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\markboth{\hfil Existence of global solutions \hfil EJDE--2002/91}
{EJDE--2002/91\hfil Mohammed Aassila \& Abbes Benaissa \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 91, pp. 1--22. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Existence of global solutions to a quasilinear
  wave equation with general nonlinear damping
 %
\thanks{ {\em Mathematics Subject Classifications:} 35B40, 35L70, 35B37.
\hfil\break\indent
{\em Key words:} Quasilinear wave equation, global
existence, asymptotic behavior, \hfil\break\indent
nonlinear dissipative term, multiplier method.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted July 02, 2002. Published October 26, 2002.} }
\date{}
%
\author{Mohammed Aassila \& Abbes Benaissa}
\maketitle

\begin{abstract}
  In this paper we prove the existence of a global solution
  and study its decay for the solutions to a quasilinear wave
  equation with a general nonlinear dissipative term by
  constructing a stable set in $H^{2}\cap H_{0}^{1}$.
\end{abstract}

\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemme}[theorem]{Lemma}
\newtheorem{remarque}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

We consider the problem
\begin{equation} \label{P}
\begin{gathered}
u''-\Phi(\|\nabla_{x}u\|_{2}^{2})\Delta_{x}u+g(u')+f(u)=0
\quad\hbox{in } \Omega\times [0, +\infty[,\\\
u=0 \quad \hbox{on } \Gamma\times [0, +\infty[,\\
u(x, 0)=u_{0}(x),\quad u'(x, 0)=u_{1}(x)\quad \hbox{ on }\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with a smooth boundary
$\partial\Omega=\Gamma$, $\Phi(s)$ is a $C^{1}$- class function on
$[0, +\infty[$ satisfying $\Phi(s)\geq m_{0}>0$ for $s\geq 0$ with $m_{0}$
constant.

For the problem \eqref{P},
when $\Phi(s)\equiv 1$ and $g(x)=\delta x\,\,(\delta >0)$, Ikehata
and Suzuki \cite{iksu} investigated the dynamics, they have shown that
for sufficiently small initial data $(u_{0}, u_{1})$, the trajectory
$(u(t),u'(t))$ tends to $(0,0)$ in $H_{0}^{1}(\Omega)\times L^{2}(\Omega)$
as $t\to +\infty$.
When $g(x)=\delta |x|^{m-1}x$ ($m\geq 1$) and
$f(y)=-\beta |y|^{p-1}y$ ($\beta>0$, $p\geq 1$), Georgiev and Todorova
{\cite{geto}} have shown that if the damping term dominates over the
source, then a global solution exists for any initial data. Quite recently,
Ikehata {\cite{ikeh}} proved that a global solution exists with no relation
between $p$ and $m$, and Todorova {\cite{todo}} proved that the energy decay
rate is $E(t)\leq (1+t)^{-2/(m-1)}$ for $t\geq 0$, she used a general method
on the energy decay introduced by Nakao {\cite{naka}}.
Unfortunately this method does not
seem to be applicable to the case of more general functions $g$.

Aassila \cite{aass1} proved the existence of a global decaying
$H^{2}$ solution when $g(x)$ has not necessarily a polynomial growth near
zero and a source term of the form $\beta |y|^{p-1}y$, but with small parameter
$\beta$. The decay rate of the global solution depends on the polynomial
growth near zero of $g(x)$ as it was proved in \cite{aass2,todo,komo}.

When $\Phi(s)$ is not a constant function, $g(x)\equiv 0$
and $f(y)\equiv 0$ the equation is often called the wave equation of
Kirchhoff type. This equation was introduced to study the nonlinear vibrations
of an elastic strings by Kirchhoff {\cite{kirc}}, and
the existence of global solutions was investigated by many authors
\cite{poho, kaya, hiro}.
In {\cite{ikma}}, the authors discussed the existence of a global
decaying solution in the case  $\Phi(s)=m_{0}+s^{\frac{(\gamma+2)}{2}}$,
$\gamma\geq 0$, $g(v)=|v|^{r}v$, $0\leq r\leq 2/(n-2)$
($0\leq r\leq \infty$ if $n=1, 2$), $f(u)=-|u|^{\alpha}u$, $0<\alpha\leq
4/(n-2)$ ($0<\alpha <\infty$ if $n=1, 2$) by use of a stable set
method due to Sattinger {\cite{satt}}. But, then, the method in
{\cite{ikma}} cannot be applied to the case $\alpha> 4/(n-2)$,
which is caused by the construction of stable set in $H_{0}^{1}$. Quite
recently, in {\cite{ikmn}} (see also {\cite{aass}}) Ikehata, Matsuyama
and Nakao have constructed a stable set in $H_{0}^{1}\cap H^{2}$ to obtain
a global decaying solution to the initial boundary value problem for
quasilinear visco-elastic wave equations.

Our purpose in this paper is to give a global solvability in the
class $H_{0}^{1}\cap H^{2}$ and energy decay estimates of the solutions to
problem \eqref{P} for a general non-linear damping $g$.
We use some new techniques introduced in {\cite{aass1}} to derive a decay
rate of the solution. So we use the argument combining the method in
{\cite{aass1}} with the concept of stable set in $H_{0}^{1}\cap H^{2}$.
We also use some ideas from {\cite{mart}} introduced in the study of the
decay rates of solutions to the wave equation $u_{tt}-\Delta u+g(u_t)=0$
in $\Omega\times \mathbb{R}^+$.

We conclude this section by stating our plan and giving some notations.
In section $2$ we shall prepare some lemmas needed for our arguments.
Section $3$ is devoted to the proof of the  global existence and decay
estimates to the problem \eqref{P}.
Section $4$ is devoted to the proof of the  global existence and decay
estimates to the problem \eqref{P} in the case $\alpha=0$, i.e.,
$f(u)=-u$. In this case the smallness of $|\Omega|$ (the volume of $\Omega$)
will play an essential role in our argument.
In the last section we shall treat the case $\Phi\equiv 1$, we prove
only the global decaying $H_{0}^{1}$ solution, but we obtain more
results than the case when $\Phi\not\equiv 1$. The condition that
$\beta$\ ($k_{1}$ in our paper) is small is removed here, also we extend
some results obtained by Ono {\cite{ono}} and Martinez {\cite{mart}}.

Throughout this paper the functions considered are all real valued. We
omit the space variable $x$ of $u(t, x)$, $u_{t}(t, x)$ and simply denote
$u(t, x)$, $u_{t}(t, x)$ by $u(t)$, $u'(t)$, respectively, when no confusion arises.
Let $l$ be a number with $2\leq l\leq \infty$. We denote by $\|\ .\ \|_{l}$ the
$L^{l}$ norm over $\Omega$. In particular, $L^{2}$ norm is denoted
$\|\ .\ \|_{2}$. $(\ .\ )$ denotes the usual $L^{2}$ inner product.
We use familiar function spaces $H_{0}^{1}$, $H^{2}$.

\section{Preliminaries}
Let us state the precise hypotheses on $\Phi$, $g$ and $f$.

\begin{enumerate}
\item[(H1)]  $\Phi$ is a $C^{1}$-class function on $\mathbb{R}^{+}$ and satisfies
\begin{equation}
\Phi(s)\geq m_{0} \quad\hbox{and}\quad |\Phi'(s)|\leq m_{1}s^{\gamma/2}
\quad \hbox{for } 0\leq s<\infty
\label{e1}
\end{equation}
for some constants $m_{0}> 0$, $m_{1}\geq 0$, and $\gamma\geq 0$.

\item[(H2)]  $g$ is a $C^{1}$ odd increasing function
and
$$
c_{2}|x|\leq |g(x)|\leq c_{3}|x|^{q}\quad\hbox{if}\quad |x|\geq 1
\quad\hbox{with}\quad 1\leq q\leq \frac{N+2}{(N-2)^{+}},
$$
where $c_{1},\,c_{2}$ and $c_{3}$ are positive constants.

\item[(H3)]  $f(.)$ is a $C^{1}(\mathbb{R})$ satisfying
\begin{equation}
|f(u)|\leq k_{2}|u|^{\alpha+1}\quad\hbox{and}\quad
 |f'(u)|\leq k_{2}|u|^{\alpha}\quad \hbox{for all} u\in \mathbb{R}
\label{e2}
\end{equation}
with some constant $k_{2}> 0$, and
\begin{equation}
0<\alpha<\frac{2}{(N-4)^{+}},
\label{e3}
\end{equation}
where $(N-4)^{+}=\max\{N-4, 0\}$. A typical
example of these functions is $f(u)=-|u|^{\alpha}u$.
\end{enumerate}
We first state three well known lemmas, and then we prove
two other lemmas that will be needed later.

\begin{lemme} [Sobolev-Poincar\'e inequality]
Let $q$ be a number with $2\leq q<+\infty\,\,(n=1,2)$ or $2\leq q\leq 2n/(n-2)\,\,(n\geq 3)$,
then there is a constant $c_{*}=c(\Omega, q)$ such that
$$
\| u\|_{q}\leq c_{*}\|\nabla u\|_2\quad
\hbox{ for }\quad u\in H_0^1(\Omega).
$$
\label{l1}
\end{lemme}

\begin{lemme} [Gagliardo-Nirenberg]
Let $1\leq r< q\leq +\infty$ and $p\leq q$.
Then, the inequality
$$
\|u\|_{W^{m, q}}\leq C \|u\|_{W^{m, p}}^{\theta}\|u\|_{r}^{1-\theta}
\quad \hbox{ for }\quad u\in W^{m, p}\cap L^{r}
$$
holds with some $C> 0$ and
$$
\theta=\big(\frac{k}{n}+\frac{1}{r}-\frac{1}{q}\big)
\big(\frac{m}{n}+\frac{1}{r}-\frac{1}{p}\big)^{-1}
$$
provided that $0<\theta\leq 1$\ (we assume $0<\theta< 1$ if $q=+\infty$).
\label{l2}
\end{lemme}

\begin{lemme}[{\cite{komo}}]
Let $E: {\mathbb{R}}_+\to {\mathbb{R}}_+$ be a non-increasing function and assume that there
are two constants $p\geq 1$ and $A>0$ such that
$$
\int_{S}^{+\infty}E^{\frac{p+1}{2}}(t)\,dt\leq AE(S),\quad 0\leq S<+\infty.
$$
Then
\begin{gather*}
E(t)\leq cE(0)(1+t)^{\frac{-2}{p-1}}\quad \forall t\geq 0,
\quad\hbox{if}\quad p>1\,,\\
E(t)\leq cE(0)e^{-\omega t}\quad \forall t\geq 0,\quad\hbox{if}\quad p=1,
\end{gather*}
where $c$ and $\omega$ are positive constants independent of the initial
energy $E(0)$.
\label{l3}
\end{lemme}

\begin{lemme}[\cite{mart}]
Let $E: {\mathbb{R}}_+\to {\mathbb{R}}_+$ be a non increasing function and
$\phi:{\mathbb{R}}_+\to {\mathbb{R}}_+$ an increasing $C^2$ function such that
$$
\phi (0)=0\quad\hbox{and}\quad \phi (t)\to +\infty\quad
\hbox{as } t\to +\infty.
$$
Assume that there exist $p\geq 1$ and $A>0$ such that
$$
\int_S^{+\infty} E(t)^{\frac{p+1}{2}}(t)\phi'(t)\,dt\leq AE(S).
\quad 0\leq S<+\infty,
$$
Then
\begin{gather*}
E(t)\leq cE(0)(1+\phi (t))^{-2/(p-1)}\quad\forall t\geq 0,
\quad\hbox{if}\quad p>1,\\
E(t)\leq cE(0)e^{-\omega \phi(t)}\quad \forall t\geq 0,
\quad\hbox{if}\quad p=1,
\end{gather*}
where $c$ and $\omega$ are positive constants independent of the
initial energy $E(0)$.
\label{l4}
\end{lemme}

\paragraph{Proof}
Let $f: {\mathbb{R}}_+\to {\mathbb{R}}_+$ be defined by $f(x):=E(\phi^{-1}(x))$,
(we remark that $\phi^{-1}$ has a sense by the hypotheses assumed on $\phi$).
$f$ is non-increasing, $f(0)=E(0)$ and if we set $x:=\phi (t)$ we obtain
\begin{align*}
\int_{\phi (S)}^{\phi (T)}f(x)^{\frac{p+1}{2}}\,dx
=& \int_{\phi (S)}^{\phi (T)}E(\phi^{-1}(x))^{(p+1)/2}\,dx\\
=& \int_{S}^{T}E(t)^{\frac{p+1}{2}}\phi'(t)\,dt\\
\leq& AE(S)= Af(\phi (S))\quad 0\leq S<T<+\infty.
\end{align*}
Setting $s:=\phi (S)$ and letting $T\to +\infty$, we deduce that
$$
\int_{s}^{+\infty}f(x)^{\frac{p+1}{2}}\,dx\leq A f(s)\quad 0\leq s<+\infty.
$$
Thanks to Lemma \ref{l3}, we deduce the desired results.
\hfill $\square$

\begin{remarque} \rm
The use of a `weight function' $\phi(t)$ to establish the decay rate of
solutions to hyperbolic PDE was successfully done by Aassila {\cite{aass2}},
Martinez {\cite{mart}}, and Mochizuki and Motai {\cite{momo}}.
\label{r1}
\end{remarque}

\begin{lemme} [\cite{mart}]
There exists a function $\phi: {\mathbb{R}}_+\to {\mathbb{R}}$ increasing and such that
$\phi$ is concave and $\phi (t)\to +\infty$ as $t\to +\infty$,
$\phi'(t)\to 0$  as $t\to +\infty$, and
$$
\int_{1}^{+\infty}\phi'(t)\left(g^{-1}(\phi'(t))\right)^2\,dt<+\infty.
$$
\label{l5}
\end{lemme}

\paragraph{Proof.}
If such a function exists, we can assume that $\phi (1)=1$. Setting
$s:=\phi (t)$ we obtain
\begin{align*}
\int_1^{+\infty}\phi'(t)\left(g^{-1}(\phi'(t))\right)\,dt
=&\int_1^{+\infty}\left(g^{-1}(\phi'(\phi^{-1}(s)))\right)^2\,ds\\
=&\int_1^{+\infty}g^{-1}\big(\frac{1}{ (\phi^{-1})'(s)}\big)^2\,ds.
\end{align*}
Let us define
$$
\psi (t):=1+\int_1^t\frac{1}{g(1/s)}\,ds,\quad t\geq 1.
$$
Note that $\psi$ is increasing, of class $C^2$, and
$$
\psi'(t)=\frac{1}{g(1/t)}\to +\infty\quad\hbox{as}\quad
t\to +\infty.
$$
Hence
$\psi (t)\to +\infty$ as $t\to +\infty$
and
$$
\int_{1}^{+\infty}\Big(g^{-1}\big(\frac{1}{\psi'(s)}\big)\Big)^2\,ds=
\int_{1}^{+\infty}\frac{1}{s^2}\, ds<+\infty.
$$
Furthermore $\psi'$ is non-decreasing, and hence $\psi$ is convex. Let us
verify that $\psi^{-1}$ is concave: from $\psi(\psi^{-1}(s))=s$ we have
$$
(\psi^{-1})''(s)
=-\frac{\psi''(\psi^{-1}(s))\left((\psi^{-1})'(s)\right)^2}{\psi'
\left(\psi^{-1}(s)\right)}
=-\frac{\psi''\left(\psi^{-1}(s)\right)}{\left(\psi'(\psi^{-1}(s))\right)^3}
\leq 0.
$$
In conclusion, if we set $\phi(t):=\psi^{-1}(t)$ for all $t\geq 1$, we see
that $\phi$ verify all the hypotheses of lemma \ref{l5}.
\hfill $\square$ \smallskip

First, we shall construct a stable set in $H_{0}^{1}\cap H^{2}$.
For this, we define the following functionals:
\begin{gather*}
J(u)\equiv \frac{1}{2}\int_{0}^{\|\nabla_{x}u\|_{2}^{2}}\Phi(s)\, ds+
\int_{\Omega}\int_{0}^{u}f(\eta)\, d\eta\, dx \quad \hbox{for } u\in H_{0}^{1},
\\
{\tilde J}(u)\equiv \Phi(\|\nabla_{x}u\|_{2}^{2})\|\nabla_{x}u\|_{2}^{2}+
\int_{\Omega}f(u)u\, dx \quad \hbox{for } u\in H_{0}^{1}
\\
E(u, v)\equiv\frac{1}{2}\|v\|_{2}^{2}+J(u)\quad
\hbox{for } (u, v)\in H_{0}^{1}\times L^{2}.
\end{gather*}

\begin{lemme}
Let $0<\alpha<4/(N-4)^{+}$. Then, for any $K>0$, there exists a
number
$\varepsilon_{0}\equiv \varepsilon_{0}(K)>0$ such that if
$\|\Delta_{x}u\|\leq K$
and $\|\nabla_{x}u\|\leq \varepsilon_{0}$, we have
\begin{equation}
J(u)\geq \frac{m_{0}}{4}\|\nabla_{x}u\|_{2}^{2}\quad \hbox{and}\quad
{\tilde J}(u)\geq \frac{m_{0}}{2}\|\nabla_{x}u\|_{2}^{2}.
\label{e4}
\end{equation}
\label{l6}
\end{lemme}

\paragraph{Proof:}
We see from the Gagliardo-Nirenberg inequality that
\begin{equation}
\|u\|_{\alpha+2}^{\alpha+2}\leq
C\|u\|_{\frac{2N}{(N-2)}}^{(\alpha+2)(1-\theta)}
\|\Delta_{x}u\|_{2}^{(\alpha+2)\theta}
\leq  C \|\nabla_{x}u\|_{2}^{(\alpha+2)(1-\theta)}
\|\Delta_{x}u\|_{2}^{(\alpha+2)\theta}
\label{e5}
\end{equation}
with
\begin{equation}
\theta=\Big(\frac{N-2}{2N}-\frac{1}{\alpha+2}\Big)^{+}
\Big(\frac{2}{N}+\frac{N-2}{2N}-\frac{1}{2}\Big)^{-1}
=\frac{((N-2)\alpha-4)^{+}}{2(\alpha+2)}\;(\leq 1).
\label{e6}
\end{equation}
Here, we note that
\begin{equation}
(\alpha+2)(1-\theta)-2=\begin{cases}
\alpha & \hbox{if } 0<\alpha\leq \frac{4}{N-2}\\
&(0<\alpha<\infty \hbox{ for } N=1, 2),\\
\frac{(4-N)\alpha+4}{2} &
\hbox{if } \frac{4}{N-2}<\alpha< \frac{4}{N-4}\\
&(\frac{4}{N-2}<\alpha< \infty \hbox{ for } N=3, 4).
\end{cases}
\label{e7}
\end{equation}
Hence, if $\|\Delta_{x}u\|_{2}\leq K$, we have
\begin{equation}
\begin{aligned}
J(u)&\geq  \frac{m_{0}}{2}\|\nabla_{x}u\|_{2}^{2}-
\frac{k_{2}}{\alpha+2}\|u\|_{\alpha+2}^{\alpha+2}\\
&\geq  \frac{m_{0}}{2}\|\nabla_{x}u\|_{2}^{2}-
C\|\nabla_{x}u\|_{2}^{(\alpha+2)(1-\theta)}
\|\Delta_{x}u\|_{2}^{(\alpha+2)\theta}\\
&\geq  \big\{\frac{m_{0}}{2}-CK^{(\alpha+2)\theta}
\|\nabla_{x}u\|_{2}^{(\alpha+2)(1-\theta)-2} \big\}
\|\nabla_{x}u\|_{2}^{2}.\end{aligned}
\label{e8}
\end{equation}
Using (\ref{e7}), we define $\varepsilon_{0}\equiv \varepsilon_{0}(K)$ by
$$
C K^{(\alpha+2)\theta}\varepsilon_{0}^{(\alpha+2)(1-\theta)-2}=
\frac{m_{0}}{4}.
$$
Thus, we obtain
\begin{equation}
J(u)\geq \frac{m_{0}}{4}\|\nabla_{x}u\|_{2}^{2}
\label{e9}
\end{equation}
if $\|\nabla_{x}u\|_{2}\leq \varepsilon_{0}$. It is clear that (\ref{e9})
is valid for ${\tilde J}(u)$.
\hfill $\square$ \smallskip

Let us define a stable in $H_{0}^{1}\cap H^{2}$ as follows: For some
$K>0$,
\begin{align*}
{\cal W}_{K}\equiv&\Big\{(u, v)\in (H_{0}^{1}\cap H^{2})\times H_{0}^{1}:
\|\Delta_{x}u\|_2<K, \\
&\|\nabla_{x}v\|_2< K \hbox{ and }\sqrt{4m_{0}^{-1}E(u, v)}<
\varepsilon_{0}\Big\}
\end{align*}

\begin{remarque} \rm
If $f(u)u\geq 0$, we do not need  $\varepsilon_{0}(K)$, and ${\cal W}_{K}$ is
replaced by
$$
{\tilde {\cal W}}_{K}\equiv\{(u, v)\in (H_{0}^{1}\cap H^{2})
\times H_{0}^{1} :\|\Delta_{x}u\|_2< K, \|\nabla_{x}v\|_2< K \}
$$
\label{r2}
\end{remarque}

\section{Global Existence and Asymptotic Behavior}

A simple computation shows that
$$
E'(t)=-\int_{\Omega} u'g(u')\,dx\leq 0,
$$
hence the energy is non-increasing and in particular $E(t)\leq E(0)$
for all $t\geq 0$.
\begin{lemme}
Let $u(t)$ be a strong solution satisfying $(u(t), u'(t))\in {\cal W}_{K}$
on $[0, T[$ for some $K>0$. Then we have
$$
E(t)\leq cE(0)\Big(G^{-1}\big(\frac{1}{t}\big)\Big)^2\quad\hbox{ on } [0, T[,
$$
where $c$ is a positive constant independent of the initial energy $E(0)$ and
$G(x)=xg(x)$.
Furthermore, if $x\mapsto g(x)/x$ is non-decreasing on $[0,\eta]$ for some
$\eta>0$, then
$$
E(t)\leq cE(0)\Big(g^{-1}\big(\frac{1}{t}\big)\Big)^2\quad\hbox{on } [0, T[,
$$
where $c$ is a positive constant independent of the initial energy $E(0)$.
\label{l7}
\end{lemme}

\paragraph{Proof of lemma \ref{l7}}
For the rest of this article, we denote by $c$ various positive constants
which may be different at different occurences. We multiply the first
equation of \eqref{P} by $E\phi' u$, where $\phi$ is a function satisfying
all the hypotheses of lemma \ref{l5}, we obtain
\begin{align*}
0=&\int_{S}^{T} E\phi'\int_{\Omega} u(u''-\Phi(\|\nabla_{x}u\|_{2}^{2})
\Delta u +g(u')+f(u))\,dx\, dt\\
=&\Big[E\phi'\int_{\Omega} uu'\,dx\Big]_{S}^{T}-\int_{S}^{T}(E'\phi'+E\phi'')
\int_{\Omega} uu'\,dx\,dt-2\int_{S}^{T} E\phi'\int_{\Omega} u'^2\,dx\,dt\\
&+\int_{S}^{T} E\phi'\int_{\Omega}\left(u'^2+\Phi(\|\nabla_{x}u\|_{2}^{2})|\nabla u|^2
+f(u)u\right)\,dx\,dt\\
&+\int_{S}^{T} E\phi'\int_{\Omega} ug(u')\,dx\,dt\,.
\end{align*}
Under the assumption $(u(t), u'(t))\in {\cal W}_{K}$, the functionals
$J(u(t))$ and ${\tilde J}(u(t))$ are both equivalent to
$\|\nabla_{x}u(t)\|_{2}^{2}$, by lemma \ref{l6}. So we deduce that
\begin{align*}
\int_{S}^{T}E^2\phi'\,dt\leq& -\Big[E\phi'\int_{\Omega} uu'\,dx\Big]_{S}^{T}
+\int_{S}^{T}(E'\phi'+E\phi'')\int_{\Omega} uu'\,dx\,dt\\
&+2\int_{S}^{T}E\phi'\int_{\Omega} u'^2\,dx\,dt-\int_{S}^{T}E\phi'\int_{\Omega}
ug(u')\,dx\,dt\\
\leq& -\Big[E\phi'\int_{\Omega} uu'\,dx\Big]_{S}^{T}+\int_{S}^{T}(E'\phi'
+E\phi'')\int_{\Omega} uu'\,dx\,dt\\
&+2\int_{S}^{T}E\phi'\int_{\Omega} u'^2\,dx\,dt+c(\varepsilon)
\int_{S}^{T}E\phi'\int_{|u'|\leq 1} g(u')^2\,dx\,dt\\
&+\varepsilon\int_{S}^{T}E\phi'\int_{|u'|\leq 1} u^2\,dx\,dt
-\int_{S}^{T}E\phi'\int_{|u'|> 1} ug(u')\,dx\,dt
\end{align*}
for all $\varepsilon >0$. Choosing $\varepsilon$ small enough, we deduce
that
\begin{align*}
&\int_{S}^{T}E^2\phi'\,dt\\
&\leq -\Big[E\phi'\int_{\Omega} uu'\,dx\Big]_{S}^{T}+\int_{S}^{T}(E'\phi'+E\phi'')
\int_{\Omega} uu'\,dx\,dt
+c\int_{S}^{T}E\phi'\int_{\Omega} u'^2\,dx\,dt\\
&\leq cE(S)-\int_{S}^{T}E\phi'\int_{|u'|> 1} ug(u')\,dx\,dt+
c\int_{S}^{T}E\phi'\int_{\Omega} u'^2\,dx\,dt.
\end{align*}
Also, we have
\begin{align*}
\int_{S}^{T}&E\phi'\int_{|u'|> 1} ug(u')\,dx\,dt\\
\leq &\int_{S}^{T}E\phi'\Big(\int_{\Omega}|u|^{q}\, dx\Big)^{1/(q+1)}
\Big(\int_{|u'|> 1}|g(u')|^{\frac{(q+1)}{q}}\, dx\Big)^{q/(q+1)}\\
\leq &c \int_{S}^{T}E^{3/2}\phi'
\Big(\int_{|u'|> 1}u'g(u')\, dx\Big)^{q/(q+1)}
\leq \int_{S}^{T}\phi' E^{3/2}(-E')^{\frac{q}{(q+1)}} \\
\leq & c\int_{S}^{T}\phi' (E^{\frac{3}{2}-\frac{q}{q+1}})
\left((-E')^{\frac{q}{(q+1)}}E^{\frac{q}{q+1}}\right) \\
\leq &c(\varepsilon')\int_{S}^{T}\phi'(-E'E)\, dt+\varepsilon'
\int_{S}^{T}\phi'E^{(q+1)(\frac{3}{2}-\frac{q}{(q+1)})}\, dt\\
\leq & c(\varepsilon')E(S)^{2}+\varepsilon' E(0)^{(q-1)/2}
\int_{S}^{T}\phi'E^{2}\, dt
\end{align*}
for every $\varepsilon' >0$. Choosing $\varepsilon'$ small enough, we obtain
$$
\int_{S}^{T}E^2\phi'\,dt
\leq cE(S)+ c\int_{S}^{T}E\phi'\int_{\Omega} u'^2\,dx\,dt
$$
We want to majorize the last term of the above inequality, we have
\begin{align*}
\int_{S}^{T}E\phi'\int_{\Omega} u'^2\,dx\,dt
=&\int_{S}^{T}E\phi'\int_{\Omega_{1}}u'^2\,dx\,dt+
\int_{S}^{T}E\phi'\int_{\Omega_{2}}u'^2\,dx\,dt \\
&+\int_{S}^{T}E\phi'\int_{\Omega_{3}} u'^2\,dx\,dt,
\end{align*}
where, for $t\geq 1$,
\begin{gather*}
\Omega_{1}:=\{x\in\Omega: |u'|\leq h(t)\},\quad
\Omega_{2}:=\{x\in\Omega:  h(t)<|u'|\leq h(1)\},\\
\Omega_{3}:=\{x\in\Omega:  |u'|>h(1)\},
\end{gather*}
and $h(t):=g^{-1}(\phi'(t))$, which is a positive non-increasing function and
satisfies $h(t)\to 0$ as $t\to +\infty$. Because
\begin{align*}
\int_{S}^{T}E\phi'\int_{\Omega_{1}}u'^{2}\,dx\,dt
\leq & c\int_{S}^{T}E(t)\phi'(t)\Big(\int_{\Omega_{1}}h(t)^{2}\,ds\Big)\,dt\\
\leq & cE(S)\int_{S}^{T}\phi'(t)(g^{-1}(\phi'(t)))^{2}\,dt
\leq cE(S),
\end{align*}
we have the following:
Since $g$ is non-decreasing, for $x\in \Omega_{2}$ we have
$\phi'(t)=g(h(t))\leq |g(u')|$, and hence
\begin{align*}
\int_{S}^{T}E\phi'\int_{\Omega_{2}}u'^{2}\,dx\,dt
\leq & \int_{S}^{T}E\int_{\Omega_{2}}|g(u')|u'^{2}\,dx\,dt\\
\leq & h(1)\int_{S}^{T}E\int_{\Omega_{2}} u'g(u')\,dx\,dt
\leq \frac{h(1)}{2}E(S)^{2}\,;
\end{align*}
and since $g(x)\geq cx$ for $x\geq h(1)$, we have
\begin{align*}
\int_{S}^{T}E\phi'\int_{\Omega_{3}}u'^{2}\,dx\,dt
\leq & c\int_{S}^{T}E\phi'\int_{\Omega} u'g(u')\,dx\,dt \\
\leq &c\int_{S}^{T} E (-E')\,dx\,dt
\leq cE(S)^{2}.
\end{align*}
Then we deduce that
$$
\int_{S}^{T}E^{2}\phi'\,dt\leq cE(S),
$$
and thanks to Lemma \ref{l5}, we obtain
$$
E(t)\leq \frac{c\ E(0)}{\phi(t)},\qquad\forall t\geq 1.
$$
Let $s_{0}$ be such that $g(1/s_{0})\leq 1$, since $g$ is non-decreasing
we have
$$
\psi (s)\leq 1+(s-1)\frac{1}{g(1/s)}\leq s\frac{1}{g(1/s)}
=\frac{1}{G(1/s)}\quad\forall s\geq s_{0},
$$
hence
$s\leq \phi\big(1/G(1/s)\big)$ and
$$
\frac{1}{\phi (t)}\leq \frac{1}{s}\quad\hbox{with}\quad
t:=\frac{1}{G(1/s)}.
$$
Thus
$$
\frac{1}{\phi(t)}\leq G^{-1}(1/t).
$$
Now define $H(x):=g(x)/x$, $H$ is non-decreasing, $H(0)=0$, then we
use the function $h(t):=H^{-1}(\phi'(t))$. On $\Omega_{2}$ it holds that
$$
\phi'(t)(u')^2\leq |H(u')|(u')^2=u'g(u').
$$
The same calculations as above with
$$
\phi^{-1}(t)=1+\int_{1}^{t}\frac{1}{H(1/s)}\, ds
$$
yield
$E(t)\leq c\ E(0)\big(g^{-1}(1/t)\big)^{2}$.
\hfill $\square$

\begin{lemme}
Let $u(t)$ be a strong solution satisfying $(u(t), u'(t))\in {\cal W}_{K}$ on
$[0, T[$ for some $K>0$. Assume that
$$
\int_0^{+\infty}\left(g^{-1}(1/t)\right)^{\min\{\gamma+1,
\alpha(1-\theta_{0}) \}}\,dt<+\infty.
$$
Then we have
$$
\|\nabla u'(t)\|_2^2+\|\Delta u(t)\|_2^2\leq Q_{1}^{2}(I_0,I_1,K),
$$
with
$\lim_{I_0\to 0}Q_{1}^{2}(I_{0},I_{1},K)=I_{1}^{2}$
and where we set
$$
I_{0}^{2}=E(0)=\frac{1}{2}\|u_{1}\|_{2}^{2}+J(u_{0}),\quad
I_{1}^{2}=\|\nabla u_{1}\|_{2}^{2}+\Phi(\|\nabla_{x}u_{0}\|_{2}^{2})
\|\Delta u_0\|_{2}^{2}
$$
\label{l8}
\end{lemme}

\paragraph{Proof}
Multiplying the first equation of \eqref{P} by $-\Delta u'(t)$ and integrating
over $\Omega$, we obtain
\begin{multline*}
\frac{1}{2}\frac{d}{dt}\Bigl[\|\nabla u'(t)\|_2^2+
\Phi(\|\nabla_{x}u\|_{2}^{2})\|\Delta u(t)\|_2^2\Bigr]+
\Bigl(\nabla g(u'(t)),\nabla u'(t)\Bigr) \\
=-\int_{\Omega} f'(u)  \nabla u.\nabla u'(t)\, dx\Bigr)
+\Phi'(\|\nabla_{x}u\|_{2}^{2})(\nabla u'(t),
\nabla u(t))\|\Delta_{x} u\|_{2}^{2}.
\end{multline*}
We set
$$
E_{1}(t)\equiv \|\nabla_{x}u'\|_{2}^{2}+\Phi(\|\nabla_{x}u\|_{2}^{2})
\|\Delta_{x}u\|_{2}^{2}
$$
Using the assumptions on $\Phi$, $g$ et $f$, we have
\begin{equation}
\begin{aligned}
\frac{d}{dt}E_{1}(t)\leq & C\|\nabla_{x}u\|_{2}^{\gamma+1}
\|\nabla_{x}u'\|_{2}\|\Delta_{x}u\|_{2}^{2}+
2k_{2}\int_{\Omega}|u|^{\alpha}|\nabla_{x}u||\nabla_{x}u'|\, dx\\
\leq & C\Big\{E(t)^{(\gamma+1)/2}K^{3}+\Big(\int_{\Omega}|u|^{2\alpha}
|\nabla_{x}u|^{2}\, dx \Big)^{1/2}
\Big(\int_{\Omega}|\nabla_{x}u'|\, dx\Big)^{1/2}\Big\}
\end{aligned}\label{e10}
\end{equation}
Here, we see from the Gagliardo-Nirenberg inequality that
\begin{equation}
\begin{aligned}
\Big(\int_{\Omega}|u|^{2\alpha}|\nabla_{x}u|^{2}\, dx\Big)^{1/2}
&\leq
\|u(t)\|_{N\alpha}^{\alpha}\|\nabla_{x}u(t)\|_{\frac{2N}{(N-2)}} \\
&\leq  C \|u(t)\|_{\frac{2N}{(N-2)}}^{\alpha(1-\theta_{0})}
\|\Delta_{x}u(t)\|_{2}^{\alpha\theta_{0}}\|\Delta_{x}u(t)\|_{2} \\
&\leq  C \|\nabla_{x}u(t)\|_{2}^{\alpha(1-\theta_{0})}
\|\Delta_{x}u(t)\|_{2}^{\alpha\theta_{0}+1}\\
&\leq  C E(t)^{\alpha(1-\theta_{0})}K^{\alpha\theta_{0}+1}
\end{aligned}\label{e11}
\end{equation}
with
$$
\theta_{0}=\Big(\frac{N-2}{2}-\frac{1}{\alpha}\Big)^{+}=
\frac{((N-2)\alpha-2)^{+}}{2\alpha}\quad (\leq 1).
$$
Hence, it follows from (\ref{e10}) and (\ref{e11}) that
\begin{equation}
\frac{d}{dt}E_{1}(t)\leq C\left\{E(t)^{\frac{(\gamma+1)}{2}}K^{3}
+E(t)^{\frac{\alpha(1-\theta_{0})}{2}}K^{\alpha\theta_{0}+2}\right\}.
\label{e12}
\end{equation}
we conclude that
\begin{align*}
&\|\Delta_{x}u(t)\|_{2}^{2}+\|\nabla_{x}u'(t)\|_{2}^{2}\\
&\leq \frac{1}{\min\{1, m_{0}\}}
\Big\{I_{1}^{2}+C K^{3}\int_{0}^{\infty}\! E(t)^{(\gamma+1)/2}\, dt
+C K^{\alpha\theta_{0}+2}\int_{0}^{\infty} \!E(t)^{\alpha(1-\theta_{0})/2}\,dt
\Big\}
\end{align*}

\paragraph{Example}
Let $g(x)$ be the inverse function of
$$
M(0)=0 \quad\hbox{and}\quad
M(x)=\frac{x^{\sigma}}{(\log(-\log x))^{\beta}}
\quad\hbox{for } 0< x< x_{0}, \quad (\beta, \sigma> 0).
$$
The function $g$ exists and satisfies the hypothesis (H2), when $0< \sigma< 1$
(see Appendix). So
$$
g^{-1}(1/t)=\frac{1}{t^{\sigma}(\log (\log t))^{\beta}}
$$
the conditions in the Lemma \ref{l8} give
\begin{gather}
\int_{t_{0}}^{\infty}
\frac{1}{t^{\sigma(\gamma+1)}(\log(\log  t))^{\beta(\gamma+1)}}
\, dt< \infty, \label{B1} \\
\int_{t_{0}}^{\infty}
\frac{1}{t^{\sigma\alpha(1-\theta_{0})}(\log(\log t))
^{\beta\alpha(1-\theta_{0})}}\, dt< \infty,
\label{B2}
\end{gather}
which are similar to Bertrand integrals. So,
when $\gamma=0$, the first integral \eqref{B1} is not finite, we obtain the following cases:
if $\sigma(\gamma+1)> 1$, the integral is finite,
if $\sigma(\gamma+1)= 1$, and $\beta(\gamma+1)> 1$, also
the integral is finite.
The second integral \eqref{B2}, is fine under the following conditions:
$$
\sigma^{-1}< \alpha\leq \frac{2}{(N-2)^{+}}\quad \hbox{for } N=1, 2, 3
$$
or
$$
\alpha> \frac{2(1-\sigma)}{\sigma} \quad\hbox{for } N=3
$$
or
$$
\alpha=\sigma^{-1}\quad\mbox{and}\quad
\beta^{-1}< \alpha\leq \frac{2}{(N-2)^{+}}\quad\hbox{for } N=1, 2, 3
$$
or
$$
\alpha=\frac{2(1-\sigma)}{\sigma}\quad\mbox{and}\quad
\alpha> \frac{2(1-\beta)}{\beta}\quad \hbox{for } N=3.
$$
Hence, we must restrict ourselves to $1\leq N\leq 3$.

\begin{remarque} \rm
When $\Phi\equiv 1$, $g(x)=|x|^{p-1}x,\,p\geq 1$, and
$f(y)=-|y|^{q-1}y$ with $q\geq 1$, we obtain
\begin{gather*}
E(t)\leq  cE(0)e^{-\omega t}\quad\forall t\geq 0,
\; c>0,\;\omega>0,\quad\hbox{if } p=1\\
E(t)\leq \frac{cE(0)}{(1+t)^{2/(p-1)}}\quad\forall t\geq 0,\;
c>0\quad\hbox{if } p>1.
\end{gather*}
Also
$$
Q_{1}^{2}(I_{0},I_{1},K) = I_{1}^{2}+cK^{2} I_{0}^{q-1},\quad
Q_{2}^{2}(I_{0},I_{1},K) = I_{1}^{2}+cK^{(q-1)\theta+2}
I_0^{(q-1)(1-\theta)}.
$$
When $g(x)=|x|^{p-1}x$, $p\geq 1$, $f(y)\equiv 0$, and $p<\gamma+2$,
we obtain the same above results (see {\cite{aass}}).
\label{r3}
\end{remarque}

\begin{theorem}
Under the hypotheses of lemma \ref{l7} and \ref{l8} there exists an
open set $S_{1}\subset (H^2(\Omega)\cap H_{0}^{1}(\Omega))
\times H_{0}^{1}(\Omega)$, which includes
$(0, 0)$ such that if $(u_{0},u_{1})\in S_{1}$, the problem \eqref{P}
has a unique global solution $u$ satisfying
$$
u\in L^{\infty}([0, \infty[; H^{2}(\Omega)\cap H_{0}^{1}(\Omega))
\cap W^{1,\infty}([0, \infty[; H_{0}^{1}(\Omega))
\cap W^{2, \infty}([0, \infty[; L^{2}(\Omega)),
$$
furthermore we have the decay estimate
\begin{equation}
E(t)\leq c\ E(0)\left(g^{-1}(1/t)\right)^2\quad\forall t>0.
\label{e13}
\end{equation}
\label{t1}
\end{theorem}

\subsection*{Proof of theorem \protect{\ref{t1}}}
Let $K>0$. Put
$$
S_{K}\equiv \{(u_{0}, u_{1})\in {\cal W}_{K}|\, Q_{1}(I_{0}, I_{1}, K)<K\},
\quad
S_{1}\equiv \bigcup_{K>0}S_{K}\,.
$$
Note that if $E_{0}$, $E_{1}$ are sufficiently small, then $S_{K}$
is not empty.

If $(u_{0}, u_{1})\in S_{K}$ for some $K> 0$, then an assumed strong solution $u(t)$
exist globally and satisfies $(u(t), u'(t))\in {\cal W}_{K}$ for all $t\geq 0$.
Let $\{w_{j}\}_{j=1}^{\infty}$ be the basis of $H_{0}^{1}$ consisted by the
eigenfunction of $-\Delta$ with Dirichlet condition. We define the
approximation solution $u_{m}$\ (m=1, 2, \ldots) in the form
$$
u_{m}=\sum_{j=1}^{m}g_{jm}w_{j}
$$
where $g_{jm}(t)$ are determined by
\begin{equation}
\begin{aligned}
(u''_{m}(t), w_{j})+\Phi(\|\nabla_{x}u_{m}(t)\|_{2}^{2})(\nabla_{x}u_{m}(t),
\nabla_{x}w_{m}) &\\
+(g(u'_{m}(t)), w_{j})+(f(u_{m}(t)), w_{j})&=0
\end{aligned}\label{e14}
\end{equation}
for $j\in\{1, 2, \ldots, m\}$ with the initial data
where $u_{m}(0)$ and $u'_{m}(0)$ are determined in such a way that
\begin{gather*}
u_{m}(0)=u_{0m}=\sum_{j=1}^{m}(u_{0}, w_{j})w_{j}\to u_{0}
\hbox{ strongly in } H_{0}^{1}\cap H^{2}\hbox{ as } m\to \infty,\\
u'_{m}(0)=u_{1m}=\sum_{j=1}^{m}(u_{1}, w_{j})w_{j}\to u_{1}
\hbox{ strongly in } H_{0}^{1} \hbox{ as } m\to \infty.
\end{gather*}

By the theory of ordinary differential equations, (\ref{e14}) has a unique
solution $u_{m}(t)$. Suppose that $(u_{0}, u_{1})\in S_{K}$ for $K>0$. Then,
$(u_{m}(0), u'_{m}(0))\in S_{K}$ for large $m$. It is clear that all the estimates
obtained above are valid for $u_{m}(t)$ and, in particular, $u_{m}(t)$ exists
on $[0, \infty[$. Thus, we conclude that $(u_{m}(t), u'_{m}(t))\in {\cal W}_{K}$
for all $t\geq 0$ and all the estimates are valid for $u_{m}(t)$ for all $t\geq 0$.

Thus, $u_{m}(t)$ converges along a subsequence to $u(t)$ in the following way:
\begin{gather*}
u_{m}(.)\to u(.) \hbox{ weakly * } \hbox{ in } L_{\rm loc}^{\infty}
([0, \infty); H_{0}^{1}\cap H^{2}), \\
u'_{m}(.)\to u_{t}(.) \hbox{ weakly * } \hbox{ in } L_{\rm loc}^{\infty}
([0, \infty); H_{0}^{1}), \\
u_{m}(.)\to u_{tt}(.) \hbox{ weakly * } \hbox{ in } L_{\rm loc}^{\infty}
([0, \infty); L^{2}),
\end{gather*}
and hence,
\begin{gather*}
\Phi(\|\nabla_{x}u_{m}(.)\|_{2}^{2})\nabla_{x} u_{m}(.)\to
\Phi(\|\nabla_{x}u(.)\|_{2}^{2})\nabla_{x} u(.) \hbox{ weakly * }
\hbox{ in } L_{\rm loc}^{\infty}([0, \infty); H_{0}^{1}),\\
g(u_{m}(.))\to  g(u(.)) \hbox{ weakly * }
\hbox{ in } L_{\rm loc}^{\infty}([0, \infty); H_{0}^{1}),
\end{gather*}
Therefore, the limit function $u(t)$ is a desired solution belonging to
$$
L^{\infty}([0, \infty[; H_{0}^{1}\cap H^{2})\cap
W^{1, \infty}([0, \infty[; H_{0}^{1})\cap W^{2, \infty}([0, \infty[; L^{2})
$$

The uniqueness can be proved by use of the monotonicity of $g$,
$n\alpha<2n/(n-4)$ and
$\sup_{0\leq t\leq T}(\|u(t)\|_{H^{2}}+\|u'(t)\|_{H_{0}^{1}})\leq C(T)< \infty$
(see {\cite{aass1}}).
\hfill $\square$

\section{The case $\alpha=0$}

In this section we shall discuss the existence of a global solution to the
problem \eqref{P} with $f(u)\equiv -u$. More precisely, we impose an
assumption on $f(u)$ instead of (H3) as follows:
\begin{itemize}
\item[(H.3)']  $f(.)$ satisfies
$f(u)=-k_{3} u$ for $u\in \mathbb{R}$ %\label{e15}
with $k_{3}C(\Omega)< m_{0}$, $k_{3}>0$, where $C(\Omega)$ is a quantity
such that
\begin{equation}
C(\Omega)=\sup_{u\in H_{0}^{1}\backslash\{0\}}\frac{\|u\|_{2}}{\|\nabla_{x}u\|_{2}}
\label{e16}
\end{equation}
\end{itemize}

\begin{remarque} \rm
The condition $k_{3}C(\Omega)< m_{0}$ implies that $|\Omega|$ is small
in some sense. On the other hand, if $f(u)=u$, we need not take
$C(\Omega)$ into consideration.
\label{r4}
\end{remarque}

Our result reads as follows.
\begin{theorem}
Under the hypotheses of Lemma \ref{l7} (we replace {\rm (H.3)} by
{\rm (H.3)'}) and \ref{l8} , there exists an open
unbounded set $S_{2}$ in $(H^{2}\cap H_{0}^{1})\times H_{0}^{1}$,
which includes $(0, 0)$,
such that if $(u_{0}, u_{1})\in S_{2}$, the problem \eqref{P} has a unique
solution u in the sense of theorem \ref{t1} which satisfies the decay
estimate (\ref{e13}).
\label{t2}
\end{theorem}

\subsection*{Proof of theorem \protect{\ref{t2}}}
This proof is also given in parallel way to the proof
of theorem \ref{t1} so se just sketch the outline.

First, let $k_{3}C(\Omega)<m_{0}$. Then, by (\ref{e16},)
\begin{equation}
J(u)=\frac{1}{2}\int_{0}^{\|\nabla_{x}u\|_{2}^{2}}\Phi(s)\, ds-
\frac{k_{3}}{2}\|u\|_{2}^{2}\geq
\frac{1}{2}(m_{0}-k_{3}C(\Omega))\|\nabla_{x}u\|_{2}^{2}.
\label{e17}
\end{equation}
We may assume ${\tilde J}(u)$ also satisfies (\ref{e17}).
If $u(t)$ is a strong solution satisfying $\|\nabla_{x}u(t)\|_{2}<K$ and
$\|\nabla_{x}u'(t)\|_{2}<K$ on $[0, T[$ for some $K> 0$,
then as in lemma \ref{l7}, we derive
the decay estimate
\begin{equation}
E(t)\leq c\left(g^{-1}(1/t)\right)^{2}\,.
\label{e18}
\end{equation}
Multiplying the equation by $-\Delta_{x}u'$, we see
\begin{equation}
\begin{gathered}
\frac{1}{2}\frac{d}{dt}E_{1}(t)\leq
|\Phi'(\|\nabla_{x}u(t)\|_{2}^{2})|(\nabla_{x}u(t), \nabla_{x}u'(t))
\|\Delta_{x}u(t)\|_{2}^{2}+\frac{k_{3}}{2}\frac{d}{dt}
\|\nabla_{x}u(t)\|_{2}^{2}\\
\leq C K^{3} E(t)^{(\gamma+1)/2}+\frac{k_{3}}{2}
\frac{d}{dt}\|\nabla_{x}u(t)\|_{2}^{2}
\end{gathered}\label{e19}
\end{equation}
where we set
$$
E_{1}(t)=\Phi(\|\nabla_{x}u(t)\|_{2}^{2})\|\Delta_{x}u(t)\|_{2}^{2}
+\|\nabla_{x}u'(t)\|_{2}^{2}.
$$
we integrate (\ref{e19}) to obtain
\begin{align*}
&\|\Delta_{x}u(t)\|_{2}^{2}+\|\nabla_{x}u(t)\|_{2}^{2}\hfill\cr
&\leq  \frac{1}{\min\{1,m_{0}\}}\Big\{I_{1}^{2}+
CK^{3}\int_{0}^{\infty}E(t)^{\frac{(\gamma+1)}{2}}\, dt+
k_{3}\|\nabla_{x}u(t)\|_{2}^{2}-k_{3}\|\nabla_{x}u_{0}\|_{2}^{2}\Big\} \\
&\leq \frac{1}{\min\{1,m_{0}\}}\Big\{I_{1}^{2}+CI_{0}^{2}+
C\ I_{0}^{\gamma+1}\ K^{3}\int_{0}^{\infty}
\left(g^{-1}(1/t)\right)^{(\gamma+1)}\, dt\Big\}\\
&\equiv  Q_{2}^{2}(I_{0}, I_{1}, K) \quad \hbox{on } [0, T[.
\end{align*}
Defining
$$
S_{K}\equiv \{(u_{0}, u_{1})\in H_{0}^{1}\cap H^{2}: Q_{2}(I_{0}, I_{1}, K)
< K\}, \quad
S_{2}\equiv \bigcup_{K> 0} S_{K}
$$
we conclude that if $(u_{0}, u_{1})\in S_{2}$, the corresponding solution
to the problem \eqref{P} exists globally and satisfies the estimate
$$
E(t)\leq c\left(g^{-1}(1/t)\right)^{2}\quad \hbox{and}\quad
 \|\Delta_{x}u(t)\|_{2}^{2}+\|\nabla_{x}u'(t)\|_{2}^{2}<K^2,
$$
for all $t> 0$. The proof of theorem \ref{t2} is complete.

\section{The case $\Phi\equiv 1$}
Usually, we study global existence for Kirchhoff equation
(i.e. when $\Phi\not\equiv 1$) in the class
$H^{2}\cap H_{0}^{1}$ (also when $f\equiv g\equiv 0$). Thus the condition
in Lemma \ref{l8} excludes some functions $g$ which verify (H2), for
example $g(x)=e^{-1/x}$ or $g(x)=e^{-e^{1/x}}$ or the example above.
We consider the case $\Phi\equiv 1$
(or a constant function) and we prove a global $H_{0}^{1}$ solution that
decays.
Here we do not need the condition of Lemma \ref{l8} and we will take only
$\alpha\leq 4/(n-2)^{+}$ because we work only in $H_{0}^{1}(\Omega)$.

Now, we consider the initial boundary-value problem
\begin{equation}
\begin{gathered}
u''-\Delta_{x}u+g(u')+f(u)=0 \quad \hbox{in } \Omega\times [0, +\infty[,\\
u=0 \quad\hbox{on } \Gamma\times [0, +\infty[,\\
u(x, 0)=u_{0}(x),\quad u'(x, 0)=u_{1}(x)\quad \hbox{on }\Omega,
\end{gathered}\label{P'}
\end{equation}
First, we shall construct a stable set in $H_{0}^{1}$.
For this, we need define the following  functionals:
\begin{gather*}
J(u)\equiv \frac{1}{2}\|\nabla_{x}u\|_{2}^{2}+
\int_{\Omega}\int_{0}^{u}f(\eta)\, d\eta\, dx \quad \hbox{for } u\in H_{0}^{1},\\
{\tilde J}(u)\equiv \|\nabla_{x}u\|_{2}^{2}+
\int_{\Omega}f(u)u\, dx \quad \hbox{ for } u\in H_{0}^{1},\\
E(u, v)\equiv\frac{1}{2}\|v\|_{2}^{2}+J(u)\quad
\hbox{for } (u, v)\in H_{0}^{1}\times L^{2}.
\end{gather*}
Then we can define the stable set
$$
{\cal W}=\{u\in H_{0}^{1}(\Omega): \|\nabla_{x}u\|_{2}^{2}- k_{1}
\|u\|_{\alpha+2}^{\alpha+2}> 0 \}\cup \{0\}
$$

\begin{lemme}
{\bf (i)} If $\alpha< 4/[n-2]^{+}$, then
${\cal W}$ is an open neighborhood of $0$ in $H_{0}^{1}(\Omega)$.
\\ %\label{e20}
{\bf (ii)} If $u\in {\cal W}$, then
\begin{equation}
\|\nabla_{x} u\|_{2}^{2}\leq d_{*} J(u)\quad\mbox{with}\quad
d_{*}=\frac{2(\alpha+2)}{\alpha}. \label{e21}
\end{equation}
\label{l10}
\end{lemme}

\paragraph{Proof.}
${\bf (i)}$ From the Sobolev-Poincar\'e inequality (see lemma \ref{l1}) we have
\begin{equation}
k_{1}\|u\|_{\alpha+2}^{\alpha+2}\leq A k_{1} \|\nabla_{x} u\|_{2}^{\alpha}
\|\nabla_{x} u\|_{2}^{2}
\label{e22}
\end{equation}
where $A=c_{*}^{\alpha+2}$.
Let
$$
U(0)\equiv \big\{u\in H_{0}^{1}(\Omega): \|\nabla_{x} u\|_{2}^{\alpha}<
\frac{1}{A k_{1}} \big\}.
$$
Then, for any $u\in U(0)\backslash \{0\}$, we deduce from (\ref{e22}) that
$$
k_{1}\|u\|_{\alpha+2}^{\alpha+2}< \|\nabla_{x} u\|_{2}^{2},
$$
that is, $K(u)> 0$. This implies $U(0)\subset {\cal W}$.

\noindent
{\bf (ii)} By the definition of $K(u)$ and $J(u)$ we have the inequality
$$
J(u)\geq \frac{1}{2}\|\nabla_{x} u\|_{2}^{2}-\frac{k_{1}}{\alpha+2}
\|u\|_{\alpha+2}^{\alpha+2}
\geq  \frac{\alpha}{2(\alpha+2)}\|\nabla_{x} u\|_{2}^{2}
$$
\hfill $\square$

\begin{lemme}
Let $u(t)$ be a strong solution of \eqref{P'}. Suppose that
\begin{equation}
u(t)\in {\overline{\cal W}}\ \hbox{ and } {\tilde J}(u(t))\geq \frac{1}{2}\|\nabla_{x}u(t)\|_{2}^{2}
\label{e23}
\end{equation}
for $0\leq t< T$. Then we have
$$
E(t)\leq cE(0)\left(G^{-1}(1/t)\right)^2\quad\hbox{on } [0, T[,
$$
where $c$ is a positive constant independent of the initial energy $E(0)$ and
$G(x)=xg(x)$.
Furthermore, if $x\mapsto g(x)/x$ is non-decreasing on $[0,\eta]$ for some
$\eta>0$, then we have
$$
E(t)\leq cE(0)\left(g^{-1}(1/t)\right)^2\quad\hbox{on } [0, T[,
$$
where $c$ is a positive constant independent of the initial energy $E(0)$.
\label{l9}
\end{lemme}

\paragraph{Examples}
\begin{itemize}
\item[1)] If $g(x)=e^{-1/x^p}$ for $0<x<1$, $p>0$, then
$E(t)\leq c/(\ln t)^{2/p}$.

\item[2)] If $g(x)=e^{-e^{1/x}}$ for $0<x<1$, then
$E(t)\leq c/(\ln (\ln t))^{2}$.
\end{itemize}

\paragraph{Proof of lemma \ref{l7}}
The functionals $J(u(t))$ and ${\tilde J}(u(t))$ are both equivalent to
$\|\nabla_{x}u(t)\|_{2}^{2}$, indeed we have
$$
\int_{\Omega}f(u)u\, dx\leq k_{1}\|u\|_{\alpha+2}^{\alpha+2}
\leq \|\nabla_{x}u(t)\|_{2}^{2}
$$
So, we have
$$
\frac{1}{2}\|\nabla_{x}u\|_{2}^{2}\leq K(u(t))\leq \frac{3}{2}
\|\nabla_{x}u\|_{2}^{2}.
$$
Also, we have
$$
|J(u(t))|\leq \frac{1}{2}\|\nabla_{x}u(t)\|_{2}^{2}+\frac{1}{\alpha+2}
\|\nabla_{x}u\|_{2}^{2} \leq \frac{\alpha+4}{2(\alpha+2)}
\|\nabla_{x}u(t)\|_{2}^{2}\,.
$$
Therefore,
\begin{equation}
K(u(t))\geq \frac{1}{2}\|\nabla_{x}u\|_{2}^{2}\geq
\frac{\alpha+2}{\alpha+4} J(u).
\label{e24}
\end{equation}
Now, we can derive the decay estimate (\ref{e13}) by similar argument as
lemma \ref{l7}.

\begin{theorem}
Suppose that
$\alpha\leq 4/(n-2)$ ($\alpha< \infty$  if $n\leq 2$),
and suppose that initial data $\{u_{0}, u_{1}\}$ belongs to ${\cal W}$,
and its initial energy $E(0)$ is sufficiently small such that
\begin{equation}
C_{4}   E(0)^{\alpha/2}< 1,
\label{e25}
\end{equation}
where $C_{4}=2 k_{1}c_{*}^{\alpha+2} d_{*}^{\alpha/2}$.
Then, Problem \eqref{P'} has a unique global solution $u\in {\cal W}$
satisfying
$$
u\in L^{\infty}([0, \infty[; H_{0}^{1}(\Omega))
\cap W^{1,\infty}([0, \infty[; L^{2}(\Omega));
$$
furthermore, we have the decay estimate
\begin{equation}
E(t)\leq c\ E(0)\left(g^{-1}(1/t)\right)^2\quad\forall t>0\,.
\label{e26}
\end{equation}
\label{t3}
\end{theorem}

\subsection*{Proof of Theorem \protect{\ref{t1}}}
Since  $u_{0}\in {\cal W}$ and ${\cal W}$ is an open set, putting
$$
T_{1}=\sup\{t\in [0, +\infty): u(s)\in {\cal W} \hbox{ for } 0\leq s\leq t\},
$$
we see that $T_{1}> 0$ and $u(t)\in {\cal W}$ for $0\leq t< T_{1}$. If $T_{1}< T_{\max}<\infty$,
where $T_{\max}$ is the lifespan of the solution, then
$u(T_{1})\in \partial {\cal W}$;
that is
\begin{equation}
K(u(T_{1}))=0 \hbox{ and } u(T_{1})\not =0.
\label{e27}
\end{equation}
We see from lemma \ref{l2} and lemma \ref{l10} that
\begin{equation}
k_{1}\|u(t)\|_{\alpha+2}^{\alpha+2}\leq \frac{1}{2} B(t) \|\nabla_{x}u(t)\|_{2}^{2}
\label{e28}
\end{equation}
for $0\leq t\leq T_{1}$, where we set
\begin{equation}
B(t)=C_{4} E(0)^{\alpha/2}
\label{e29}
\end{equation}
with $C_{4}=2 k_{1}c_{*}^{\alpha+2} d_{*}^{\alpha/2}$.
Next, we put
$$
T_{2}\equiv \sup\{t\in [0, +\infty):    B(s)< 1 \hbox{ for } 0\leq s< t\},
$$
and then we see that $T_{2}> 0$  and $T_{2}=T_{1}$ because $B(t)< 1$
by (\ref{e25}). Then
\begin{equation}
K(u(t))\geq \|\nabla_{x}u(t)\|_{2}^{2}-\frac{1}{2}B(t)
\|\nabla_{x}u(t)\|_{2}^{2}
\geq \frac{1}{2}\|\nabla_{x}u(t)\|_{2}^{2}
\label{e30}
\end{equation}
for $0\leq t\leq T_{1}$.
Moreover, (\ref{e27}) and (\ref{e30}) imply
$$
K(u(T_{1}))\geq \frac{1}{2}\|\nabla_{x}u(T_{1})\|_{2}^{2}> 0
$$
which is a contradiction, and hence, it might be $T_{1}=T_{\max}$.
Therefore, (\ref{e26}) hold true for $0\leq T\leq T_{\max}$,
and such estimate give the desired a priori estimate; that is, the local
solution u can be extended globally \ (i.e., $T_{\max}=\infty$).
The proof of theorem \ref{t3} is now complete.
\hfill $\square$

\paragraph{Remarks:}
${\bf a)}$ By a similar argument as the proof of Theorem \ref{t2}, we can
extend Theorem \ref{t3} to the case $\alpha=0$.

\noindent
${\bf b)}$ It seems to be interesting to study a global decaying $H^{2}$ solution
for Kirchhoff equation with nonlinear source and boundary damping terms
or with nonlinear boundary damping and source terms, also in the case
of polynomial damping term i.e. the following problems
\begin{gather*}
u''-\Phi(\|\nabla_{x}u\|_{2}^{2})\Delta_{x}u+f(u)=0
\quad \hbox{in } \Omega\times [0, +\infty[,\\
u=0 \quad \hbox{on } \Gamma_{0}\times [0, +\infty[,\\
\frac{\partial u}{\partial \nu}=-Q(x)g(u') \quad\hbox{on }
\Gamma_{1}\times [0, +\infty[,\\
u(x, 0)=u_{0}(x),\quad u'(x, 0)=u_{1}(x)\quad\hbox{on }\Omega,
\end{gather*}
and
\begin{gather*}
u''-\Phi(\|\nabla_{x}u\|_{2}^{2})\Delta_{x}u=0
\quad\hbox{in } \Omega\times [0, +\infty[,\\
u=0 \quad\hbox{on } \Gamma_{0}\times [0, +\infty[,\\
\frac{\partial u}{\partial \nu}=-Q(x)g(u')+f(u) \quad\hbox{on }
\Gamma_{1}\times [0, +\infty[,\\
u(x, 0)=u_{0}(x),\quad u'(x, 0)=u_{1}(x)\quad\hbox{on }\Omega,
\end{gather*}
We plan to address these questions in a future investigation.

\subsection*{Appendix}
Let $g(x)$ be the inverse of the function $M(x)$ defined by
$$
M(0)=0,\quad M(x)= \frac{x^{\sigma}}{(\log(-\log  x))^{\beta}}
\quad\hbox{for } 0< x< x_{0},\quad (\sigma, \beta> 0).
$$
For $x=1/t (0< x< x_{0})$ we have
$$
g^{-1}(1/t)=\frac{1}{t^{\sigma}(\log(\log t))^{\beta}}\quad (t\geq t_{0}).
$$
Now, we prove that the function $g(x)$ exists and verifies the hypothesis
(H2). Indeed,
$$
(M(x))'=\frac{x^{\sigma}\Big[\sigma(\log(-\log x))-
\frac{\beta}{\log x}\Big]}{(\hbox{log}(-\log x))^{\beta+1}},\quad (
\sigma, \beta > 0).
$$
When $x$ is near $0$ ($0< x< x_{0}$), it is clear that $(M(x))'\geq 0$, so
$M(x)$ is an increasing continuous function. Thus the function $g$ exists.
We have also
$$
\frac{x}{M(x)}=\frac{(\log(-\log x))^{\beta}}{x^{\sigma-1}}\to 0
$$
as $x\to 0$ if $0<\sigma <1$, so $M(x) \to 0$
(as $x\to 0$) not faster than $x$ (near $0$). We deduce that
$g(x)\to 0$ as $x\to 0$  faster than $x$ i.e. $|g(x)|\leq c|x|$.
We obtain hypothesis (H2). Now,
$M(x)/x$ is a decreasing function; indeed,
$$
\Big(\frac{M(x)}{x}\Big)'
= \frac{x^{\sigma-2}\Big[(\sigma-1)(\log(-\log x))-
\frac{\beta}{\log x}\Big]}{(\log(-\log x))^{\beta+1}}.
$$
For $x=e^{-n}$, and $n$ big, we see that $\left(M(x)/x\right)'\leq 0$.
$g$ is a bijective and decreasing function, so for each $x$ and $y$ near $0$,
 such that $x\leq y$, we have
$M(x)/x\geq M(y)/y$, also there exist unique $x'$ and $y'$ such that
$M(x)=x'$ and $M(y)=y'$ (because $M$ is a bijective function), also
$M(x)$ is an increasing function, thus, we have
$$
x\leq y \Longleftrightarrow M(x)=x'\leq M(y)=y'
$$
Therefore,
\begin{align*}
x'\leq y' &\Longleftrightarrow \frac{x'}{g(x')}\geq \frac{y'}{g(y')}\\
&\Longleftrightarrow \frac{g(x')}{x'}\leq \frac{g(y')}{y'} \quad
\hbox{for } 0< x< x_{0}.
\end{align*}

\paragraph{Acknowledgments.}
The authors wish to thank the anonymous referee for his/her valuable
suggestions.

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\noindent\textsc{Mohammed Aassila }\\
Mathematisches Institut, Universit\"at zu K\"oln \\
Weyertal 86-90, D-50931 K\"oln, Germany.\\
e-mail: aassila@mi.uni-koeln.de
\smallskip

\noindent\textsc{Abbes Benaissa }\\
Universit\'e Djillali Liabes, Facult\'e des Sciences,  \\
D\'epartement de Math\'ematiques,  \\
B. P. 89, Sidi Bel Abbes 22000, Algeria  \\
e-mail: benaissa\_abbes@yahoo.com

\end{document}