# Description of the lack of compactness in Orlicz spaces and applications

Ines Ben Ayed  Université de Tunis El Manar, Faculté des Sciences de Tunis,
LR03ES04 Équations aux dérivées partielles et applications, 2092 Tunis,
Tunisie<EMAIL_ADDRESS>and  Mohamed Khalil Zghal  Université de Tunis El
Manar, Faculté des Sciences de Tunis, LR03ES04 Équations aux dérivées
partielles et applications, 2092 Tunis, Tunisie<EMAIL_ADDRESS>
###### Abstract.

In this paper, we investigate the lack of compactness of the Sobolev embedding
of $H^{1}(\mathbb{R}^{2})$ into the Orlicz space
$L^{{\phi}_{p}}(\mathbb{R}^{2})$ associated to the function $\phi_{p}$ defined
by
$\phi_{p}(s):={\rm{e}^{s^{2}}}-\displaystyle\sum_{k=0}^{p-1}\frac{s^{2k}}{k!}\cdot$
We also undertake the study of a nonlinear wave equation with exponential
growth where the Orlicz norm $\|.\|_{L^{\phi_{p}}}$ plays a crucial role. This
study includes issues of global existence, scattering and qualitative study.

##  1\. Introduction

###  1.1. Critical $2D$ Sobolev embedding

It is well known (see for instance [7]) that $H^{1}(\mathbb{R}^{2})$ is
continuously embedded in all Lebesgue spaces $L^{q}(\mathbb{R}^{2})$ for
$2\leq q<\infty$, but not in $L^{\infty}(\mathbb{R}^{2})$. It is also known
that (for more details, we refer the reader to [21])

(1) $H^{1}(\mathbb{R}^{2})\hookrightarrow
L^{{\phi}_{p}}(\mathbb{R}^{2}),\quad\forall p\in\mathbb{N}^{*},$

where $L^{{\phi}_{p}}(\mathbb{R}^{2})$ denotes the Orlicz space associated to
the function

(2)
$\displaystyle\phi_{p}(s)=\rm{e}^{s^{2}}-\sum_{k=0}^{p-1}\frac{s^{2k}}{k!}\,\cdot$

The embedding (1) is a direct consequence of the following sharp Trudinger-
Moser type inequalities (see [1, 20, 22, 26]):

######  Proposition 1.1.

(3) $\sup_{\|u\|_{H^{1}}\leq
1}\;\;\displaystyle\int_{\mathbb{R}^{2}}\,\left({\rm
e}^{4\pi|u|^{2}}-1\right)\,dx:=\kappa<\infty,$

and states as follows:

(4) $\|u\|_{L^{\phi_{p}}}\leq\frac{1}{\sqrt{4\pi}}\|u\|_{H^{1}},$

where the norm $\|.\|_{L^{{\phi}_{p}}}$ is given by:

$\|u\|_{L^{\phi_{p}}}=\inf\,\left\\{\,\lambda>0,\int_{\mathbb{R}^{d}}\,\phi_{p}\left(\frac{|u(x)|}{\lambda}\right)\;dx\leq\kappa\,\right\\}.$

For our purpose, we shall resort to the following Trudinger-Moser inequality,
the proof of which is postponed in the appendix.

######  Proposition 1.2.

Let $\alpha\in[0,4\pi[$ and $p$ an integer larger than $1$. There is a
constant $c(\alpha,p)$ such that

(5) $\displaystyle\int_{\mathbb{R}^{2}}\left({\rm
e}^{\alpha|u(x)|^{2}}-\displaystyle\sum_{k=0}^{p-1}\frac{\alpha^{k}|u(x)|^{2k}}{k!}\right)\,dx\leq
c(\alpha,p)\|u\|_{L^{2p}(\mathbb{R}^{2})}^{2p},$

for all $u\in H^{1}(\mathbb{R}^{2})$ satisfying $\|\nabla
u\|_{L^{2}(\mathbb{R}^{2})}\leq 1$.

###  1.2. Development on the lack of compactness of Sobolev embedding in the
Orlicz space in the case $p=1$

In [3], [4] and [5], H. Bahouri, M. Majdoub and N. Masmoudi characterized the
lack of compactness of $H^{1}(\mathbb{R}^{2})$ into the Orlicz space
$L^{\phi_{1}}(\mathbb{R}^{2})$. To state their result in a clear way, let us
recall some definitions.

######  Definition 1.3.

We shall designate by a scale any sequence $(\alpha_{n})$ of positive real
numbers going to infinity, a core any sequence $(x_{n})$ of points in
$\mathbb{R}^{2}$ and a profile any function $\psi$ belonging to the set

${{\mathcal{P}}}:=\Big{\\{}\;\psi\in L^{2}(\mathbb{R},{\rm
e}^{-2s}ds);\;\;\;\psi^{\prime}\in
L^{2}(\mathbb{R}),\;\psi_{|]-\infty,0]}=0\,\Big{\\}}.$

Given two scales $(\alpha_{n})$, $(\tilde{\alpha}_{n})$, two cores $(x_{n})$,
$(\tilde{x}_{n})$ and tow profiles $\psi$, $\tilde{\psi}$, we say that the
triplets $\big{(}(\alpha_{n}),(x_{n}),\psi\big{)}$ and
$\big{(}(\tilde{\alpha}_{n}),(\tilde{x}_{n}),\tilde{\psi}\big{)}$ are
orthogonal if

$\mbox{either}\quad\quad\Big{|}\log\left(\tilde{\alpha}_{n}/{\alpha}_{n}\right)\Big{|}\to\infty,$

or $\tilde{\alpha}_{n}=\alpha_{n}$ and

$-\frac{\log|x_{n}-\tilde{x}_{n}|}{\alpha_{n}}\longrightarrow a\geq
0\,\,\mbox{with}\,\,\psi\,\,\mbox{or}\,\,{\tilde{\psi}}\,\,\mbox{null
for}\,\,s<a\,.$

######  Remarks 1.4.

  * •

The profiles belong to the Hölder space $C^{\frac{1}{2}}$. Indeed, for any
profile $\psi$ and real numbers $s$ and $t$, we have by Cauchy-Schwarz
inequality

$|\psi(s)-\psi(t)|=\left|\int_{s}^{t}\psi^{\prime}(\tau)\;d\tau\right|\leq\|\psi^{\prime}\|_{L^{2}(\mathbb{R})}|s-t|^{\frac{1}{2}}.$

  * •

Note also that (see [3])

(6) $\frac{\psi(s)}{\sqrt{s}}\rightarrow 0\quad as\quad s\rightarrow 0\quad
and\quad as\quad s\rightarrow\infty.$

The asymptotically orthogonal decomposition derived in [4] is formulated in
the following terms:

######  Theorem 1.5.

Let $(u_{n})$ be a bounded sequence in $H^{1}(\mathbb{R}^{2})$ such that

(7) $u_{n}\rightharpoonup 0,$ (8)
$\limsup_{n\to\infty}\|u_{n}\|_{L^{\phi_{1}}}=A_{0}>0\quad\quad\mbox{and}$ (9)
$\lim_{R\to\infty}\;\limsup_{n\to\infty}\,\|u_{n}\|_{L^{\phi_{1}}(|x|>R)}=0.$

Then, there exist a sequence of scales $({\alpha}_{n}^{(j)})$, a sequence of
cores $({x}_{n}^{(j)})$ and a sequence of profiles $(\psi^{(j)})$ such that
the triplets $({\alpha}_{n}^{(j)},{x}_{n}^{(j)},\psi^{(j)})$ are pairwise
orthogonal and, up to a subsequence extraction, we have for all $\ell\geq 1$,

(10)
$u_{n}(x)=\displaystyle\sum_{j=1}^{\ell}\,\sqrt{\frac{\alpha_{n}^{(j)}}{2\pi}}\;\psi^{(j)}\left(\frac{-\log|x-x_{n}^{(j)}|}{\alpha_{n}^{(j)}}\right)+{\rm
r}_{n}^{(\ell)}(x),\quad\limsup_{n\to\infty}\;\|{\rm
r}_{n}^{(\ell)}\|_{L^{\phi_{1}}}\stackrel{{\scriptstyle\ell\to\infty}}{{\longrightarrow}}0.$

Moreover, we have the following stability estimate

(11) $\|\nabla
u_{n}\|_{L^{2}}^{2}=\displaystyle\sum_{j=1}^{\ell}\,\|{\psi^{(j)}}^{\prime}\|_{L^{2}}^{2}+\|\nabla
r_{n}^{(\ell)}\|_{L^{2}}^{2}+\circ(1),\quad n\to\infty.$

######  Remarks 1.6.

  * •

It will be useful later on to point out that for any $q\geq 2$, we have

(12) $\|g_{n}^{(j)}\|_{L^{q}}\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}0,$

where $g_{n}^{(j)}$ is the elementary concentration involving in Decomposition
(10) defined by

(13)
$g_{n}^{(j)}(x):=\sqrt{\frac{\alpha_{n}^{(j)}}{2\pi}}\;\psi^{(j)}\left(\frac{-\log|x-x_{n}^{(j)}|}{\alpha_{n}^{(j)}}\right).$

Since the Lebesgue measure is invariant under translations, we have

$\|g_{n}^{(j)}\|_{L^{q}}^{q}=(2\pi)^{-\frac{q}{2}}(\alpha_{n}^{(j)})^{\frac{q}{2}}\int_{\mathbb{R}^{2}}\bigg{|}\psi^{(j)}\bigg{(}-\frac{\log|x|}{\alpha_{n}^{(j)}}\bigg{)}\bigg{|}^{q}dx.$

Performing the change of variable $s=-\frac{\log|x|}{\alpha_{n}^{(j)}}$,
yields

$\|g_{n}^{(j)}\|_{L^{q}}^{q}=(2\pi)^{1-\frac{q}{2}}(\alpha_{n}^{(j)})^{\frac{q}{2}+1}\int^{\infty}_{0}\big{|}\psi^{(j)}(s)\big{|}^{q}{\rm
e}^{-2\alpha_{n}^{(j)}s}\;ds.$

Fix $\varepsilon>0$. Then in view of (6), there exist two real numbers $s_{0}$
and $S_{0}$ such that $0<s_{0}<S_{0}$ and

$\left|\psi^{(j)}(s)\right|\leq\varepsilon\sqrt{s},\quad\forall\,s\in[0,s_{0}]\cup[S_{0},\infty[.$

This implies, by the change of variable $u=\alpha_{n}^{(j)}s$, that

$\displaystyle(\alpha_{n}^{(j)})^{\frac{q}{2}+1}\int_{0}^{s_{0}}\left|\psi^{(j)}(s)\right|^{q}{\rm
e}^{-2\alpha_{n}^{(j)}s}\;ds$ $\displaystyle\leq$
$\displaystyle{\varepsilon}^{q}\int_{0}^{\alpha_{n}^{(j)}s_{0}}u^{\frac{q}{2}}{\rm
e}^{-2u}\;du$ $\displaystyle\leq$ $\displaystyle C_{q}\,\varepsilon^{q}.$

In the same way, we obtain

$\displaystyle(\alpha_{n}^{(j)})^{\frac{q}{2}+1}\int_{S_{0}}^{\infty}\left|\psi^{(j)}(s)\right|^{q}{\rm
e}^{-2\alpha_{n}^{(j)}s}\;ds$ $\displaystyle\leq$ $\displaystyle
C_{q}\,\varepsilon^{q}.$

Finally taking advantage of the continuity of $\psi^{(j)}$, we deduce that

$\displaystyle(\alpha_{n}^{(j)})^{\frac{q}{2}+1}\int_{s_{0}}^{S_{0}}\left|\psi^{(j)}(s)\right|^{q}{\rm
e}^{-2\alpha_{n}^{(j)}s}\;ds$ $\displaystyle\lesssim$
$\displaystyle(\alpha_{n}^{(j)})^{\frac{q}{2}+1}\int_{s_{0}}^{S_{0}}{\rm
e}^{-2\alpha_{n}^{(j)}s}\;ds$ $\displaystyle\lesssim$
$\displaystyle(\alpha_{n}^{(j)})^{\frac{q}{2}}\left({\rm
e}^{-2\alpha_{n}^{(j)}s_{0}}-{\rm
e}^{-2\alpha_{n}^{(j)}S_{0}}\right)\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}0,$

which ends the proof of the assertion (12).

  * •

Recall that it was proved in [5] that

$\|g_{n}^{(j)}\|_{L^{\phi_{1}}}\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}\frac{1}{\sqrt{4\pi}}\,\max_{s>0}\;\frac{|\psi^{(j)}(s)|}{\sqrt{s}}\,$

and

(14)
$\big{\|}\displaystyle\sum_{j=1}^{\ell}\,g_{n}^{(j)}\big{\|}_{L^{\phi_{1}}}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\sup_{1\leq
j\leq\ell}\,\left(\lim_{n\to\infty}\,\|g_{n}^{(j)}\|_{L^{\phi_{1}}}\right)\,,$

in the case when the scales $(\alpha_{n}^{(j)})_{1\leq j\leq\ell}$ are
pairwise orthogonal. Note that Property (14) does not necessarily remain true
in the case when we have the same scales and the pairwise orthogonality of the
couples $\big{(}(x^{(j)}_{n}),\psi^{(j)}\big{)}$ (see Lemma $3.6$ in [5]).

###  1.3. Study of the lack of compactness of Sobolev embedding in the Orlicz
space in the case $p>1$

Our first goal in this paper is to describe the lack of compactness of the
Sobolev embedding (1) for $p>1$. Our result states as follows:

######  Theorem 1.7.

Let $p>1$ be an integer larger than $1$ and $(u_{n})$ be a bounded sequence in
$H^{1}(\mathbb{R}^{2})$ such that

(15) $u_{n}\rightharpoonup 0,$ (16)
$\limsup_{n\to\infty}\|u_{n}\|_{L^{\phi_{p}}}=A_{0}>0\quad\quad\mbox{and}$
(17)
$\lim_{R\to\infty}\;\limsup_{n\to\infty}\,\|u_{n}\|_{L^{\phi_{p}}(|x|>R)}=0.$

Then, there exist a sequence of scales $({\alpha}_{n}^{(j)})$, a sequence of
cores $({x}_{n}^{(j)})$ and a sequence of profiles $(\psi^{(j)})$ such that
the triplets $({\alpha}_{n}^{(j)},{x}_{n}^{(j)},\psi^{(j)})$ are pairwise
orthogonal in the sense of Definition 1.3 and, up to a subsequence extraction,
we have for all $\ell\geq 1$,

(18)
$u_{n}(x)=\displaystyle\sum_{j=1}^{\ell}\,\sqrt{\frac{\alpha_{n}^{(j)}}{2\pi}}\;\psi^{(j)}\left(\frac{-\log|x-x_{n}^{(j)}|}{\alpha_{n}^{(j)}}\right)+{\rm
r}_{n}^{(\ell)}(x),$

with $\displaystyle\limsup_{n\to\infty}\;\|{\rm
r}_{n}^{(\ell)}\|_{L^{\phi_{p}}}\stackrel{{\scriptstyle\ell\to\infty}}{{\longrightarrow}}0.$
Moreover, we have the following stability estimate

(19) $\|\nabla
u_{n}\|_{L^{2}}^{2}=\displaystyle\sum_{j=1}^{\ell}\,\|{\psi^{(j)}}^{\prime}\|_{L^{2}}^{2}+\|\nabla
r_{n}^{(\ell)}\|_{L^{2}}^{2}+\circ(1),\quad n\to\infty.$

######  Remarks 1.8.

  * •

Arguing as in [5], we can easily prove that

(20) $\|g_{n}\|_{L^{\phi_{p}}}\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}\frac{1}{\sqrt{4\pi}}\,\max_{s>0}\;\frac{|\psi(s)|}{\sqrt{s}},$

where

$g_{n}(x):=\sqrt{\frac{\alpha_{n}}{2\pi}}\;\psi\left(\frac{-\log|x-x_{n}|}{\alpha_{n}}\right)\cdot$

Indeed setting
$L=\displaystyle\liminf_{n\rightarrow\infty}\|g_{n}\|_{L^{\phi_{p}}}$, we have
for fixed $\varepsilon>0$ and $n$ sufficiently large (up to subsequence
extraction)

$\displaystyle\int_{\mathbb{R}^{2}}\Big{(}{\rm
e}^{\big{|}\frac{g_{n}(x+x_{n})}{L+\varepsilon}\big{|}^{2}}-\displaystyle\sum_{k=0}^{p-1}\displaystyle\frac{|g_{n}(x+x_{n})|^{2k}}{(L+\varepsilon)^{2k}k!}\Big{)}\,dx\leq\kappa.$

Therefore,

$\displaystyle\int_{\mathbb{R}^{2}}\Big{(}{\rm
e}^{\big{|}\frac{g_{n}(x+x_{n})}{L+\varepsilon}\big{|}^{2}}-1\Big{)}\,dx\lesssim\kappa+\displaystyle\sum_{k=1}^{p-1}\|g_{n}\|_{L^{2k}}^{2k},$

which implies in view of (12) that

$\displaystyle\int_{\mathbb{R}^{2}}\Big{(}{\rm
e}^{\big{|}\frac{g_{n}(x+x_{n})}{L+\varepsilon}\big{|}^{2}}-1\Big{)}\,dx=2\pi\displaystyle\int_{0}^{+\infty}\alpha_{n}{\rm
e}^{2\alpha_{n}s\Big{[}\frac{1}{4\pi(L+\varepsilon)^{2}}\big{(}\frac{\psi(s)}{\sqrt{s}}\big{)}^{2}-1\Big{]}}\,ds-\pi\lesssim
1.$

Using the fact that $\psi$ is a continuous function, we deduce that

$L+\varepsilon\geq\frac{1}{\sqrt{4\pi}}\displaystyle\max_{s>0}\displaystyle\frac{|\psi(s)|}{\sqrt{s}},$

which ensures that

$L\geq\frac{1}{\sqrt{4\pi}}\displaystyle\max_{s>0}\displaystyle\frac{|\psi(s)|}{\sqrt{s}}\cdot$

To end the proof of (20), it suffices to establish that for any $\delta>0$

$\displaystyle\int_{\mathbb{R}^{2}}\Big{(}{\rm
e}^{\big{|}\frac{g_{n}(x+x_{n})}{\lambda}\big{|}^{2}}-\displaystyle\sum_{k=0}^{p-1}\displaystyle\frac{|g_{n}(x+x_{n})|^{2k}}{(\lambda)^{2k}k!}\Big{)}\,dx\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}0,$

where
$\lambda=\frac{1+\delta}{\sqrt{4\pi}}\displaystyle\max_{s>0}\frac{|\psi(s)|}{\sqrt{s}}\cdot$
Since

$\displaystyle\int_{\mathbb{R}^{2}}\Big{(}{\rm
e}^{\big{|}\frac{g_{n}(x+x_{n})}{\lambda}\big{|}^{2}}-\displaystyle\sum_{k=0}^{p-1}\displaystyle\frac{|g_{n}(x+x_{n})|^{2k}}{(\lambda)^{2k}k!}\Big{)}\,dx\leq\displaystyle\int_{\mathbb{R}^{2}}\Big{(}{\rm
e}^{\big{|}\frac{g_{n}(x+x_{n})}{\lambda}\big{|}^{2}}-1\Big{)}\,dx,$

the result derives immediately from Proposition $1.15$ in [5], which achieves
the proof of the result.

  * •

Applying the same lines of reasoning as in the proof of Proposition 1.19 in
[5], we obtain the following result:

######  Proposition 1.9.

Let $\big{(}(\alpha_{n}^{(j)}),(x_{n}^{(j)}),\psi^{(j)}\big{)}_{1\leq
j\leq\ell}$ be a family of triplets of scales, cores and profiles such that
the scales are pairwise orthogonal. Then for any integer $p$ larger than $1$,
we have

$\Big{\|}\displaystyle\sum_{j=1}^{\ell}\,g_{n}^{(j)}\Big{\|}_{L^{\phi_{p}}}\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}\sup_{1\leq
j\leq\ell}\,\left(\lim_{n\to\infty}\,\big{\|}g_{n}^{(j)}\big{\|}_{L^{{\phi}_{p}}}\right)\;,$

where the functions $g^{(j)}_{n}$ are defined by (13).

As we will see in Section 2, it turns out that the heart of the matter in the
proof of Theorem 1.7 is reduced to the following result concerning the radial
case:

######  Theorem 1.10.

Let $p$ be an integer strictly larger than $1$ and $(u_{n})$ be a bounded
sequence in $H_{rad}^{1}(\mathbb{R}^{2})$ such that

(21) $u_{n}\rightharpoonup 0\quad\quad\mbox{and}$ (22)
$\limsup_{n\to\infty}\|u_{n}\|_{L^{\phi_{p}}}=A_{0}>0.$

Then, there exist a sequence of pairwise orthogonal scales
$({\alpha}_{n}^{(j)})$ and a sequence of profiles $(\psi^{(j)})$ such that up
to a subsequence extraction, we have for all $\ell\geq 1$,

(23)
$u_{n}(x)=\displaystyle\sum_{j=1}^{\ell}\,\sqrt{\frac{\alpha_{n}^{(j)}}{2\pi}}\;\psi^{(j)}\left(\frac{-\log|x|}{\alpha_{n}^{(j)}}\right)+{\rm
r}_{n}^{(\ell)}(x),\quad\limsup_{n\to\infty}\;\|{\rm
r}_{n}^{(\ell)}\|_{L^{\phi_{p}}}\stackrel{{\scriptstyle\ell\to\infty}}{{\longrightarrow}}0.$

Moreover, we have the following stability estimate

$\|\nabla
u_{n}\|_{L^{2}}^{2}=\displaystyle\sum_{j=1}^{\ell}\,\|{\psi^{(j)}}^{\prime}\|_{L^{2}}^{2}+\|\nabla
r_{n}^{(\ell)}\|_{L^{2}}^{2}+\circ(1),\quad n\to\infty.$

######  Remarks 1.11.

  * •

Compared with the analogous result concerning the Sobolev embedding of
$H_{rad}^{1}(\mathbb{R}^{2})$into $L^{\phi_{1}}$ established in [5], the
hypothesis of compactness at infinity is not required. This is justified by
the fact that $H^{1}_{rad}(\mathbb{R}^{2})$ is compactly embedded in
$L^{q}(\mathbb{R}^{2})$ for any $2<q<\infty$ which implies that

(24)
$\displaystyle\lim_{n\rightarrow\infty}\|u_{n}\|_{L^{q}(\mathbb{R}^{2})}=0,\quad\forall\,2<q<\infty.$

  * •

In view of Proposition 1.9, Theorem 1.10 yields to

$\|u_{n}\|_{L^{\phi_{p}}}\to\sup_{j\geq
1}\,\left(\lim_{n\to\infty}\,\|g_{n}^{(j)}\|_{L^{\phi_{p}}}\right),$

which implies that the first profile in Decomposition (23) can be chosen such
that up to extraction

(25)
$A_{0}:=\displaystyle\limsup_{n\rightarrow\infty}\|u_{n}\|_{L^{\phi_{p}}}=\displaystyle\lim_{n\rightarrow\infty}\left\|\sqrt{\frac{\alpha_{n}^{(1)}}{2\pi}}\psi^{(1)}\left(-\frac{\log|x|}{\alpha_{n}^{(1)}}\right)\right\|_{L^{\phi_{p}}}.$

Note that the description of the lack of compactness in other critical Sobolev
embeddings was achieved in [8, 10, 14] and has been at the origin of several
prospectus. Among others, one can mention [2, 6, 9, 11, 19].

###  1.4. Layout of the paper

Our paper is organized as follows: in Section 2, we establish the algorithmic
construction of the decomposition stated in Theorem 1.7. Then, we study in
Section 3 a nonlinear two-dimensional wave equation with the exponential
nonlinearity $u\,\phi_{p}(\sqrt{4\pi}u)$. Firstly, we prove the global well-
posedness and the scattering in the energy space both in the subcritical and
critical cases, and secondly we compare the evolution of this equation with
the evolution of the solutions of the free Klein-Gordon equation in the same
space.

We mention that $C$ will be used to denote a constant which may vary from line
to line. We also use $A\lesssim B$ to denote an estimate of the form $A\leq
CB$ for some absolute constant $C$ and $A\approx B$ if $A\lesssim B$ and
$B\lesssim A$. For simplicity, we shall also still denote by $(u_{n})$ any
subsequence of $(u_{n})$ and designate by $\circ(1)$ any sequence which tends
to $0$ as $n$ goes to infinity.

##  2\. Proof of Theorem 1.7

###  2.1. Strategy of the proof

The proof of Theorem 1.7 uses in a crucial way capacity arguments and is done
in three steps: in the first step, we begin by the study of $u^{\ast}_{n}$ the
symmetric decreasing rearrangement of $u_{n}$. This led us to establish
Theorem 1.10. In the second step, by a technical process developed in [4], we
reduce ourselves to one scale and extract the first core $(x_{n}^{(1)})$ and
the first profile $\psi^{(1)}$ which enables us to extract the first element
$\sqrt{\frac{\alpha_{n}^{(1)}}{2\pi}}\;\psi^{(1)}\left(\frac{-\log|x-x_{n}^{(1)}|}{\alpha_{n}^{(j)}}\right)$.
The third step is devoted to the study of the remainder term. If the limit of
its Orlicz norm is null we stop the process. If not, we prove that this
remainder term satisfies the same properties as the sequence we start with
which allows us to extract a second elementary concentration concentrated
around a second core $(x_{n}^{(2)})$. Thereafter, we establish the property of
orthogonality between the first two elementary concentrations and finally we
prove that this process converges.

###  2.2. Proof of Theorem 1.10

The main ingredient in the proof of Theorem 1.10 consists to extract a scale
and a profile $\psi$ such that

(26) $\|\psi^{\prime}\|_{L^{2}(\mathbb{R})}\geq CA_{0},$

where $C$ is a universal constant. To go to this end, let us for a bounded
sequence $(u_{n})$ in $H^{1}_{rad}(\mathbb{R}^{2})$ satisfying the assumptions
(21) and (22), set $v_{n}(s)=u_{n}({\rm e}^{-s})$. Combining (24) with the
following well-known radial estimate:

$|u(r)|\leq\frac{C}{r^{\frac{1}{p+1}}}\|u\|_{L^{2p}}^{\frac{p}{p+1}}\|\nabla
u\|_{L^{2}}^{\frac{1}{p+1}}$

where $r=|x|$, we infer that

(27)
$\displaystyle\lim_{n\rightarrow\infty}\|v_{n}\|_{L^{\infty}(]-\infty,M])}=0,\quad\forall
M\in\mathbb{R}.$

This gives rise to the following result:

######  Proposition 2.1.

For any $\delta>0$, we have

(28) $\sup_{s\geq
0}\left(\Big{|}\frac{v_{n}(s)}{A_{0}-\delta}\Big{|}^{2}-s\right)\to\infty,\quad
n\to\infty.$

###### Proof.

We proceed by contradiction. If not, there exists $\delta>0$ such that, up to
a subsequence extraction

(29) $\sup_{s\geq
0,n\in\mathbb{N}}\;\;\left(\Big{|}\frac{v_{n}(s)}{A_{0}-\delta}\Big{|}^{2}-s\right)\leq
C<\infty.$

On the one hand, thanks to (27) and (29), we get by virtue of Lebesgue theorem

$\displaystyle\int_{|x|<1}\;\left({\rm
e}^{|\frac{u_{n}(x)}{A_{0}-\delta}|^{2}}-\displaystyle\sum_{k=0}^{p-1}\frac{|u_{n}(x)|^{2k}}{(A_{0}-\delta)^{2k}k!}\right)\,dx$
$\displaystyle\leq$ $\displaystyle\int_{|x|<1}\;\left({\rm
e}^{|\frac{u_{n}(x)}{A_{0}-\delta}|^{2}}-1\right)\,dx$ $\displaystyle\leq$
$\displaystyle 2\pi\,\int_{0}^{\infty}\;\left({\rm
e}^{|\frac{v_{n}(s)}{A_{0}-\delta}|^{2}}-1\right)\,{\rm
e}^{-2s}\,ds\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}0.$

On the other hand, using Property (27) and the simple fact that for any
positive real number $M$, there exists a finite constant $C_{M,p}$ such that

$\sup_{|t|\leq M}\,\left(\frac{{\rm
e}^{t^{2}}-\sum_{k=0}^{p-1}\frac{t^{2k}}{k!}}{t^{2p}}\right)<C_{M,p},$

we deduce in view of (24) that

$\int_{|x|\geq 1}\;\left({\rm
e}^{|\frac{u_{n}(x)}{A_{0}-\delta}|^{2}}-\displaystyle\sum_{k=0}^{p-1}\frac{|u_{n}(x)|^{2k}}{(A_{0}-\delta)^{2k}k!}\right)\,dx\lesssim\|u_{n}\|_{L^{2p}}^{2p}\to
0\,.$

Consequently,

$\displaystyle\limsup_{n\to\infty}\,\|u_{n}\|_{L^{\phi_{p}}}\leq
A_{0}-\delta,$

which is in contradiction with Hypothesis (22). ∎

An immediate consequence of the previous proposition is the following
corollary whose proof is identical to the proof of Corollaries 2.4 and 2.5 in
[5].

######  Corollary 2.2.

Under the above notations, there exists a sequence $(\alpha_{n}^{(1)})$ in
$\mathbb{R}_{+}$ tending to infinity such that

(30)
$4\,\Big{|}\frac{v_{n}(\alpha_{n}^{(1)})}{A_{0}}\Big{|}^{2}-\alpha_{n}^{(1)}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\infty$

and for $n$ sufficiently large, there exists a positive constant $C$ such that

(31) $\frac{A_{0}}{2}\sqrt{\alpha_{n}^{(1)}}\leq|v_{n}(\alpha_{n}^{(1)})|\leq
C\sqrt{\alpha_{n}^{(1)}}+\circ(1).$

Now, setting

$\psi_{n}(y)=\sqrt{\frac{2\pi}{\alpha_{n}^{(1)}}}v_{n}(\alpha_{n}^{(1)}y),$

we obtain along the same lines as in Lemma 2.6 in [5] the following result:

######  Lemma 2.3.

Under notations of Corollary 2.2, there exists a profile
$\psi^{(1)}\in\mathcal{P}$ such that, up to a subsequence extraction

$\psi_{n}^{\prime}\rightharpoonup(\psi^{(1)})^{\prime}\;in\;L^{2}(\mathbb{R})\quad
and\quad\|(\psi^{(1)})^{\prime}\|_{L^{2}}\geq\sqrt{\frac{\pi}{2}}A_{0}.$

To achieve the proof of Theorem 1.10, let us consider the remainder term

(32) $r_{n}^{(1)}(x)=u_{n}(x)-g_{n}^{(1)}(x),$

where

$g_{n}^{(1)}(x)=\sqrt{\frac{\alpha_{n}^{(1)}}{2\pi}}\psi^{(1)}\left(\frac{-\log|x|}{\alpha_{n}^{(1)}}\right).$

By straightforward computations, we can easily prove that $(r^{(1)}_{n})$ is
bounded in $H^{1}_{rad}(\mathbb{R}^{2})$ and satisfies the hypothesis (21)
together with the following property:

(33) $\displaystyle\lim_{n\rightarrow\infty}\|\nabla
r_{n}^{(1)}\|_{L^{2}(\mathbb{R}^{2})}^{2}=\underset{n\rightarrow\infty}{\lim}\|\nabla
u_{n}\|_{L^{2}(\mathbb{R}^{2})}^{2}-\big{\|}(\psi^{(1)})^{\prime}\big{\|}_{L^{2}(\mathbb{R})}^{2}.$

Let us now define
$A_{1}=\underset{n\rightarrow\infty}{\limsup}\|r_{n}^{(1)}\|_{L^{\phi_{p}}}$.
If $A_{1}=0$, we stop the process. If not, arguing as above, we prove that
there exist a scale $(\alpha_{n}^{(2)})$ satisfying the statement of Corollary
2.2 with $A_{1}$ instead of $A_{0}$ and a profile $\psi^{(2)}$ in
$\mathcal{P}$ such that

$r_{n}^{(1)}(x)=\sqrt{\frac{\alpha_{n}^{(2)}}{2\pi}}\psi^{(2)}\left(\frac{-\log|x|}{\alpha_{n}^{(2)}}\right)+r_{n}^{(2)}(x),$

with $\|(\psi^{(2)})^{\prime}\|_{L^{2}}\geq\frac{\sqrt{2\pi}}{2}A_{1}$ and

$\displaystyle\lim_{n\rightarrow\infty}\|\nabla
r_{n}^{(2)}\|_{L^{2}(\mathbb{R}^{2})}^{2}=\underset{n\rightarrow\infty}{\lim}\|\nabla
r_{n}^{(1)}\|_{L^{2}(\mathbb{R}^{2})}^{2}-\big{\|}(\psi^{(2)})^{\prime}\big{\|}_{L^{2}(\mathbb{R})}^{2}.$

Moreover, as in [5] we can show that $(\alpha_{n}^{(1)})$ and
$(\alpha_{n}^{(2)})$ are orthogonal. Finally, iterating the process, we get at
step $\ell$

$u_{n}(x)=\displaystyle\sum_{j=1}^{\ell}\,\sqrt{\frac{\alpha_{n}^{(j)}}{2\pi}}\;\psi^{(j)}\left(\frac{-\log|x|}{\alpha_{n}^{(j)}}\right)+{\rm
r}_{n}^{(\ell)}(x),$

with

$\limsup_{n\to\infty}\,\|r^{(\ell)}_{n}\|_{H^{1}}^{2}\lesssim
1-A_{0}^{2}-A_{1}^{2}-\cdots-A_{\ell-1}^{2}\,,$

which implies that $A_{\ell}\to 0$ as $\ell\to\infty$ and ends the proof of
the theorem.

###  2.3. Extraction of the cores and profiles

This step is performed as the proof of Theorem 1.16 in [3]. We sketch it here
briefly for the convenience of the reader. Let $u_{n}^{\ast}$ be the symmetric
decreasing rearrangement of $u_{n}$. Since $u^{\ast}_{n}\in
H^{1}_{rad}(\mathbb{R}^{2})$ and satisfies the assumptions of Theorem 1.10, we
infer that there exist a sequence $(\alpha_{n}^{(j)})$ of pairwise orthogonal
scales and a sequence of profiles $(\varphi^{(j)})$ such that, up to
subsequence extraction,

$u_{n}^{*}(x)=\displaystyle\sum_{j=1}^{\ell}\,\sqrt{\frac{\alpha_{n}^{(j)}}{2\pi}}\;\varphi^{(j)}\left(\frac{-\log|x|}{\alpha_{n}^{(j)}}\right)+{\rm
r}_{n}^{(\ell)}(x),\quad\limsup_{n\to\infty}\;\|{\rm
r}_{n}^{(\ell)}\|_{L^{\phi_{p}}}\stackrel{{\scriptstyle\ell\to\infty}}{{\longrightarrow}}0.$

Besides, in view of (25), we can assume that

$A_{0}=\underset{n\rightarrow\infty}{\lim}\left\|\sqrt{\frac{\alpha_{n}^{(1)}}{2\pi}}\varphi^{(1)}\left(-\frac{\log|x|}{\alpha_{n}^{(1)}}\right)\right\|_{L^{{\Phi}_{p}}}.$

Now to extract the cores and profiles, we shall firstly reduce to the case of
one scale according to Section 2.3 in [4], where a suitable truncation of
$u_{n}$ was introduced. Then assuming that

$u_{n}^{*}(x)=\sqrt{\frac{\alpha_{n}^{(1)}}{2\pi}}\;\varphi^{(1)}\left(\frac{-\log|x|}{\alpha_{n}^{(1)}}\right),$

we apply the strategy developed in Section 2.4 in [4] to extract the cores and
the profiles. This approach is based on capacity arguments: to carry out the
extraction process of mass concentrations, we prove by contradiction that if
the mass responsible for the lack of compactness of the Sobolev embedding in
the Orlicz space is scattered, then the energy used would exceed that of the
starting sequence. This main point can be formulated in the following terms:

######  Lemma 2.4 ( Lemma 2.5 in [4]).

There exist $\delta_{0}>0$ and $N_{1}\in\mathbb{N}$ such that for any $n\geq
N_{1}$ there exists $x_{n}$ such that

(34) $\frac{|E_{n}\cap
B(x_{n},\rm{e}^{-b\alpha_{n}^{(1)}})|}{|E_{n}|}\geq\delta_{0}A_{0}^{2},$

where
$E_{n}:=\\{x\in\mathbb{R}^{2};|u_{n}(x)|\geq\sqrt{2\alpha_{n}^{(1)}}(1-\frac{\varepsilon_{0}}{10})A_{0}\\}$
with $0<\varepsilon_{0}<\frac{1}{2}$, $B(x_{n},\rm{e}^{-b\,\alpha_{n}^{(1)}})$
designates the ball of center $x_{n}$ and radius
$\rm{e}^{-b\,\alpha_{n}^{(1)}}$ with $b=1-2\varepsilon_{0}$ and $|.|$ denotes
the Lebesgue measure.

Once extracting the first core $(x_{n}^{(1)})$ making use of the previous
lemma, we focus on the extraction of the first profile. For that purpose, we
consider the sequence

$\psi_{n}(y,\theta)=\sqrt{\frac{2\pi}{\alpha_{n}^{(1)}}}v_{n}(\alpha_{n}^{(1)}y,\theta),$

where
$v_{n}(s,\theta)=(\tau_{x_{n}^{(1)}}u_{n})(\rm{e}^{-s}\cos\theta,\rm{e}^{-s}\sin\theta)$
and $(x_{n}^{(1)})$ satisfies

$\frac{|E_{n}\cap
B(x_{n},\rm{e}^{-(1-2\varepsilon_{0})\alpha_{n}^{(1)}}|}{|E_{n}|}\geq\delta_{0}A_{0}^{2}.$

Taking advantage of the invariance of Lebesgue measure under translations, we
deduce that

$\displaystyle\|\nabla u_{n}\|_{L^{2}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{\mathbb{R}}\int_{0}^{2\pi}|\partial_{y}\psi_{n}(y,\theta)|^{2}dyd\theta$
$\displaystyle+$
$\displaystyle\frac{\alpha_{n}^{(1)}}{2\pi}\int_{\mathbb{R}}\int_{0}^{2\pi}|\partial_{\theta}\psi_{n}(y,\theta)|^{2}dyd\theta.$

Since the scale $\alpha_{n}^{(1)}$ tends to infinity and the sequence
$(u_{n})$ is bounded in $H^{1}(\mathbb{R}^{2})$, this implies that up to a
subsequence extraction
$\partial_{\theta}\psi_{n}\underset{n\rightarrow\infty}{\rightarrow}0$ and
$\partial_{y}\psi_{n}\underset{n\rightarrow\infty}{\rightharpoonup}g$ in
$L^{2}(\mathbb{R}\times[0,2\pi])$, where $g$ only depends on the variable $y$.
Thus introducing the function

$\psi^{(1)}(y)=\int_{0}^{y}g(\tau)d\tau,$

we obtain along the same lines as in Proposition 2.8 in [4] the following
result:

######  Proposition 2.5.

The function $\psi^{(1)}$ belongs to the set of profiles $\mathcal{P}$.
Besides for any $y\in\mathbb{R}$, we have

(35)
$\frac{1}{2\pi}\int^{2\pi}_{0}\psi_{n}(y,\theta)\,d\theta\to\psi^{(1)}(y),$

as $n$ tends to infinity and there exists an absolute constant $C$ such that

(36) $\|{\psi^{(1)}}^{\prime}\|_{L^{2}}\geq C\,A_{0}.$

###  2.4. End of the proof

To achieve the proof of the theorem, we argue exactly as in Section 2.5 in [4]
by iterating the process exposed in the previous section. For that purpose, we
set

$r_{n}^{(1)}(x)=u_{n}(x)-g_{n}^{(1)}(x),$

where

$g_{n}^{(1)}(x)=\sqrt{\frac{\alpha_{n}^{(1)}}{2\pi}}\psi^{(1)}\left(-\frac{\log|x-x_{n}^{(1)}|}{\alpha^{(1)}_{n}}\right).$

One can easily check that the sequence $(r_{n}^{(1)})$ weakly converges to $0$
in $H^{1}(\mathbb{R}^{2})$. Moreover, since $\psi^{(1)}_{|]-\infty,0]}=0$, we
have for any $R\geq 1$

(37) $\|r_{n}^{(1)}\|_{L^{\Phi_{p}}(|x-x_{n}^{(1)}|\geq
R)}=\|u_{n}\|_{L^{\Phi_{p}}(|x-x_{n}^{(1)}|\geq R)}.$

But by assumption, the sequence $(u_{n})$ is compact at infinity in the Orlicz
space $L^{\Phi_{p}}$. Thus the core $(x_{n}^{(1)})$ is bounded in
$\mathbb{R}^{2}$, which ensures in view of (37) that $(r_{n}^{(1)})$ satisfies
the hypothesis of compactness at infinity (17). Finally, taking advantage of
the weak convergence of $(\partial_{y}\psi_{n})$ to ${\psi^{(1)}}^{\prime}$ in
$L^{2}(y,\theta)$ as $n$ goes to infinity, we get

$\underset{n\rightarrow\infty}{\lim}\|\nabla
r_{n}^{(1)}\|_{L^{2}}^{2}=\underset{n\rightarrow\infty}{\lim}\|\nabla
u_{n}^{(1)}\|_{L^{2}}^{2}-\|{\psi^{(1)}}^{\prime}\|_{L^{2}}^{2}.$

Now, let us define
$A_{1}:=\underset{n\rightarrow\infty}{\limsup}\|r_{n}^{(1)}\|_{L^{\Phi_{p}}}$.
If $A_{1}=0$, we stop the process. If not, knowing that $(r_{n}^{(1)})$
verifies the assumptions of Theorem 1.7, we apply the above reasoning, which
gives rise to the existence of a scale $(\alpha_{n}^{(2)})$, a core
$(x^{(2)}_{n})$ satisfying the statement of Lemma 2.4 with $A_{1}$ instead of
$A_{0}$ and a profile $\psi^{(2)}$ in $\mathcal{P}$ such that

$r_{n}^{(1)}(x)=\sqrt{\frac{\alpha_{n}^{(2)}}{2\pi}}\psi^{(2)}\left(-\frac{\log|x-x_{n}^{(2)}|}{\alpha^{(2)}_{n}}\right)+r_{n}^{(2)}(x),$

with $\|{\psi^{(2)}}^{\prime}\|_{L^{2}}\geq C\,A_{1}$ and

$\underset{n\rightarrow\infty}{\lim}\|\nabla
r_{n}^{(2)}\|_{L^{2}}^{2}=\underset{n\rightarrow\infty}{\lim}\|\nabla
r_{n}^{(1)}\|_{L^{2}}^{2}-\|{\psi^{(2)}}^{\prime}\|_{L^{2}}^{2}.$

Arguing as in [4], we show that the triplets
$\big{(}\alpha_{n}^{(1)},x_{n}^{(1)},\psi^{(1)}\big{)}$ and
$\big{(}\alpha_{n}^{(2)},x_{n}^{(2)},\psi^{(2)}\big{)}$ are orthogonal in the
sense of Definition 1.3 and prove that the process of extraction of the
elementary concentration converges. This ends the proof of Decomposition (10).
The orthogonality equality (11) derives immediately from Proposition 2.10 in
[4]. The proof of Theorem 1.7 is then achieved.

##  3\. Nonlinear wave equation

###  3.1. Statement of the results

In this section, we investigate the initial value problem for the following
nonlinear wave equation:

(41) $\displaystyle\left\\{\begin{array}[]{lll}\square u+u+u\,\left({\rm
e}^{4\pi
u^{2}}-\displaystyle\sum_{k=0}^{p-1}\frac{(4\pi)^{k}u^{2k}}{k!}\right)=0,\\\
\\\ u(0)=u_{0}\in H^{1}(\mathbb{R}^{2}),\quad\partial_{t}u(0)=u_{1}\in
L^{2}(\mathbb{R}^{2}),\end{array}\right.$

where $p\geq 1$ is an integer, $u=u(t,x)$ is a real-valued function of
$(t,x)\in\mathbb{R}\times\mathbb{R}^{2}$ and $\square=\partial^{2}_{t}-\Delta$
is the wave operator.

Let us recall that in [17, 18], the authors proved the global well-posedness
for the Cauchy problem (41) when $p=1$ and the scattering when $p=2$ in the
subcritical and critical cases (i.e when the energy is less or equal to some
threshold). Note also that in [24, 25], M. Struwe constructed global smooth
solutions to (41) with smooth data of arbitrary size in the case $p=1$.

Formally, the solutions of the Cauchy problem (41) satisfy the following
conservation law:

$\displaystyle\quad\quad E_{p}(u,t)$ $\displaystyle:=$
$\displaystyle\|\partial_{t}u(t)\|_{L^{2}}^{2}+\|\nabla
u(t)\|_{L^{2}}^{2}+\frac{1}{4\pi}\left\|{\rm e}^{4\pi
u(t)^{2}}-1-\displaystyle\sum_{k=2}^{p}\frac{(4\pi)^{k}}{k!}u(t)^{2k}\right\|_{L^{1}}$
$\displaystyle=$ $\displaystyle E_{p}(u,0):=E_{p}^{0}.$

This conducts us, as in [17], to define the notion of criticality in terms of
the size of the initial energy $E_{p}^{0}$ with respect to $1$.

######  Definition 3.1.

The Cauchy problem (41) is said to be subcritical if

$E_{p}^{0}<1.$

It is said to be critical if $E_{p}^{0}=1$ and supercritical if $E_{p}^{0}>1$.

We shall prove the following result:

######  Theorem 3.2.

Assume that $E_{p}^{0}\leq 1.$ Then the Cauchy problem (41) has a unique
global solution $u$ in the space

$\mathcal{C}(\mathbb{R},H^{1}(\mathbb{R}^{2}))\cap\mathcal{C}^{1}(\mathbb{R},L^{2}(\mathbb{R}^{2})).$

Moreover, $u\in L^{4}(\mathbb{R},\mathcal{C}^{1/4})$ and scatters.

###  3.2. Technical tools

The proof of Theorem 3.2 is based on priori estimates. This requires the
control of the nonlinear term

(43) $F_{p}(u):=u\,\left({\rm e}^{4\pi
u^{2}}-\displaystyle\sum_{k=0}^{p-1}\frac{(4\pi)^{k}u^{2k}}{k!}\right)$

in $L^{1}_{t}(L^{2}_{x})$. To achieve our goal, we will resort to Strichartz
estimates for the 2D Klein-Gordon equation. These estimates, proved in [15],
state as follows:

######  Proposition 3.3.

Let $T>0$ and $(q,r)\in[4,\infty]\times[2,\infty]$ an admissible pair, i.e

$\frac{1}{q}+\frac{2}{r}=1.$

Then,

(44)
$\|v\|_{L^{q}([0,T],{\mathrm{B}}^{1}_{r,2}(\mathbb{R}^{2}))}\lesssim\Big{[}\|v(0)\|_{H^{1}(\mathbb{R}^{2})}+\|\partial_{t}v(0)\|_{L^{2}(\mathbb{R}^{2})}+\|\square
v+v\|_{L^{1}([0,T],L^{2}(\mathbb{R}^{2}))}\Big{]},$

where ${\mathrm{B}}^{1}_{r,2}(\mathbb{R}^{2})$ stands for the usual
inhomogeneous Besov space (see for example [12] or [23] for a detailed
exposition on Besov spaces).

Noticing that $(q,r)=(4,8/3)$ is an admissible pair and recalling that

${\mathrm{B}}^{1}_{8/3,2}(\mathbb{R}^{2})\hookrightarrow\mathcal{C}^{1/4}(\mathbb{R}^{2}),$

we deduce that

(45)
$\|v\|_{L^{4}([0,T],\mathcal{C}^{1/4}(\mathbb{R}^{2}))}\lesssim\Big{[}\|v(0)\|_{H^{1}(\mathbb{R}^{2})}+\|\partial_{t}v(0)\|_{L^{2}(\mathbb{R}^{2})}+\|\square
v+v\|_{L^{1}([0,T],L^{2}(\mathbb{R}^{2}))}\Big{]}.$

To control the nonlinear term $F_{p}(u)$ in $L^{1}_{t}(L^{2}_{x})$, we will
make use of the following logarithmic inequalities proved in [16, Theorem
1.3].

######  Proposition 3.4.

For any $\lambda>\frac{2}{\pi}$ and any $0<\mu\leq 1$, a constant
$C_{\lambda}>0$ exists such that for any function $u$ in
$H^{1}(\mathbb{R}^{2})\cap\mathcal{C}^{1/4}(\mathbb{R}^{2})$, we have

(46)
$\|u\|^{2}_{L^{\infty}}\leq\lambda\|u\|_{\mu}^{2}\log\left(C_{\lambda,\mu}+\frac{2\|u\|_{{\mathcal{C}}^{1/4}}}{\|u\|_{\mu}}\,\right),$

where $\|u\|_{\mu}^{2}:=\|\nabla u\|_{L^{2}}^{2}+\mu^{2}\|u\|_{L^{2}}^{2}$.

###  3.3. Proof of Theorem 3.2

The proof of this result, divided into three steps, is inspired from the
proofs of Theorems $1.8$, $1.11$, $1.12$ in [17] and Theorem $1.3$ in [18].

####  3.3.1. Local existence

Let us start by proving the local existence to the Cauchy problem (41) in the
case where $\|\nabla u_{0}\|_{L^{2}(\mathbb{R}^{2})}<1$. To do so, we use a
standard fixed-point argument and introduce for any nonnegative time $T$ the
following space:

$\mathcal{E}_{T}=\mathcal{C}([0,T],H^{1}(\mathbb{R}^{2}))\cap\mathcal{C}^{1}([0,T],L^{2}(\mathbb{R}^{2}))\cap
L^{4}([0,T],\mathcal{C}^{1/4}(\mathbb{R}^{2}))$

endowed with the norm

$\|u\|_{T}:=\underset{0\leq t\leq
T}{\sup}\Big{[}\|u(t)\|_{H^{1}}+\|\partial_{t}u(t)\|_{L^{2}}\Big{]}+\|u\|_{L^{4}([0,T],\mathcal{C}^{1/4})}.$

For a positive time $T$ and a positive real number $\delta$, we denote by
$\mathcal{E}_{T}(\delta)$ the ball in the space $\mathcal{E}_{T}$ of radius
$\delta$ and centered at the origin. On this ball, we define the map $\Phi$ by

$v\longmapsto\Phi(v)=\widetilde{v},$

where

$\square\widetilde{v}+\widetilde{v}=-F_{p}(v+v_{0}),\quad\widetilde{v}(0)=\partial_{t}\widetilde{v}(0)=0$

and $v_{0}$ is the solution of the free Klein-Gordon equation

$\square v_{0}+v_{0}=0,\quad v_{0}(0)=u_{0},\quad
and\quad\partial_{t}v_{0}(0)=u_{1}.$

Now, the goal is to show that if $\delta$ and $T$ are small enough, then the
map $\Phi$ is well-defined from $\mathcal{E}_{T}(\delta)$ into itself and it
is a contraction. To prove that $\Phi$ is well-defined, it suffices in view of
the Strichartz estimates (44) to estimate $F_{p}(v+v_{0})$ in the space
$L^{1}([0,T],L^{2}(\mathbb{R}^{2}))$. Arguing as in [17] and using the Hölder
inequality and the Sobolev embedding, we obtain for any $\epsilon>0$

$\displaystyle\displaystyle\int_{\mathbb{R}^{2}}|F_{p}(v+v_{0})|^{2}\;dx$
$\displaystyle\leq$
$\displaystyle\displaystyle\int_{\mathbb{R}^{2}}|F_{1}(v+v_{0})|^{2}\;dx$
$\displaystyle\lesssim$ $\displaystyle\|v+v_{0}\|_{H^{1}}^{2}\,{\rm
e}^{4\pi\|v+v_{0}\|_{L^{\infty}}^{2}}\left\|{\rm
e}^{4\pi(v+v_{0})^{2}}-1\right\|_{L^{1+\epsilon}}.$

Since $\|\nabla u_{0}\|_{L^{2}}<1$, we can choose $\mu>0$ such that
$\|u_{0}\|_{\mu}<1$. Since $v_{0}$ is continuous in time, there exist a time
$T_{0}$ and a constant $0<c<1$ such that for any $t$ in $[0,T_{0}]$ we have

$\|v_{0}(t)\|_{\mu}\leq c.$

According to Proposition 3.4, we infer that

${\rm
e}^{4\pi\|v+v_{0}\|_{L^{\infty}}^{2}}\lesssim\left(1+\frac{\|v+v_{0}\|_{\mathcal{C}^{1/4}}}{\delta+c}\right)^{8\eta},$

for some $0<\eta<1$. Besides, applying the Trudinger-Moser inequality (5) for
$p=1$, the fact that

$4\pi(1+\epsilon)(\delta+c)^{2}\longrightarrow 4\pi
c<4\pi\quad\mbox{as}\;\epsilon,\,\delta\rightarrow
0\quad\mbox{and}\quad\left\|\nabla\left(\frac{v+v_{0}}{\delta+c}\right)\right\|_{L^{2}}\leq
1$

ensures that

$\displaystyle\left\|{\rm
e}^{4\pi(v+v_{0})^{2}}-1\right\|_{L^{1+\epsilon}}^{1+\epsilon}$
$\displaystyle\leq$ $\displaystyle C_{\epsilon}\left\|{\rm
e}^{4\pi(1+\epsilon)(v+v_{0})^{2}}-1\right\|_{L^{1}}$ $\displaystyle\leq$
$\displaystyle C_{\epsilon,\delta}\|v+v_{0}\|_{L^{2}}^{2}$ $\displaystyle\leq$
$\displaystyle
C_{\epsilon,\delta}(1+\|u_{0}\|_{H^{1}}+\|u_{1}\|_{L^{2}})^{2}.$

Therefore, for any $0<T\leq T_{0}$, we obtain that

$\|F_{p}(v+v_{0})\|_{L^{1}([0,T],L^{2}(\mathbb{R}^{2}))}\lesssim
T^{1-\eta}(1+\|u_{0}\|_{H^{1}}+\|u_{1}\|_{L^{2}})^{4\eta}.$

Now, to prove that $\Phi$ is a contraction (at least for $T$ small), let us
consider two elements $v_{1}$ and $v_{2}$ in $\mathcal{E}_{T}(\delta)$. Notice
that, for any $\epsilon>0$,

$\displaystyle|F_{p}(v_{1}+v_{0})-F_{p}(v_{2}+v_{0})|$ $\displaystyle=$
$\displaystyle|v_{1}-v_{2}|(1+8\pi\overline{v}^{2})\left({\rm
e}^{4\pi\overline{v}^{2}}-\displaystyle\sum_{k=0}^{p-2}\frac{(4\pi)^{k}\overline{v}^{2k}}{k!}\right)$
$\displaystyle\leq$ $\displaystyle C_{\epsilon}|v_{1}-v_{2}|\left({\rm
e}^{4\pi(1+\epsilon)\overline{v}^{2}}-1\right),$

where $\overline{v}=(1-\theta)(v_{0}+v_{1})+\theta(v_{0}+v_{2}),$ for some
$\theta=\theta(t,x)\in[0,1].$ Using a convexity argument, we get

$\displaystyle|F_{p}(v_{1}+v_{0})-F_{p}(v_{2}+v_{0})|$ $\displaystyle\leq$
$\displaystyle C_{\epsilon}\left|(v_{1}-v_{2})\left({\rm
e}^{4\pi(1+\epsilon)(v_{1}+v_{0})^{2}}-1\right)\right|$ $\displaystyle+$
$\displaystyle C_{\epsilon}\left|(v_{1}-v_{2})\left({\rm
e}^{4\pi(1+\epsilon)(v_{2}+v_{0})^{2}}-1\right)\right|.$

This implies, in view of Strichartz estimates (45), that

$\displaystyle\|\Phi(v_{1})-\Phi(v_{2})\|_{T}$ $\displaystyle\lesssim$
$\displaystyle\|F_{p}(v_{1}+v_{0})-F_{p}(v_{2}+v_{0})\|_{L^{1}([0,T],L^{2}(\mathbb{R}^{2}))}$
$\displaystyle\leq$ $\displaystyle
C_{\epsilon}\displaystyle\int_{0}^{T}\left\|(v_{1}-v_{2})\left({\rm
e}^{4\pi(1+\epsilon)(v_{1}+v_{0})^{2}}-1\right)\right\|_{L^{2}}\,dt$
$\displaystyle+$ $\displaystyle
C_{\epsilon}\displaystyle\int_{0}^{T}\left\|(v_{1}-v_{2})\left({\rm
e}^{4\pi(1+\epsilon)(v_{2}+v_{0})^{2}}-1\right)\right\|_{L^{2}}\,dt,$

which leads along the same lines as above to

$\displaystyle\|\Phi(v_{1})-\Phi(v_{2})\|_{T}$ $\displaystyle\lesssim$
$\displaystyle
T^{1-(1+\epsilon)\eta}(1+\|u_{0}\|_{H^{1}}+\|u_{1}\|_{L^{2}})^{4(1+\epsilon)\eta}\|v_{1}-v_{2}\|_{T}.$

If the parameter $\epsilon$ is small enough, then $(1+\epsilon)\eta<1$ and
therefore, for $T$ small enough, $\Phi$ is a contraction map. This implies the
uniqueness of the solution in $v_{0}+\mathcal{E}_{T}(\delta)$.
Now, we shall prove the uniqueness in the energy space. The idea here is to
establish that, if $u=v_{0}+v$ is a solution of (41) in
$\mathcal{C}([0,T],H^{1}(\mathbb{R}^{2}))\cap\mathcal{C}^{1}([0,T],L^{2}(\mathbb{R}^{2}))$,
then necessarily $v\in\mathcal{E}_{T}(\delta)$ at least for $T$ small.
Starting from the fact that $v$ satisfies

$\square v+v=-F_{p}(v+v_{0}),\quad v(0)=\partial_{t}v(0)=0,$

we are reduced, thanks to the Strichartz estimates (44), to control the term
$F_{p}(v+v_{0})$ in the space $L^{1}([0,T],L^{2}(\mathbb{R}^{2})).$ But
$|F_{p}(v+v_{0})|\leq|F_{1}(v+v_{0})|$, which leads to the result arguing
exactly as in [17].

####  3.3.2. Global existence

In this section, we shall establish that our solution is global in time both
in subcritical and critical cases. Firstly, let us notice that the assumption
$E_{p}^{0}\leq 1$ implies that $\|\nabla u_{0}\|_{L^{2}(\mathbb{R}^{2})}<1$,
which ensures in view of Section 3.3.1 the existence of a unique maximal
solution $u$ defined on $[0,T^{*})$ where $0<T^{*}\leq\infty$ is the lifespan
of $u$. We shall proceed by contradiction assuming that $T^{*}<\infty$. In the
subcritical case, the conservation law (3.1) implies that

$\displaystyle\sup_{t\in(0,T^{*})}\|\nabla u(t)\|_{L^{2}(\mathbb{R}^{2})}<1.$

Let then $0<s<T^{*}$ and consider the following Cauchy problem:

(47) $\square v+v+F_{p}(v)=0,\quad
v(s)=u(s),\quad\mbox{and}\quad\partial_{t}v(s)=\partial_{t}u(s).$

As in the first step of the proof, a fixed-point argument ensures the
existence of $\tau>0$ and a unique solution $v$ to (47) on the interval
$[s,s+\tau]$. Noticing that $\tau$ does not depend on $s$, we can choose $s$
close to $T^{*}$ such that $T^{*}-s<\tau$. So, we can prolong the solution $u$
after the time $T^{*}$, which is a contradiction.
In the critical case, we cannot apply the previous argument because it is
possible that the following concentration phenomenon holds:

(48) $\displaystyle\limsup_{t\rightarrow T^{*}}\|\nabla
u(t)\|_{L^{2}(\mathbb{R}^{2})}=1.$

In fact, we shall show that (48) cannot hold in this case. To go to this end,
we argue as in the proof of Theorem $1.12$ in [17]. Firstly, since the first
equation of the Cauchy problem (41) is invariant under time translation, we
can assume that $T^{*}=0$ and that the initial time is $t=-1$. Similarly to
[17, Proposition 4.2, Corollary 4.4], it follows that the maximal solution $u$
satisfies

(49) $\displaystyle\limsup_{t\rightarrow 0^{-}}\|\nabla
u(t)\|_{L^{2}(\mathbb{R}^{2})}=1,$ (50) $\displaystyle\lim_{t\rightarrow
0^{-}}\|u(t)\|_{L^{2}(\mathbb{R}^{2})}=0,$ (51)
$\displaystyle\lim_{t\rightarrow
0^{-}}\displaystyle\int_{|x-x^{*}|\leq-t}|\nabla
u(t,x)|^{2}\;dx=1,\quad\mbox{and}$ (52) $\forall
t<0,\quad\displaystyle\int_{|x-x^{*}|\leq-t}e_{p}(u)(t,x)\;dx=1,$

for some $x^{*}\in\mathbb{R}^{2}$, where $e_{p}(u)$ denotes the energy density
defined by

$e_{p}(u)(t,x):=(\partial_{t}u)^{2}+|\nabla u|^{2}+\frac{1}{4\pi}\left({\rm
e}^{4\pi
u^{2}}-1-\displaystyle\sum_{k=2}^{p}\displaystyle\frac{(4\pi)^{k}u^{2k}}{k!}\right).$

Without loss of generality, we can assume that $x^{*}=0$, then multiplying the
equation of the problem (41) respectively by $\partial_{t}u$ and $u$, we
obtain formally

(53) $\partial_{t}e_{p}(u)-div_{x}(2\partial_{t}u\nabla u)=0,$ (54)
$\partial_{t}(u\partial_{t}u)-div_{x}(u\nabla u)+|\nabla
u|^{2}-|\partial_{t}u|^{2}+u^{2}{\rm e}^{4\pi
u^{2}}-\displaystyle\sum_{k=1}^{p-1}\displaystyle\frac{(4\pi)^{k}u^{2k+2}}{k!}=0.$

Integrating the conservation laws (53) and (54) over the backward truncated
cone

$K_{S}^{T}:=\Big{\\{}(t,x)\in\mathbb{R}\times\mathbb{R}^{2}\;\mbox{ such
that}\;S\leq t\leq T\;\mbox{and}\;|x|\leq-t\Big{\\}}$

for $S<T<0$, we get

(55)
$\displaystyle\int_{B(-T)}e_{p}(u)(T,x)\;dx-\displaystyle\int_{B(-S)}e_{p}(u)(S,x)\;dx$
$=\frac{-1}{\sqrt{2}}\displaystyle\int_{M_{S}^{T}}\left[\left|\partial_{t}u\frac{x}{|x|}+\nabla
u\right|^{2}+\frac{1}{4\pi}\left({\rm e}^{4\pi
u^{2}}-1-\displaystyle\sum_{k=2}^{p}\frac{(4\pi)^{k}u^{2k}}{k!}\right)\;dx\,dt\right],$
(56)
$\displaystyle\int_{B(-T)}\partial_{t}u(T)u(T)\;dx-\displaystyle\int_{B(-S)}\partial_{t}u(S)u(S)\;dx+\frac{1}{\sqrt{2}}\displaystyle\int_{M_{S}^{T}}\left(\partial_{t}u+\nabla
u.\frac{x}{|x|}\right)u\;dx\,dt$ $+\displaystyle\int_{K_{S}^{T}}\left(|\nabla
u|^{2}-|\partial_{t}u|^{2}+u^{2}{\rm e}^{4\pi
u^{2}}-\displaystyle\sum_{k=1}^{p-1}\displaystyle\frac{(4\pi)^{k}u^{2k+2}}{k!}\right)\;dx\,dt=0,$

where $B(r)$ is the ball centered at $0$ and of radius $r$ and

$M_{S}^{T}:=\Big{\\{}(t,x)\in\mathbb{R}\times\mathbb{R}^{2}\;\mbox{ such
that}\;S\leq t\leq T\;\mbox{and}\;|x|=-t\Big{\\}}.$

According to (52) and (55), we infer that

$\displaystyle\int_{M_{S}^{T}}\left[\left|\partial_{t}u\frac{x}{|x|}+\nabla
u\right|^{2}+\frac{1}{4\pi}\left({\rm e}^{4\pi
u^{2}}-1-\displaystyle\sum_{k=2}^{p}\frac{(4\pi)^{k}u^{2k}}{k!}\right)\right]\;dx\,dt=0.$

This implies, using (56) and Cauchy-Schwarz inequality, that

(57)
$\displaystyle\int_{B(-T)}\partial_{t}u(T)u(T)\;dx-\displaystyle\int_{B(-S)}\partial_{t}u(S)u(S)\;dx$
$+\displaystyle\int_{K_{S}^{T}}\left(|\nabla
u|^{2}-|\partial_{t}u|^{2}+u^{2}{\rm e}^{4\pi
u^{2}}-\displaystyle\sum_{k=1}^{p-1}\displaystyle\frac{(4\pi)^{k}u^{2k+2}}{k!}\right)\;dx\,dt=0,$

By virtue of Identities (49) and (50) and the conservation law (3.1), it can
be seen that

(58) $\partial_{t}u(t)\underset{t\rightarrow
0}{\longrightarrow}0\quad\mbox{in}\;L^{2}(\mathbb{R}^{2}),$

which ensures by Cauchy-Schwarz inequality that

(59) $\displaystyle\int_{B(-T)}\partial_{t}u(T)u(T)\;dx\rightarrow 0.$

Letting $T\rightarrow 0$ in (57), we deduce from (59) and the fact that
$u^{2}{\rm e}^{4\pi
u^{2}}-\displaystyle\sum_{k=1}^{p-1}\displaystyle\frac{(4\pi)^{k}u^{2k+2}}{k!}$
is positive

(60)
$-\displaystyle\int_{B(-S)}\partial_{t}u(S)u(S)\;dx\leq-\displaystyle\int_{K_{S}^{0}}|\nabla
u|^{2}\,dx\,dt+\displaystyle\int_{K_{S}^{0}}|\partial_{t}u|^{2}\,dx\,dt.$

Multiplying Inequality (60) by the positive number $-\frac{1}{S}$, we deduce
that

(61)
$\displaystyle\int_{B(-S)}\partial_{t}u(S)\frac{u(S)}{S}\;dx\leq\frac{1}{S}\displaystyle\int_{K_{S}^{0}}|\nabla
u|^{2}\;dx\;dt-\frac{1}{S}\displaystyle\int_{K_{S}^{0}}|\partial_{t}u|^{2}\;dx\;dt.$

Now, Identity (58) leads to

(62) $\displaystyle\lim_{S\rightarrow
0^{-}}\frac{1}{S}\displaystyle\int_{K_{S}^{0}}|\partial_{t}u|^{2}\;dx\;dt=0.$

Moreover, using (51), it is clear that

(63) $\displaystyle\lim_{S\rightarrow
0^{-}}\frac{1}{S}\displaystyle\int_{K_{S}^{0}}|\nabla u|^{2}\;dx\;dt=-1.$

Finally, since

$\frac{u(S)}{S}=\frac{1}{S}\int_{0}^{S}\partial_{t}u(\tau)d\tau,$

then $(\frac{u(S)}{S})$ is bounded in $L^{2}(\mathbb{R}^{2})$ and hence

(64) $\displaystyle\lim_{S\rightarrow
0^{-}}\displaystyle\int_{B(-S)}\partial_{t}u(S)\frac{u(S)}{S}\;dx=0.$

The identities (62), (63) and (64) yield a contradiction in view of (61). This
achieves the proof of the global existence in the critical case.

####  3.3.3. Scattering

Our concern now is to prove that, in the subcritical and critical cases, the
solution of the equation (41) approaches a solution of a free wave equation
when the time goes to infinity. Using the fact that

(65) $|F_{p}(u)|\leq|F_{2}(u)|,\quad\forall p\geq 2,$

we can apply the arguments used in [18]. More precisely, in the subcritical
case the key point consists to prove that there exists an increasing function
$C:[0,1[\longrightarrow[0,\infty[$ such that for any $0\leq E<1$, any global
solution $u$ of the Cauchy problem (41) with $E_{p}(u)\leq E$ satisfies

(66) $\|u\|_{X(\mathbb{R})}\leq C(E),$

where $X(\mathbb{R})=L^{8}(\mathbb{R},L^{16}(\mathbb{R}^{2}))$. Now, denoting
by

$E^{*}:=\sup\Big{\\{}0\leq E<1;\;\displaystyle\sup_{E_{p}(u)\leq
E}\|u\|_{X(\mathbb{R})}<\infty\Big{\\}},$

and arguing as in [18, Lemma 4.1], we can show that Inequality (66) is
satisfied if $E_{p}(u)$ is small, which implies that $E^{*}>0$. Now our goal
is to prove that $E^{*}=1$. To do so, let us proceed by contradiction and
assume that $E^{*}<1$. Then, for any $E\in]E^{*},1[$ and any $n>0$, there
exists a global solution $u$ to (41) such that $E_{p}(u)\leq E$ and
$\|u\|_{X(\mathbb{R})}>n$. By time translation, one can reduce to

(67) $\|u\|_{X(]0,\infty[)}>\frac{n}{2}.$

Along the same lines as the proof of Proposition 5.1 in [18], we can show
taking advantage of (65) that if $E$ is close enough to $E^{*}$, then $n$
cannot be arbitrarily large which yields a contradiction and ends the proof of
the result in the subcritical case.
The proof of the scattering in the critical case is done as in Section 6 in
[18] once we observed Inequality (65). It is based on the notion of
concentration radius $r_{\epsilon}(t)$ introduced in [18].

###  3.4. Qualitative study

In this section we shall investigate the feature of solutions of the two-
dimensional nonlinear Klein-Gordon equation (41) taking into account the
different regimes. As in [5], the approach that we adopt here is the one
introduced by P. Gérard in [13] which consists in comparing the evolution of
oscillations and concentration effects displayed by sequences of solutions of
the nonlinear Klein-Gordon equation (41) and solutions of the free linear
Klein-Gordon equation.

(68) $\displaystyle\square v+v=0.$

More precisely, let $(\varphi_{n},\psi_{n})$ be a sequence of data in
$H^{1}\times L^{2}$ supported in some fixed ball and satisfying

(69) $\varphi_{n}\rightharpoonup
0\quad\mbox{in}\;H^{1},\quad\psi_{n}\rightharpoonup 0\quad\mbox{in}\;L^{2},$

such that

(70) $E_{p}^{n}\leq 1,\quad n\in\mathbb{N}$

where $E_{p}^{n}$ stands for the energy of $(\varphi_{n},\psi_{n})$ given by

$E_{p}^{n}=\|\psi_{n}\|_{L^{2}}^{2}+\|\nabla\varphi_{n}\|_{L^{2}}^{2}+\frac{1}{4\pi}\;\Big{\|}{\rm
e}^{4\pi\varphi_{n}^{2}}-1-\displaystyle\sum_{k=2}^{p}\frac{(4\pi)^{k}}{k!}\varphi_{n}^{2k}\Big{\|}_{L^{1}},$

and let us consider $(u_{n})$ and $(v_{n})$ the sequences of finite energy
solutions of (41) and (68) such that

$(u_{n},\partial_{t}u_{n})(0)=(v_{n},\partial_{t}v_{n})(0)=(\varphi_{n},\psi_{n}).$

Arguing as in [13], the notion of linearizability is defined as follows:

######  Definition 3.5.

Let $T$ be a positive time. We shall say that the sequence $(u_{n})$ is
linearizable on $[0,T]$, if

$\displaystyle\sup_{t\in[0,T]}E_{c}(u_{n}-v_{n},t)\longrightarrow
0\quad\mbox{as}\quad n\rightarrow\infty,$

where $E_{c}(w,t)$ denotes the kinetic energy defined by:

$E_{c}(w,t)=\displaystyle\int_{{\mathbb{R}}^{2}}\left[|\partial_{t}w|^{2}+|\nabla_{x}w|^{2}+|w|^{2}\right](t,x)\;dx.$

For any time slab $I\subset\mathbb{R}$, we shall denote

$\|v\|_{\mbox{\tiny ST}(I)}:=\sup_{(q,r)\;\mbox{\tiny
admissible}}\;\|v\|_{L^{q}(I;{\mathrm{B}}^{1}_{r,2}(\mathbb{R}^{2}))}\,.$

By interpolation argument, this Strichartz norm is equivalent to

$\|v\|_{L^{\infty}(I;H^{1}(\mathbb{R}^{2}))}+\|v\|_{L^{4}(I;{\mathrm{B}}^{1}_{8/3,2}(\mathbb{R}^{2}))}\,.$

As ${\mathrm{B}}^{1}_{r,2}(\mathbb{R}^{2})\hookrightarrow
L^{p}(\mathbb{R}^{2})$ for all $r\leq p<\infty$ (and $r\leq p\leq\infty$ if
$r>2$), it follows that

(71) $\|v\|_{L^{q}(I;L^{p})}\lesssim\|v\|_{\mbox{\tiny
ST}(I)},\quad\frac{1}{q}+\frac{2}{p}\leq 1\,.$

As in [5], in the subcritical case, i.e
$\displaystyle\limsup_{n\rightarrow\infty}\;E_{p}^{n}<1$, the nonlinearity
does not induce any effect on the behavior of the solutions. But, in the
critical case i.e $\displaystyle\limsup_{n\rightarrow\infty}\;E_{p}^{n}=1$, it
turns out that a nonlinear effect can be produced. More precisely, we have the
following result:

######  Theorem 3.6.

Let $T$ a strictly positive time. Then

  1. (1)

If $\underset{n\rightarrow\infty}{\limsup}\,E_{p}^{n}<1$, the sequence
$(u_{n})$ is linearizable on $[0,T]$.

  2. (2)

If $\underset{n\rightarrow\infty}{\limsup}\,E_{p}^{n}=1$, the sequence
$(u_{n})$ is linearizable on $[0,T]$ provided that the sequence $(v_{n})$
satisfies

(72)
$\displaystyle\limsup_{n\to\infty}\;\|v_{n}\|_{L^{\infty}([0,T];{L^{\Phi_{p}}})}<\frac{1}{\sqrt{4\pi}}\cdot$

###### Proof.

The proof of Theorem 3.6 is similar to the one of Theorems 3.3 and 3.5 in [5].
Denoting by $w_{n}=u_{n}-v_{n}$, it is clear that $w_{n}$ is the solution of
the nonlinear wave equation

$\square w_{n}+w_{n}=-F_{p}(u_{n})$

with null Cauchy data.
Under energy estimate, we obtain

$\|w_{n}\|_{T}\lesssim\|F_{p}(u_{n})\|_{L^{1}([0,T],L^{2}(\mathbb{R}^{2}))},$

where $\|w_{n}\|^{2}_{T}\buildrel\hbox{\footnotesize
def}\over{=}\sup_{t\in[0,T]}E_{c}(w_{n},t)$. Therefore, it suffices to prove
in the subcritical and critical cases that

(73) $\|F_{p}(u_{n})\|_{L^{1}([0,T],L^{2}(\mathbb{R}^{2}))}\longrightarrow
0\quad\mbox{as}\quad n\rightarrow\infty.$

Let us begin by the subcritical case. Our goal is to prove that the nonlinear
term does not affect the behavior of the solutions. By hypothesis, there
exists some nonnegative real $\rho$ such that
$\displaystyle\limsup_{n\rightarrow\infty}E_{p}^{n}=1-\rho$. The main point
for the proof is based on the following lemma, the proof of which is similar
to the proof of Lemma 3.16 in [5] once we observed Inequality (65).

######  Lemma 3.7.

For every $T>0$ and $E^{0}_{p}<1$, there exists a constant $C(T,E^{0}_{p})$,
such that every solution $u$ of the nonlinear Klein-Gordon equation (41) of
energy $E_{p}(u)\leq E^{0}_{p}$, satisfies

(74) $\displaystyle\|u\|_{L^{4}([0,T];{{\mathcal{C}}}^{1/4})}\leq
C(T,E^{0}_{p}).$

Now to establish (73), it suffices to prove that the sequence $(F_{p}(u_{n}))$
is bounded in $L^{1+\epsilon}([0,T],L^{2+\epsilon}(\mathbb{R}^{2}))$ for some
nonnegative $\epsilon$ and converges to $0$ in measure in
$[0,T]\times\mathbb{R}^{2}$. This can be done exactly as in [5] using the fact
that $|F_{p}(u_{n})|\leq|F_{1}(u_{n})|$.

Let us now prove (73) in the critical case. For that purpose, let $T>0$ and
assume that

(75)
$L:=\limsup_{n\to\infty}\;\|v_{n}\|_{L^{\infty}([0,T];{L^{\Phi_{p}}})}<\frac{1}{\sqrt{4\pi}}\cdot$

Applying Taylor’s formula, we obtain

$F_{p}(u_{n})=F_{p}(v_{n}+w_{n})=F_{p}(v_{n})+F_{p}^{\prime}(v_{n})\,w_{n}+\frac{1}{2}\;F_{p}^{\prime\prime}(v_{n}+\theta_{n}\,w_{n})\,w_{n}^{2},$

for some $0\leq\theta_{n}\leq 1$. Strichartz estimates (44) yields

$\|w_{n}\|_{\mbox{\tiny ST}([0,T])}\lesssim I_{n}+J_{n}+K_{n},$

where

$\displaystyle I_{n}$ $\displaystyle=$
$\displaystyle\|F_{p}(v_{n})\|_{L^{1}([0,T];L^{2}(\mathbb{R}^{2}))},$
$\displaystyle J_{n}$ $\displaystyle=$
$\displaystyle\|F_{p}^{\prime}(v_{n})\,w_{n}\|_{L^{1}([0,T];L^{2}(\mathbb{R}^{2}))},\quad\mbox{and}$
$\displaystyle K_{n}$ $\displaystyle=$
$\displaystyle\|F_{p}^{\prime\prime}(v_{n}+\theta_{n}\,w_{n})\,w_{n}^{2}\|_{L^{1}([0,T];L^{2}(\mathbb{R}^{2}))}.$

As in [5], we have

$\displaystyle I_{n}$
$\displaystyle\underset{n\rightarrow\infty}{\longrightarrow}$ $\displaystyle
0\quad\mbox{and}$ $\displaystyle J_{n}$ $\displaystyle\leq$
$\displaystyle\varepsilon_{n}\|w_{n}\|_{ST([0,T])},$

where $\varepsilon_{n}\rightarrow 0$. Besides, provided that

(76)
$\limsup_{n\to\infty}\,\|w_{n}\|_{L^{\infty}([0,T];H^{1})}\leq\frac{1-L\,\sqrt{4\pi}}{2},$

we get

$K_{n}\leq\varepsilon_{n}\|w_{n}\|_{ST([0,T])}^{2},\quad\varepsilon_{n}\rightarrow
0.$

Since $\|w_{n}\|_{ST([0,T])}\lesssim
I_{n}+\varepsilon_{n}\|w_{n}\|_{ST([0,T])}^{2}$, wet obtain by bootstrap
argument

$\|w_{n}\|_{ST([0,T])}\lesssim\varepsilon_{n},$

which ends the proof of the result. ∎

##  4\. Appendix: Proof of Proposition 1.2

The proof uses in a crucial way the rearrangement of functions (for a complete
presentation and more details, we refer the reader to [20]). By virtue of
density arguments and the fact that for any function $f\in
H^{1}(\mathbb{R}^{2})$ and $f^{*}$ the rearrangement of f, we have

$\displaystyle\|\nabla f\|_{L^{2}}$ $\displaystyle\geq$ $\displaystyle\|\nabla
f^{*}\|_{L^{2}},$ $\displaystyle\|f\|_{L^{p}}$ $\displaystyle=$
$\displaystyle\|f^{*}\|_{L^{p}},$ $\displaystyle\|f\|_{L^{\phi_{p}}}$
$\displaystyle=$ $\displaystyle\|f^{*}\|_{L^{\phi_{p}}}\,,$

one can reduce to the case of a nonnegative radially symmetric and non-
increasing function $u$ belonging to ${{\mathcal{D}}}(\mathbb{R}^{2})$. With
this choice, let us introduce the function

$w(t)=(4\pi)^{\frac{1}{2}}u(|x|),\quad\mbox{where}\quad|x|={\rm
e}^{-\frac{t}{2}}.$

It is then obvious that the functions $w(t)$ and $w^{\prime}(t)$ are
nonnegative and satisfy

$\displaystyle\int_{\mathbb{R}^{2}}|\nabla u(x)|^{2}\,dx$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}|{w}^{\prime}(t)|^{2}\,dt,$
$\displaystyle\int_{\mathbb{R}^{2}}|u(x)|^{2p}\,dx$ $\displaystyle=$
$\displaystyle\frac{1}{4^{p}\,\pi^{p-1}}\int_{-\infty}^{+\infty}|w(t)|^{2p}\leavevmode\nobreak\
{\rm e}^{-t}\,dt,$ $\displaystyle\displaystyle\int_{\mathbb{R}^{2}}\left({\rm
e}^{\alpha|u(x)|^{2}}-\sum_{k=0}^{p-1}\frac{\alpha^{k}|u(x)|^{2k}}{k!}\right)dx$
$\displaystyle=$
$\displaystyle\pi\displaystyle\int_{-\infty}^{+\infty}\left({\rm
e}^{\frac{\alpha}{4\pi}|w(t)|^{2}}-\sum_{k=0}^{p-1}\frac{\alpha^{k}|w(t)|^{2k}}{(4\pi)^{k}k!}\right){\rm
e}^{-t}\,dt.$

So we are reduced to prove that for any $\beta\in[0,1[$, there exists
$C_{\beta}\geq 0$ so that

$\displaystyle\int_{-\infty}^{+\infty}\left({\rm
e}^{\beta|w(t)|^{2}}-\sum_{k=0}^{p-1}\frac{\beta^{k}|w(t)|^{2k}}{k!}\right){\rm
e}^{-t}dt\leq C({\beta,p})\displaystyle\int_{-\infty}^{+\infty}|w(t)|^{2p}{\rm
e}^{-t}\,dt,\quad\forall\,\beta\in[0,1[,$

when $\displaystyle\int_{-\infty}^{+\infty}|{w}^{\prime}(t)|^{2}dt\leq 1.$ For
that purpose, let us set

$T_{0}=\sup\bigg{\\{}{t\in\mathbb{R},\leavevmode\nobreak\ w(t)\leq
1\bigg{\\}}}.$

The existence of a real number $t_{0}$ such that $w(t_{0})=0$ ensures that the
set $\bigg{\\{}{t\in\mathbb{R},\leavevmode\nobreak\ w(t)\leq 1\bigg{\\}}}$ is
non empty. Then

$T_{0}\in]-\infty,+\infty].$

Knowing that $w$ is nonnegative and increasing function, we deduce that

$w:]-\infty,T_{0}]\longrightarrow[0,1].$

Therefore, observing that $\displaystyle{\rm
e}^{s}-\sum_{k=0}^{p-1}\frac{s^{k}}{k!}\leq c_{p}\,s^{p}\,{\rm e}^{s}$ for any
nonnegative real $s$, we obtain

$\displaystyle\int_{-\infty}^{T_{0}}\left({\rm
e}^{\beta|w(t)|^{2}}-\sum_{k=0}^{p-1}\frac{\beta^{k}|w(t)|^{2k}}{k!}\right){\rm
e}^{-t}dt\leq c_{p}\,\beta^{p}\,{\rm
e}^{\beta}\displaystyle\int_{-\infty}^{T_{0}}|w(t)|^{2p}{\rm e}^{-t}dt.$

To estimate the integral on $[T_{0},+\infty[$, let us first notice that in
view of the definition of $T_{0}$, we have for all $t\geq T_{0}$

$\displaystyle w(t)$ $\displaystyle=$ $\displaystyle
w(T_{0})+\int_{T_{0}}^{t}{w}^{\prime}(\tau)d\tau$ $\displaystyle\leq$
$\displaystyle
w(T_{0})+(t-T_{0})^{\frac{1}{2}}\left(\int_{T_{0}}^{+\infty}{w^{\prime}}(\tau)^{2}d\tau\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle 1+(t-T_{0})^{\frac{1}{2}}.$

Thus, using the fact that for any $\varepsilon>0$ and any $s\geq 0$, we have

$(1+s^{\frac{1}{2}})^{2}\leq(1+\varepsilon)s+1+\frac{1}{\varepsilon}=(1+\varepsilon)s+C_{\varepsilon},$

we infer that for for any $\varepsilon>0$ and all $t\geq T_{0}$

(77) $|w(t)|^{2}\leq(1+\varepsilon)(t-T_{0})+C_{\varepsilon}.$

Now $\beta$ being fixed in $[0,1[$, let us choose $\varepsilon>0$ so that
$\beta(1+\varepsilon)<1$. Then by virtue of (77)

$\displaystyle\int_{T_{0}}^{+\infty}\Big{(}{\rm
e}^{\beta|w(t)|^{2}}-\sum_{k=0}^{p-1}\frac{\beta^{k}|w(t)|^{2k}}{k!}\Big{)}{\rm
e}^{-t}\,dt$ $\displaystyle\leq$ $\displaystyle\int_{T_{0}}^{+\infty}{\rm
e}^{\beta|w(t)|^{2}}{\rm e}^{-t}\,dt$ $\displaystyle\leq$
$\displaystyle\frac{{\rm e}^{\beta
C_{\varepsilon}-T_{0}}}{1-\beta(1+\varepsilon)}\cdot$

But

${\rm e}^{-T_{0}}=\int_{T_{0}}^{+\infty}{\rm
e}^{-t}\,dt\leq\int_{T_{0}}^{+\infty}|w(t)|^{2p}\,{\rm e}^{-t}\,dt,$

which gives rise to

$\int_{T_{0}}^{+\infty}\Big{(}{\rm
e}^{\beta|w(t)|^{2}}-\sum_{k=0}^{p-1}\frac{\beta^{k}|w(t)|^{2k}}{k!}\Big{)}{\rm
e}^{-t}dt\leq\frac{{\rm e}^{\beta
C_{\varepsilon}}}{1-\beta(1+\varepsilon)}\int_{T_{0}}^{\infty}|w(t)|^{2p}{\rm
e}^{-t}\,dt.$

Choosing $C({\beta,p})=\max\Big{(}c_{p}{\rm
e}^{\beta}\beta^{p},\displaystyle\frac{{\rm e}^{\beta
C_{\varepsilon}}}{1-\beta(1+\varepsilon)}\Big{)}$ ends the proof of the
proposition.

## References

  * [1] S. Adachi and K. Tanaka, Trudinger type inequalities in $\mathbb{R}^{N}$ and their best exponents, Proc. Amer. Math. Soc. 128 (2000), no. 7, 2051–2057.
  * [2] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math, 121 (1999), 131–175.
  * [3] H. Bahouri, M. Majdoub and N. Masmoudi, Lack of compactness in the 2D critical Sobolev embedding, the general case, Notes aux Comptes-Rendus de l’Académie des Sciences de Paris, 350 (2012), Série I, 177–181.
  * [4] H. Bahouri, M. Majdoub and N. Masmoudi, Lack of compactness in the 2D critical Sobolev embedding, the general case, to appear in Journal de Mathématiques Pures et Appliquées.
  * [5] H. Bahouri, M. Majdoub and N. Masmoudi, On the lack of compactness in the 2D critical Sobolev embedding, Journal of Functional Analysis, 260 (2011), 208–252.
  * [6] H. Bahouri and I. Gallagher On the stability in weak topology of the set of global solutions to the Navier-Stokes equations, Archiv for Rationnal Mechanics and Analysis, 209 (2) (2013), 569–629.
  * [7] H. Bahouri, J.-Y. Chemin and R. Danchin : Fourier analysis and applications to nonlinear partial differential equations, Grundlehren der Mathematischen Wisserchaften, Springer Verlag, 343, (2011).
  * [8] H. Bahouri, A. Cohen and G. Koch, A general wavelet-based profile decomposition in the critical embedding of function spaces, Confluentes Mathematici, 3 (2011), 1–25.
  * [9] H. Bahouri, S. Ibrahim and G. Perelman : Scattering for the critical 2-D NLS with exponential growth, arXiv:1302.1269.
  * [10] I. Ben Ayed and M. K. Zghal, Characterization of the lack of compactness of $H^{2}_{rad}(\mathbb{R}^{4})$ into the Orlicz space, to appear in Communications in Contemporary Mathematics.
  * [11] H. Brézis and J.-M. Coron, Convergence of solutions of H-Systems or how to blow bubbles, Archive for Rational Mechanics and Analysis, 89 (1985), 21–86.
  * [12] J.-Y. Chemin: Fluides parfaits incompressibles, Astérisque, 230, 1995.
  * [13] P. Gérard, Oscillations and concentration effects in semilinear dispersive wave equations, J. Funct. Anal. 133 (1996), 50–68.
  * [14] P. Gérard, Description du défaut de compacité de l’injection de Sobolev, ESAIM Control Optim. Calc. Var. 3 (1998), 213–233 (electronic, URL: http://www.emath.fr/cocv/).
  * [15] J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995), no. 1, 50–68.
  * [16] S. Ibrahim, M. Majdoub and N. Masmoudi, Double logarithmic inequality with a sharp constant, Proc. Amer. Math. Soc. 135 (2007), no. 1, 87–97.
  * [17] S. Ibrahim, M. Majdoub and N. Masmoudi, Global solutions for a semilinear, two-dimensional Klein-Gordon equation with exponential-type nonlinearity, Comm. Pure Appl. Math. (2006), 1–20.
  * [18] S. Ibrahim, M. Majdoub, N. Masmoudi and K. Nakanishi, Scattering for the two-dimentional energy-critical wave equation, Duke Mathematical Journal 150 (2009), no. 2, 287–329.
  * [19] C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation, Acta Math., 201 (2008), 147–212.
  * [20] J. Moser, A sharp form of an inequality of N. Trudinger, Ind. Univ. Math. J. 20 (1971), 1077–1092.
  * [21] M.-M. Rao and Z.-D. Ren, Applications of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, 250 (2002), Marcel Dekker Inc.
  * [22] B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^{2}$, J. Funct. Anal. 219 (2005), no. 2, 340–367.
  * [23] T. Runst and W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, de Gruyter Series in Nonlinear Analysis and Applications, Vol.3, Berlin, 1996.
  * [24] M. Struwe, A Super-Critical Nonlinear Wave Equation in 2 Space Dimensions, Milan J. Math. (2011), 129–143.
  * [25] M. Struwe, Global well-posedness of the Cauchy problem for a super-critical nonlinear wave equation in two space dimensions, Math. Ann. (2011), 707–719.
  * [26] N.S. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–484.