m}{2m}}}(1+|y|^{j(M+1)}).$

Therefore,

$\displaystyle\Pi_{k}(B(q))=\sum_{j=2}^{M+1}\sum_{l=0}^{M}\frac{1}{s^{l}}\int_{|y|\leq
s^{\frac{1}{2m}}}q_{<}^{j}B_{j,l}\psi_{k}^{*}dy+\mathcal{O}\left(\frac{A^{j(M+1)}}{s^{\frac{M+4+2m}{2m}}}\right).$

We only handle the case $k=2m$, which is the most delicate. The case $k\neq
2m$ can be processed similarly and we omit it. From (4.13), we have

$\sum_{j=3}^{M}\sum_{l=0}^{M}\frac{1}{s^{l}}\int_{|y|\leq
s^{\frac{1}{2m}}}\big{|}q_{<}^{j}B_{j,l}\psi_{2m}^{*}\big{|}dy+\sum_{l=1}^{M}\frac{1}{s^{l}}\int_{|y|\leq
s^{\frac{1}{2m}}}\big{|}q_{<}^{2}B_{j,l}\psi_{2m}^{*}\big{|}dy\leq\frac{CA^{j(M+1)}}{s^{3+\frac{1}{m}}}\leq\frac{C}{s^{3}}.$

It remains to control the term $I:=\int_{|y|\leq
s^{\frac{1}{2m}}}q_{<}^{2}\psi_{2m}^{*}dy$. By the definition of $q_{<}$, we
expand

$\displaystyle q_{<}^{2}$
$\displaystyle=\left(\sum_{i=0}^{2m}q_{i}\psi_{i}\right)^{2}+2\left(\sum_{i=0}^{2m}q_{i}\psi_{i}\right)\left(\sum_{i^{\prime}=2m+1}^{M}q_{i^{\prime}}\psi_{i^{\prime}}\right)+\left(\sum_{i^{\prime}=2m+1}^{M}q_{i^{\prime}}\psi_{i^{\prime}}\right)^{2}$
$\displaystyle=I_{1}+I_{2}+I_{3}.$

From Definition 3.1, we have
$|I_{1}|\leq\frac{CA^{4}\log^{2}}{s^{4}}(1+|y|^{4m})$ and
$|I_{2}|\leq\frac{CA^{M+4}\log s}{s^{3+1/m}}(1+|y|^{M+2m})$. Hence,

$\int_{|y|\leq
s^{\frac{1}{2m}}}\big{|}(I_{1}+I_{2})\psi_{2m}^{*}dy\big{|}\leq\frac{CA^{4}\log^{2}}{s^{4}}+\frac{CA^{M+4}\log
s}{s^{3+1/m}}\leq\frac{1}{s^{3}}.$

Note from the orthogonality relation (2.11) that if $i^{\prime}+l^{\prime}\neq
J2m$ for $J\in\mathbb{N}^{*}$, we have

$\displaystyle\big{<}\psi_{i^{\prime}}\psi_{l^{\prime}},\psi_{2m}^{*}\big{>}$
$\displaystyle=\left<\left(\sum_{i=0}^{\lfloor\frac{i^{\prime}}{2m}\rfloor}c_{i^{\prime},i}\partial_{y}^{2im}y^{i^{\prime}}\right)\left(\sum_{l=0}^{\lfloor\frac{l^{\prime}}{2m}\rfloor}c_{l^{\prime},l}\partial_{y}^{2lm}y^{l^{\prime}}\right),\psi_{2m}^{*}\right>$
$\displaystyle=\left<\sum_{j=0}^{\lfloor\frac{i^{\prime}}{2m}\rfloor+\lfloor\frac{l^{\prime}}{2m}\rfloor}\hat{c}_{i^{\prime}+l^{\prime},j}y^{i^{\prime}+l^{\prime}-2jm},\psi_{2m}^{*}\right>=\left<\sum_{j=0}^{\lfloor\frac{i^{\prime}}{2m}\rfloor+\lfloor\frac{l^{\prime}}{2m}\rfloor}\tilde{c}_{i^{\prime}+l^{\prime},j}\psi_{i^{\prime}+l^{\prime}-2jm},\psi_{2m}^{*}\right>=0.$

Therefore, we estimate

$\displaystyle\int_{|y|\leq s^{\frac{1}{2m}}}I_{3}\psi_{2m}^{*}dy$
$\displaystyle=\sum_{i^{\prime},l^{\prime}=2m+1}^{M}c_{i^{\prime},l^{\prime}}q_{i^{\prime}}(s)q_{l^{\prime}}(s)\int_{\mathbb{R}}\psi_{i^{\prime}}\psi_{l^{\prime}}\psi_{2m}^{*}dy+\mathcal{O}(e^{-cs})$
$\displaystyle\quad=\sum_{i^{\prime},l^{\prime}=2m+1,i^{\prime}+l^{\prime}=J2m}^{M}c_{i^{\prime},l^{\prime}}q_{i^{\prime}}(s)q_{l^{\prime}}(s)\int_{\mathbb{R}}\psi_{i^{\prime}}\psi_{l^{\prime}}\psi_{2m}^{*}dy+\mathcal{O}(e^{-cs}).$

for $J\in\mathbb{N}^{*}$. Since $i^{\prime}+l^{\prime}\geq 4m+2$, it follows
that $J\geq 3$. From Definition 3.1, we have the bound
$|q_{i^{\prime}}q_{l^{\prime}}|\leq\frac{A^{i^{\prime}+l^{\prime}}}{s^{\frac{i^{\prime}+l^{\prime}+2}{2m}}}\leq\frac{A^{i^{\prime}+l^{\prime}}}{s^{3+\frac{1}{m}}}\leq\frac{1}{s^{3}}$.
Therefore, $\left|\int_{|y|\leq
s^{\frac{1}{2m}}}I_{3}\psi_{2m}^{*}dy\right|\leq\frac{1}{s^{3}}$. This
concludes the proof of Lemma 4.6. ∎

We now deal with the generated error term $R$. We begin with the following
expansion.

######  Lemma 4.7 (Expansion of $R$).

The function $R$ defined by (3.5) satisfies

$\|R(s)\|_{L^{\infty}}\leq\frac{C}{s},$

and admits the following expansion: for all $n\in\mathbb{N}^{*}$, $|y|\leq
s^{\frac{1}{2m}}$ and $s\geq 1$:

$\left|R(y,s)-\sum_{j=1}^{n-1}\frac{1}{s^{j+1}}R_{j}(y)\right|\leq\frac{C(1+|y|^{2mn})}{s^{n+1}},$
(4.16)

where $R_{j}$’s are polynomials of degree $2mj$ of the form
$R_{j}(y)=\sum_{i=0}^{j}d_{i}y^{2mi}$. Moreover, the coefficient of degree
$2m$ of $R_{1}$ is identically zero, hence,
$\big{<}R_{1},\psi^{*}_{2m}\big{>}=0$.

###### Proof.

Let $z=ys^{-\frac{1}{2m}}$ and note that $\Phi(z)$ satisfies equation (2.21).
Therefore, we rewrite (3.5) as follows:

$R(y,s)=\frac{z}{2ms}\cdot\nabla\Phi+\frac{A_{m,p}}{s^{2}}+\frac{1}{s}\mathbf{A}_{m}\Phi-\frac{A_{m,p}}{(p-1)s}+Q\left(\Phi+\frac{A_{m,p}}{s}\right)-Q(\Phi),$
(4.17)

where $Q(h)=h^{p}$. Since $c_{0}\leq\Phi(z)\leq C_{0}$ for all $|z|\leq 1$
with $c_{0},C_{0}$ some positive constants, we have the following expansion of
$Q$,

$\left|Q\left(\Phi+\frac{A_{m,p}}{s}\right)-Q(\Phi)-\sum_{j=1}^{J}\frac{1}{s^{j}}Q_{j}(\Phi)\right|\leq\frac{C}{s^{J+1}}\quad\textup{for}\;\;J\in\mathbb{N}^{*}.$

Then, we expand $Q_{j}$ and all the remaining terms in (4.17) in power series
of $Z=z^{2m}$ to obtain the desired result. Note that the coefficient of
$\frac{1}{s}$ in the expansion of $R$ (after an elementary computation) is
given by

$A_{m,p}-\frac{(2m)!\kappa B_{m,p}}{p-1}=0,$

where we used (2.24). Moreover, $R_{1}(y)=C_{1}y^{2m}+C_{2}$, where
$C_{2}=C_{2}(p,m)$ and

$C_{1}=\frac{\kappa
B_{m,p}}{p-1}\left(\frac{(4m)!}{(2m)!}\frac{pB_{m,p}}{2(p-1)}-\frac{pA_{m,p}}{\kappa}-1\right)=0.$

Again, the precise values of $B_{m,p}$ and $A_{m,p}$ given in (2.24) are
crucial in deriving that $C_{1}$ is identically zero. Therefore, the
orthogonality relation (2.11) yields $\big{<}R_{1},\psi_{2m}^{*}\big{>}=0$.
This concludes the proof of Lemma 4.7. ∎

As a direct consequence of Lemma 4.7, we have the following.

######  Lemma 4.8 (Estimate of $\Pi_{k}(R)$).

Under the assumption of Proposition 4.1, we have

$\displaystyle\Pi_{k}(R)=\mathcal{O}(s^{-M})\quad\textup{for}\quad(k\;\textup{mod}\;2m)\neq
0,$
$\displaystyle\big{|}\Pi_{0}(R)\big{|}\leq\frac{C}{s^{2}},\quad\big{|}\Pi_{2m}(R)\big{|}\leq\frac{C}{s^{3}},\quad\big{|}\Pi_{2mi}(R)\big{|}\leq\frac{C}{s^{i+1}}\quad\textup{for}\;\;i\in\mathbb{N}^{*}.$

###### Proof.

The proof directly follows from the expansion (4.16) and the fact that

$\big{<}y^{2mj},\psi_{k}^{*}\big{>}=\sum_{i=0}^{j}a_{i}\big{<}\psi_{2mi},\psi_{k}^{*}\big{>}=0\quad\textup{for}\quad(k\;\textup{mod}\;2m)\neq
0.$

For $k\in\mathbb{N}$ with $(k\;\textup{mod}\;2m)\neq 0$, we use the expansion
(4.16) with $n=M-1$ (we can replace $M$ by any positive integer $L\gg 1$) and
write

$\displaystyle|\Pi_{k}(R)|$
$\displaystyle=\left|\int_{\mathbb{R}}R(y,s)\psi_{k}^{*}(y)dy\right|=\left|\int_{|y|\leq
s^{\frac{1}{2m}}}R(y,s)\psi_{k}^{*}(y)dy+\int_{|y|\geq
s^{\frac{1}{2m}}}R(y,s)\psi_{k}^{*}(y)dy\right|$
$\displaystyle\leq\sum_{j=1}^{M-2}\int_{|y|\geq
s^{\frac{1}{2m}}}\frac{|R_{j}(y)|}{s^{j+1}}|\psi_{k}^{*}(y)|dy+\frac{C}{s^{M}}\int_{\mathbb{R}}(1+|y|^{2m(M-1)})|\psi_{k}^{*}(y)|dy$
$\displaystyle\quad\qquad+\frac{C}{s}\int_{|y|\geq
s^{\frac{1}{2m}}}|\psi_{k}^{*}(y)|dy\leq\frac{C}{s^{M}}+Ce^{-cs}\leq\frac{C}{s^{M}}.$

For $(k\;\textup{mod}\;2m)\neq 0$, i.e. $k=2mi$ for some $i\in\mathbb{N}$, we
use (4.16) with $n=i+1$ to get the conclusion. This ends the proof of Lemma
4.8. ∎

A collection of all estimates given in Lemmas 4.4, 4.6, 4.8 and equation (4.9)
yields the conclusion of part $(i)$ of Proposition 4.1.

###  4.2. Control of the infinite dimensional and the outer part.

We prove item $(ii)$ of Proposition in this part. We first deal with the
infinite dimensional part $q_{{}_{M,\bot}}$, then the outer part $q_{e}$.

####  Control of $q_{{}_{M,\bot}}$:

Applying $\Pi_{{}_{M,\bot}}$ to equation (1.15) and using the fact that
$\Pi_{{}_{M,\bot}}\psi_{n}=0$ for all $n\leq M$ (see (3.10)), we obtain

$\partial_{s}q_{{}_{M,\bot}}=\mathscr{L}_{m}q_{{}_{M,\bot}}+\Pi_{{}_{M,\bot}}\left(Vq+B(q)+R\right).$
(4.18)

By definition, we have $\Pi_{{}_{M,\bot}}(y^{k})=0$ for all $k\leq M$. Using
the uniform bound $\|R(s)\|_{L^{\infty}}\leq\frac{C}{s}$ and the expansion
(4.16) with $n=\frac{M}{2m}$ (see (3.11) for the definition of $M$) and noting
that $\Pi_{{}_{M,\bot}}(R_{j})=0$ for $j\leq\frac{M}{2m}$, we obtain

$\displaystyle\left|\Pi_{{}_{M,\bot}}(R)\right|$
$\displaystyle\leq\frac{C(1+|y|^{M+2m})}{s^{\frac{M}{2m}+2}}\mathbf{1}_{|y|\leq
s^{\frac{1}{2m}}}+|R(y,s)|\mathbf{1}_{|y|\geq
s^{\frac{1}{2m}}}\leq\frac{C(1+|y|^{M+1})}{s^{\frac{M+1}{2m}+1}}.$ (4.19)

As for the estimate of $\Pi_{{}_{M,\bot}}(Vq)$, we claim the following.

######  Lemma 4.9 (Estimate of $\Pi_{{}_{M,\bot}}(Vq)$).

Under the assumption of Proposition 4.1, we have

$\left\|\frac{\Pi_{{}_{M,\bot}}(Vq)}{1+|y|^{M+1}}\right\|_{L^{\infty}(\mathbb{R})}\leq\left(\|V(s)\|_{L^{\infty}(\mathbb{R})}+\frac{C}{s}\right)\left\|\frac{q_{{}_{M,\bot}}}{1+|y|^{M+1}}\right\|_{L^{\infty}(\mathbb{R})}+\frac{CA^{M}}{s^{\frac{M+2}{2m}}}.$
(4.20)

###### Proof.

By definition, we write

$\Pi_{{}_{M,\bot}}(Vq)=Vq_{{}_{M,\bot}}-\Pi_{{}_{M,<}}\big{(}Vq_{{}_{M,\bot}}\big{)}+\Pi_{{}_{M,\bot}}\big{(}Vq_{{}_{M,<}}\big{)},$

where
$q_{{}_{M,<}}=\Pi_{{}_{M,<}}(q)=\big{(}\textup{Id}-\Pi_{{}_{M,\bot}}\big{)}(q)$.
Using estimate (4.10), we derive

$\left\|\frac{Vq_{{}_{M,\bot}}}{1+|y|^{M+1}}\right\|_{L^{\infty}}+\left\|\frac{\Pi_{{}_{M,<}}\big{(}Vq_{{}_{M,\bot}}\big{)}}{1+|y|^{M+1}}\right\|_{L^{\infty}}\leq\left(\|V(s)\|_{L^{\infty}(\mathbb{R})}+\frac{1}{s}\right)\left\|\frac{q_{{}_{M,\bot}}}{1+|y|^{M+1}}\right\|_{L^{\infty}}.$

As for the control of $\Pi_{{}_{M,\bot}}\big{(}Vq_{{}_{M,<}})=\sum_{i\leq
M}\Pi_{{}_{M,\bot}}\big{(}q_{i}V\psi_{i})$, we argue as follows:
\- If $M-i=0\;\textup{mod}\;2m$, we take $n=\frac{M-i}{2m}$ in the expansion
(4.11) and write

$\displaystyle\Pi_{{}_{M,\bot}}\big{(}Vq_{i}\psi_{i}\big{)}=\sum_{j=1}^{n}\frac{q_{i}}{s^{j}}\Pi_{{}_{M,\bot}}\big{(}V_{j}\psi_{i}\big{)}+\Pi_{{}_{M,\bot}}\Big{(}\tilde{V}_{n}q_{i}\psi_{i}\Big{)},$

where $|\tilde{V}_{n}(y,s)|\leq\frac{C(1+|y|^{2m(n+1)})}{s^{n+1}}$. Using the
fact that $\Pi_{{}_{M,\bot}}(y^{n})=0$ for $n\leq M$, we have
$\sum_{j=1}^{n}\Pi_{{}_{M,\bot}}(V_{j}q_{i})=0$. Hence,

$\left|\Pi_{{}_{M,\bot}}\big{(}Vq_{i}\psi_{i}\big{)}\right|\leq\frac{C|q_{i}|}{s^{\frac{M-i}{2m}+1}}\leq\frac{CA^{M}}{s^{\frac{M+2}{2m}}}.$

\- If $M-i=l\;\textup{mod}\;2m$ for some $l\in(1,2m)$, we take
$n=\frac{M-i-l}{2m}$ in the expansion (4.11) to deduce that

$\left|\Pi_{{}_{M,\bot}}\big{(}Vq_{i}\psi_{i}\big{)}\right|\leq\frac{C|q_{i}|}{s^{\frac{M-i-l}{2m}+1}}\leq\frac{CA^{M}}{s^{\frac{M+2}{2m}}}.$

This concludes the proof of Lemma 4.9. ∎

Concerning the control of $\Pi_{{}_{M,\bot}}(B(q))$, we claim the following.

######  Lemma 4.10 (Estimate of $\Pi_{{}_{M,\bot}}(B(q))$).

Under the assumption of Proposition 4.1, we have

$\left\|\frac{\Pi_{{}_{M,\bot}}(B(q))}{1+|y|^{M+1}}\right\|_{L^{\infty}(\mathbb{R})}\leq\frac{CA^{(M+2)^{2}}}{s^{\frac{M+1+\bar{p}}{2m}}}\quad\textup{with}\;\bar{p}=\min\\{2,p\\}.$
(4.21)

###### Proof.

Let $B_{M}$ be defined by

$B_{M}(y,s)=\Pi_{{}_{M,<}}\left[\sum_{j=2}^{M+1}q^{j}\sum_{l=0}^{M}\frac{1}{s^{l}}B_{j,l}(ys^{-\frac{1}{2m}})\right],$
(4.22)

where $\Pi_{{}_{M,<}}=\textup{Id}-\Pi_{{}_{M,\bot}}$ and $B_{j,l}$’s are
introduced in Lemma 4.5. Then we claim the following: for all $y\in\mathbb{R}$
and $s\geq 1$,

$\left|B(q)-B_{M}\right|\leq\frac{CA^{(M+2)^{2}}}{s^{\frac{M+1+\bar{p}}{2m}}}(1+|y|^{M+1}).$
(4.23)

The estimate (4.21) simply follows from (4.23) since
$\Pi_{{}_{M,\bot}}(B_{M})=0$. Let us prove (4.23). In the region $|y|\geq
s^{\frac{1}{2m}}$, we use (4.15) and the bounds given in Definition 3.1 to
estimate

$|B(q)|\leq\frac{CA^{\bar{p}(M+2)}}{s^{\frac{\bar{p}}{2m}}}\leq\frac{CA^{\bar{p}(M+2)}}{s^{\frac{M+1+\bar{p}}{2m}}}(1+|y|^{M+1})\quad\textup{for}\;\;|y|\geq
s^{\frac{1}{2m}}.$

In the region $|y|\leq s^{\frac{1}{2m}}$, we write from Lemma 4.5 the
expansion

$\displaystyle\left|B(q)-\Pi_{{}_{M,<}}\left(\sum_{j=2}^{M+1}q^{j}\sum_{l=0}^{M}\frac{1}{s^{l}}B_{j,l}(ys^{-\frac{1}{2m}})\right)\right|$
$\displaystyle\quad\leq\left|\Pi_{{}_{M,\bot}}\left(\sum_{j=2}^{M+1}q^{j}\sum_{l=0}^{M}\frac{1}{s^{l}}B_{j,l}(ys^{-\frac{1}{2m}})\right)\right|+\left|\sum_{j=2}^{M+1}q^{j}\sum_{l=0}^{M}\frac{1}{s^{l}}\tilde{B}_{j,l}(y,s)\right|+|q|^{M+2}+\frac{C}{s^{M+1}}.$

A similar argument as in the proof of Lemma 4.6 shows that the coefficient of
degree $k\geq M+1$ of the polynomial
$\sum_{j=0}^{M+1}q^{j}\sum_{l=0}^{M}\frac{1}{s^{l}}B_{j,l}(ys^{-\frac{1}{2m}})$
is bounded by $\frac{A^{k}}{s^{\frac{k+2}{2m}}}$, hence,

$\left|\Pi_{{}_{M,\bot}}\left(\sum_{j=2}^{M+1}q^{j}\sum_{l=0}^{M}\frac{1}{s^{l}}B_{j,l}(ys^{-\frac{1}{2m}})\right)\right|\leq\frac{CA^{2(M+1)}}{s^{\frac{M+3}{2m}}}(1+|y|^{M+1}),\quad\forall|y|\leq
s^{\frac{1}{2m}}.$

From (4.13) and Remark 3.2, we obtain

$\left|\sum_{j=2}^{M+1}q^{j}\sum_{l=0}^{M}\frac{1}{s^{l}}\tilde{B}_{j,l}(y,s)\right|\leq\frac{CA^{(M+2)(M+1)}}{s^{\frac{1}{m}+\frac{M+1}{2m}}}(1+|y|^{M+1})\leq\frac{CA^{(M+2)^{2}}}{s^{\frac{M+1+\bar{p}}{2m}}}(1+|y|^{M+1}),$

and

$|q|^{M+2}\leq
C\left(\frac{A^{M+2}}{s^{\frac{1}{2m}}}\right)^{M+1}\frac{A^{M+2}}{s^{1+\frac{1}{m}}}(1+|y|^{m+1})\leq\frac{CA^{(M+2)^{2}}}{s^{\frac{M+1+\bar{p}}{2m}}}(1+|y|^{M+1}).$

This completes the proof of (4.23) as well as the proof of Lemma 4.10. ∎

Plugging (4.19), (4.20) and (4.21) into equation (4.18) yields

$\partial_{s}q_{{}_{M,\bot}}=\mathscr{L}_{m}q_{{}_{M,\bot}}+G_{{}_{M,\bot}},$

where $G_{{}_{M,\bot}}$ satisfies for $s$ large enough,

$\left\|\frac{G_{{}_{M,\bot}}(y,s)}{1+|y|^{M+1}}\right\|_{L^{\infty}(\mathbb{R})}\leq\|V(s)\|_{L^{\infty}(\mathbb{R})}\left\|\frac{q_{{}_{M,\bot}}(y,s)}{1+|y|^{M+1}}\right\|_{L^{\infty}(\mathbb{R})}+\frac{CA^{M}}{s^{\frac{M+2}{2m}}}.$

Using the semigroup representation by $\mathscr{L}_{m}$, we write for all
$s\in[\tau,\tau_{1}]$,

$q_{{}_{M,\bot}}(s)=e^{(s-\tau)\mathscr{L}_{m}}q_{{}_{M,\bot}}(\tau)+\int_{\tau}^{s}e^{(s-s^{\prime})\mathscr{L}_{m}}G_{{}_{M,\bot}}(s^{\prime})ds^{\prime},$

where $e^{s\mathscr{L}_{m}}$ is defined in Lemma 2.3. Letting
$\lambda(s)=\left\|\frac{q_{{}_{M,\bot}}(s)}{1+|y|^{M+1}}\right\|_{L^{\infty}(\mathbb{R})}$
and using $(iii)$ of Lemma (2.3), we have

$\displaystyle\lambda(s)$ $\displaystyle\leq
e^{-\frac{M}{2m}(s-\tau)}\lambda(\tau)+\int_{\tau}^{s}e^{-\frac{M}{2m}(s-s^{\prime})}\left\|\frac{G_{{}_{M,\bot}}(s^{\prime})}{1+|y|^{M+1}}\right\|_{L^{\infty}(\mathbb{R})}ds^{\prime}$
$\displaystyle\quad\leq
e^{-\frac{M}{2m}(s-\tau)}\lambda(\tau)+\int_{\tau}^{s}e^{-\frac{M}{2m}(s-s^{\prime})}\left(\|V(s^{\prime})\|_{L^{\infty}(\mathbb{R})}\lambda(s^{\prime})+\frac{CA^{M}}{s^{\prime\frac{M+2}{2m}}}\right)ds^{\prime}.$

Since we have fixed $M$ large such that
$\|V\|_{L^{\infty}_{y,s}}\leq\frac{M}{4m}$ (see (3.11)), a standard Gronwall’s
argument applied to the function $e^{\frac{Ms}{2m}}\lambda(s)$ yields

$e^{\frac{Ms}{2m}}\lambda(s)\leq
e^{\frac{M(s-\tau)}{4m}}e^{\frac{M\tau}{2m}}\lambda(\tau)+Ce^{\frac{Ms}{2m}}\frac{A^{M}}{s^{\frac{M+2}{2}}},$

from which we conclude the proof of (4.4).

####  Control of $q_{e}$.

We write from (1.15) the equation satisfied by $q_{e}=(1-\chi(y,s))q$,

$\displaystyle\partial_{s}q_{e}$
$\displaystyle=\left(\mathscr{L}_{m}-\frac{p}{p-1}\right)q_{e}+(1-\chi)(Q+R)+E_{1}+E_{2},$

where $R$ is defined by (3.5),

$\displaystyle Q$
$\displaystyle=|q+\varphi|^{p-1}(q+\varphi)-\varphi^{p},\quad
E_{1}=-q\big{(}\partial_{s}\chi+\mathbf{A}_{m}\chi+\frac{1}{2m}y\cdot\nabla\chi\big{)},$
$\displaystyle E_{2}$ $\displaystyle=\mathbf{A}_{m}(\chi
q)+q\mathbf{A}_{m}\chi-\chi\mathbf{A}_{m}q=\sum_{j=1}^{2m-1}c_{j}\nabla^{j}\big{(}q\nabla^{2m-j}\chi\big{)}.$

Using the semigroup representation by $\mathscr{L}_{m}$ and items $(i)-(ii)$
of Lemma 2.3, we write for all $s\in[\tau,\tau_{1}]$,

$\displaystyle\|q_{e}(s)\|_{L^{\infty}}$ $\displaystyle\leq
e^{-\frac{s-\tau}{p-1}}\|q_{e}(\tau)\|_{L^{\infty}}+\int_{\tau}^{s}e^{-\frac{s-s^{\prime}}{p-1}}\left\|(1-\chi)(Q(s^{\prime})+R(s^{\prime}))+E_{1}(s^{\prime})\right\|_{L^{\infty}}ds^{\prime}$
$\displaystyle\quad+C\sum_{j=1}^{2m-1}\int_{\tau}^{s}e^{-\frac{s-s^{\prime}}{p-1}}\big{(}1-e^{(s-s^{\prime})}\big{)}^{-\frac{j}{2m}}\left\|q(s^{\prime})\nabla^{2m-j}\chi(s^{\prime})\right\|_{L^{\infty}}ds^{\prime}.$

For $K$ large enough, we have

$\|(1-\chi(y,s^{\prime})Q(s^{\prime}))\|_{L^{\infty}}\leq
C\|\varphi(s^{\prime})\|_{L^{\infty}}^{p-1}\|q_{e}(s^{\prime})\|_{L^{\infty}}\leq\frac{1}{2(p-1)}\|q_{e}(s^{\prime})\|_{L^{\infty}}.$

Recall from Lemma 4.7 that
$\|R(s^{\prime})\|_{L^{\infty}}\leq\frac{C}{s^{\prime}}$. As for $E_{1}$, we
use the definition of $\chi$ given in (3.6) and the bounds given in Definition
3.1 to obtain the estimate

$\|E_{1}(s^{\prime})\|_{L^{\infty}}\leq
C\|q(s^{\prime})\|_{L^{\infty}\big{(}Ks^{\prime\frac{1}{2m}}\leq|y|\leq
2Ks^{\prime\frac{1}{2m}}\big{)}}\leq\frac{CA^{M+1}}{s^{\prime\frac{1}{2m}}}.$

Since $\|\nabla^{2m-j}\chi(s^{\prime})\|_{L^{\infty}}\leq
Cs^{\prime-\frac{2m-j}{2m}}$, using the fact that $\nabla^{2m-j}\chi$ is
compactly supported, we estimate

$\sum_{j=1}^{2m-1}\left\|q(s^{\prime})\nabla^{2m-j}\chi(s^{\prime})\right\|_{L^{\infty}}\leq\frac{CA^{M+1}}{s^{\prime\frac{1}{m}}}.$

A collection of these above estimates yields

$\displaystyle\|q_{e}(s)\|_{L^{\infty}}$ $\displaystyle\leq
e^{-\frac{s-\tau}{p-1}}\|q_{e}(\tau)\|_{L^{\infty}}$
$\displaystyle+\int_{\tau}^{s}e^{-\frac{s-s^{\prime}}{p-1}}\left(\frac{1}{2(p-1)}\|q_{e}(s^{\prime})\|_{L^{\infty}}+\frac{CA^{M+1}}{s^{\prime\frac{1}{2m}}}+\frac{CA^{M+1}}{\Big{[}s^{\prime}\sqrt{1-e^{(s-s^{\prime})}}\,\Big{]}^{\frac{1}{m}}}\right)ds^{\prime}.$

Applying the standard Gronwall’s inequality to the function
$e^{\frac{s}{p-1}}\|q_{e}(s)\|_{L^{\infty}}$ yields

$e^{\frac{s}{p-1}}\|q_{e}(s)\|_{L^{\infty}}\leq
e^{\frac{s-\tau}{2(p-1)}}e^{\frac{\tau}{p-1}}\lambda(\tau)+\frac{CA^{M+1}}{s^{\frac{1}{2m}}}(s-\tau+1),$

from which the estimate (4.5) follows. This completes the proof of Proposition
4.1. $\square$

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