\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 109, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/109\hfil Energy decay for solutions]
{Energy decay for wave equations of
$\phi$-Laplacian type with weakly nonlinear dissipation}

\author[A. Benaissa, A. Guesmia\hfil EJDE-2008/109\hfilneg]
{Abbes Benaissa, Aissa Guesmia}  % in alphabetical order

\address{Abbes Benaissa \newline
 Department of Mathematics,
 Djillali Liabes University \\
 P. O. Box 89, Sidi Bel Abbes 22000, Algeria}
\email{benaissa\_abbes@yahoo.com}

\address{Aissa Guesmia \newline
 LMAM, ISGMP, Bat. A, UFR MIM,
 Universit\'e Paul Verlaine - Metz \\
 Ile du Saulcy, 57045 Metz Cedex 01, France}
\email{guesmia@univ-metz.fr}

\thanks{Submitted September 19, 2007. Published August 11, 2008.}
\subjclass[2000]{35B40, 35L70}
\keywords{Wave equation; global existence;
general dissipative term; \hfill\break\indent
rate of decay; multiplier method; integral inequalities}

\begin{abstract}
 In this paper, first we prove the existence of global
 solutions in Sobolev spaces for the initial boundary value problem
 of the wave equation of $\phi$-Laplacian with a general
 dissipation of the form
 $$
 (|u'|^{l-2}u')'-\Delta_{\phi}u+\sigma(t) g(u')=0 \quad\text{in }
 \Omega\times \mathbb{R}_+ ,
 $$
 where $\Delta_{\phi}=\sum_{i=1}^n \partial_{x_i}\bigl(\phi
 (|\partial_{x_i}|^2)\partial_{x_i}\bigr)$. Then we prove general
 stability estimates using multiplier method and general weighted
 integral inequalities proved by the second author in
 \cite{gues1}. Without imposing any growth condition at the
 origin on $g$ and $\phi$, we show that the energy of the system is
 bounded above by a quantity, depending on $\phi$, $\sigma$ and
 $g$, which tends to zero (as time approaches infinity). These
 estimates allows us to consider large class of functions $g$ and
 $\phi$ with general growth at the origin. We give some
 examples to illustrate how to derive from our general estimates
 the polynomial, exponential or logarithmic decay. The results of
 this paper improve and generalize many existing results in the
 literature, and generate some interesting open problems.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{remark}{Remark}[section]
\allowdisplaybreaks

\section{Introduction} In this paper we investigate the
existence of global solutions and their decay properties for the initial
boundary value problem of the wave equation with weak
dissipation
\begin{equation} \label{eP}
\begin{gathered}
(|u'|^{l-2}u')'-\Delta_{\phi} u+\sigma(t) g(u')=0
\quad  \text{in }\Omega\times \mathbb{R}_+\\
u=0\quad \text{on } \Gamma \times \mathbb{R}_+ \\
u(x, 0)=u_{0}(x),\quad  u'(x, 0)=u_{1}(x)\quad  \text{on }\Omega
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$,
$n\in {\mathbb{N}}^*$, with a smooth boundary $\partial\Omega=\Gamma$,
$l\geq 2$, $\phi$, $\sigma$ and $g$ are given functions, and
$\Delta_{\phi}=\sum_{i=1}^n \partial_{x_i}\bigl(\phi
(|\partial_{x_i}|^2)\partial_{x_i}\bigr)$. The functions
$(u_0,u_1)$ are the given initial data.

Concrete examples of \eqref{eP} include the dissipative wave equation
\begin{equation} \label{eP1}
\begin{gathered}
u''-\Delta_{x} u+ g(u')=0 \quad  \text{in }\Omega\times \mathbb{R}_+\\
u=0\quad  \text{on } \Gamma \times \mathbb{R}_+\\
u(x, 0)=u_{0}(x),\quad  u'(x,0)=u_{1}(x)\quad  \text{on }\Omega
\end{gathered}
\end{equation}
where $l=2$, $\phi\equiv 1$ and $\sigma\equiv$const.
The degenerate Laplace operator
\begin{equation} \label{eP2}
\begin{gathered}
u''-\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)+ g(u')=0 \quad
\text{in }\Omega\times \mathbb{R}_+\\
u=0\quad \text{on } \Gamma \times \mathbb{R}_+\\
u(x,0)=u_{0}(x),\quad u'(x, 0)=u_{1}(x)\quad  \text{on }\Omega
\end{gathered}
\end{equation}
where $l=2$, $\phi=s^{{p-2}\over 2}$ with $p\geq 2$ and
$\sigma\equiv$const. And the
quasilinear wave equation
\begin{equation} \label{eP3}
\begin{gathered}
u''-\mathop{\rm div}\Big(\frac{\nabla u}{\sqrt{1+|\nabla
u|^{2}}}\Big)+ g(u')=0 \quad  \text{in }\Omega\times \mathbb{R}_+\\
u=0\quad \text{on } \Gamma \times \mathbb{R}_+\\
u(x, 0)=u_{0}(x),\quad u'(x,0)=u_{1}(x)\quad  \text{on }\Omega
\end{gathered}
\end{equation}
when $l=2$, $\phi=1/\sqrt{1+s}$ and $\sigma\equiv$const.
Problem
\eqref{eP3}, with $-\Delta u'$ instead of $g(u')$, describe the motion of
fixed membrane with strong viscosity. This problem with $n=1$ was
proposed by Greenberg \cite{gree} and Greenberg-MacCamy-Mizel
\cite{grma} as a model of quasilinear wave equation which admits a
global solution for large data. Quite recently,
Kobayashi-Pecher-Shibata \cite{kope} have treated such
nonlinearity and proved the global existence of smooth solutions.
Subsequently, Nakao \cite{nakaa} derived a decay estimate of
the solutions under the assumption that the mean curvature of
$\partial \Omega$ is non positive.

Our purpose in this paper is: firstly to give an existence and
uniqueness theorem for global solutions in Orlitz-Sobolev spaces to the
problem \eqref{eP}.

Secondly (for the stabilization problem), the aim of this paper is
to obtain an explicit and general decay rate, depending on
$\sigma$, $g$ and $\phi$, for the energy of solutions of \eqref{eP}
without any growth assumption on $g$ and $\phi$ at the origin, and
on $\sigma$ at infinity. More precisely, we intend to obtain a
general relation between the decay rate for the energy (when $t$
goes to infinity), the functions $\sigma$, $\phi$ and $g$. The
proof is based on some general weighted integral inequalities
proved by the second author in \cite{gues1} and some properties of convex
functions, in particular, the dual function of convex
function to use the general Young's inequality and
Jensen's inequality (instead of H{\"o}lder
inequality widely used in the classical case of linear or
polynomial growth of $g$ at the origin) in objective to
prove our general decay estimate under a general growth of $g$ at the origin. These
arguments of convexity were used for the first time (in our knowledge) by Liu and Zuazua
{\bf\cite{lizu}}, and then by Eller, Lagnese and Nicaise {\bf\cite{elle}}
and Alabau-Boussouira {\bf\cite{alab}}.

In particular, we can consider the cases where $g$ and $\phi$
degenerate near the origin polynomially, between polynomially and
exponentially, exponentially or faster than exponentially.
This kind of growth was considered by Liu and Zuazua
{\bf\cite{lizu}} and Alabau-Boussouira {\bf\cite{alab}}
for the wave equation, and Eller, Lagnese and Nicaise {\bf\cite{elle}} for Maxwell system.
So we complement the results obtained in \cite{bemi} and \cite{bena1}.

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

The paper is organized as follow: in section 2, we give some
hypotheses and we announce the main results of this paper. In
section 3 and section 4, we prove all the announced results. In
section 5, we give some applications. Finally, we conclude and give
some comments and open questions in section 6.

\section{Preliminaries and main results}
We use the following hypotheses:
\begin{itemize}
\item[(H1)] $\sigma:\mathbb{R}_+\to ]0,+\infty[$
is a non increasing function of class $C^{1} (\mathbb{R}_+)$
satisfying
\begin{equation}
\int_{0}^{+\infty} \sigma(\tau)\, d\tau=+\infty .
\label{m1}
\end{equation}

\item[(H2)] $\phi:\mathbb{R}_+\to \mathbb{R}_+$ is
of class $C^{1}(]0,+\infty[)\cap C([0,+\infty[)$ satisfying: $\phi
(s)> 0$ on $]0,+\infty[$ and $\phi$ is non decreasing.

\item[(H3)] $g:\mathbb{R}\to \mathbb{R}$ is a non
decreasing function of class $C (\mathbb{R})$ such that there
exist $\epsilon_1,\,c_1,\,c_2> 0$, $l-1\leq r$, $(n-2)r\leq n+2$
and a convex and increasing function $G:\mathbb{R}_+\to
\mathbb{R}_+$ of class $C^1 (\mathbb{R}_+)\cap C^2 (]0,+\infty[)$
satisfying $G(0)=0$, and $G'(0)=0$ or $G$ is linear on
$[0,\epsilon_1]$ such that
\begin{gather}
c_1 |s|^{l-1}\leq |g(s)|\leq c_2 |s|^r\quad \text{if}\,\,|s|\geq
\epsilon_1 , \label{x2} \\
|s|^l +g^2 (s)\leq G^{-1} (sg(s))\quad\text{if}\,\, |s|\leq
\epsilon_1. \label{x3}
\end{gather}
\end{itemize}

\begin{remark}  \label{ddr} \rm
1. We have $\int_0^{+\infty}\phi (\tau)d\tau=+\infty$, and
$s\mapsto\int_0^{s}\phi (\tau)d\tau$ is a bijection from
$\mathbb{R}_+$ to $\mathbb{R}_+$.

\noindent 2. The function ${\tilde \phi} (s)={1\over 2}\int_0^s \phi
(\tau)\,d\tau$ is a convex function. Indeed, let $x_{1}\not=0$ and
$x_{1}\not=0$ such that $x_{1}< x_{2}$, as $\phi$ is of class
$C^{1} ([x_{1}, x_{2}])$ and a non decreasing function, then
${\tilde \phi}$ is a convex function. Now if $x_{1}=0$, we have for
all $0\leq \lambda\leq 1$
$$
{\tilde \phi}(\lambda x_{2})={1\over 2}\int_{0}^{\lambda x_{2}}
\phi(s)\, ds ={1\over 2}\lambda\int_{0}^{x_{2}} \phi(\lambda z)\, dz
$$
where we have make the  change of variable $s=\lambda
z$. As $\phi$ is a non decreasing function and $\lambda x_{2}\leq
x_{2}$ for all $\lambda \in [0,  1]$, then
$$
{\tilde \phi}(\lambda x_{2})\leq \lambda {\tilde \phi}(x_{2}).
$$

\noindent 3. If $g$ satisfies
$$
H (|s|)\leq |g(s)|\leq H^{-1} (|s|)\quad \text{if}\,\,|s|\leq
\epsilon_1
$$
for a function $H:\mathbb{R}_+\to\mathbb{R}_+$
satisfying $H'(0)=0$ or $H$ being linear on
$[0,{\sqrt{{{\epsilon_1}\over{\delta}}}}]$ where
$\delta=2\max\{1,\epsilon_1^{l-2}\}$ such that the function
$s\mapsto {\sqrt{s}}H({\sqrt{s}})$ is convex and increasing
function from $\mathbb{R}_+$ to $\mathbb{R}_+$ of class $C^1
(\mathbb{R}_+)\cap C^2 (]0,+\infty[$, then the condition
(\ref{x3}) is satisfied for
$$
G (s)={\sqrt{{s\over {\delta}}}}H({\sqrt{{s\over {\delta}}}}).
$$
In the other hand, $g$ satisfies (H3) for any $\epsilon'_1\in
]0,\epsilon_1]$ (with some $c'_1,\,c'_2 >0$ instead of $c_1,\,
c_2$, respectively).
\end{remark}

Now we define (as before) $\tilde{\phi}(s)={1\over
2}\int_0^s\phi(\tau)d\tau$
and the energy associated to the solution
of the system \eqref{eP} by the following formula:
\begin{equation}
E(t)=\frac{l-1}{l}\int_{\Omega}|u'|^{l}
dx+\int_{\Omega}\sum_{i=1}^n \tilde{\phi}(|\partial_{x_i} u|^{2})dx.
\label{x4}
\end{equation}
By a simple computation, we have
\begin{equation}
E'(t)=-\sigma(t)\int_{\Omega}u'g(u') dx, \label{x5}
\end{equation}
so $E$ is non negative and non increasing function. We first state
two lemmas which will be needed later.

\begin{lemma}[Sobolev-Poincar\'e's inequality] \label{dl1}
Let $p>1$ and
$q>1$ with $(n-p)q\leq np$, then there is a constant
$c_{*}=c_{*}(\Omega, p, q)$ such that
$$
\| u\|_{q}\leq c_{*}\|\nabla u\|_{p}\quad \text{for }\quad u\in
W_{0}^{1, p}(\Omega).
$$
The case $p=q=2$ gives the known Poincar\'e's inequality.
\end{lemma}

\begin{lemma}[Guesmia \cite{gues1}] \label{ll1}
Let $E:\mathbb{R}_+\to \mathbb{R}_+$ differentiable function,
$\lambda\in \mathbb{R}_+$ and $\Psi:\mathbb{R}_+\to \mathbb{R}_+$
convex and increasing function such that $\Psi (0)=0$. Assume that
\begin{gather*}
 \int_s^{+\infty} \Psi(E(t))\,dt\leq E (s),\quad   \forall s\geq 0\\
 E'(t)\leq\lambda E(t),\quad   \forall t\geq 0.
\end{gather*}
Then $E$ satisfies the estimate
$$
E(t)\leq e^{\tau_0\lambda T_0} d^{-1}
\Bigl(e^{{\lambda}(t-h(t))} \Psi \Bigl(\psi^{-1} \Bigl(h(t)+\psi
(E(0))\Bigr)\Bigr)\Bigr),\quad \forall t\geq 0
$$
where
\begin{gather*}
\psi(t)=\int_t^1 {1\over{\Psi (s)}}\,ds,\quad \forall t> 0, \\
d(t)=\begin{cases} \Psi (t) & \text{if } \lambda =0,\\
\int_0^t {{\Psi (s)}\over{s}}\,ds  & \text{if } \lambda >0,
\end{cases}
\quad \forall t\geq 0,\\
h(t)=\begin{cases} K^{-1}(D(t)), & \text{if } t> T_0,\\
0 &\text{if } t\in [0,T_0]
\end{cases}
\\
K(t)=D(t)+{{\psi^{-1} \bigl(t+\psi (E(0))\bigr)}\over
{\Psi \bigl(\psi^{-1} \bigl(t+\psi
(E(0))\bigr)\bigr)}}e^{{\lambda}t},\quad \forall t\geq 0,\\
D(t)=\int_{0}^{t}e^{{\lambda}s}\, ds,\quad\forall t\geq 0,
\\
T_0=D^{-1}\Bigl({{E(0)}\over{\Psi (E(0))}}\Bigr),\quad
\tau_0=\begin{cases} 0,& \text{if } t> T_0,\\
1,& \text{if } t\in [0,T_0].
\end{cases}
\end{gather*} %\label{ll1}
\end{lemma}


\begin{remark} \label{re1} \rm
If $\lambda =0$ (that is $E$ is non increasing), then we have
\begin{equation}
E(t)\leq \psi^{-1}\Bigl( h(t)+\psi (E(0))\Bigr),\quad \forall t\geq 0
\label{x6}
\end{equation}
where $\psi (t)=\int_t^1 {1\over{\Psi (s)}}\,ds$ for $t> 0$,
$h(t)=0$ for $0\leq t\leq {{E(0)}\over{\Psi (E(0))}}$ and
$$
h^{-1}(t)=t+{{\psi^{-1} \Bigl(t+\psi (E(0))\Bigr)}\over{\Psi
\Bigl(\psi^{-1} \Bigl(t+\psi (E(0))\Bigr)\Bigr)}},\quad
 t> 0.
$$
This particular result generalizes the one obtained by  Martinez
\cite{mart} in the particular case $\Psi(t)=dt^{p+1}$ with $p\geq
0$ and $d>0$, and improves the one obtained by  Eller, Lagnese and
 Nicaise \cite{elle}.
\end{remark}

\begin{proof}[Proof of Lemma \ref{ll1}]
Because $E'(t)\leq \lambda E(t)$ implies
 $E(t)\leq e^{\lambda (t-t_0)}E(t_0)$ for all $t\geq
t_0\geq 0$, then, if $E(t_0)=0$ for some
$t_0\geq 0$, then $E(t)=0$ for all $t\geq t_0$, and then there is
nothing to prove in this case. So we assume that $E(t)> 0$ for all
$t\geq 0$ without lose of generality. Let
$$
L (s)=\int_s^{+\infty} \Psi (E (t))\, dt,\quad\forall s\geq 0.
$$
We have
$ L (s)\leq E (s)$, for all $s\geq 0$.
The function $L$ is positive, decreasing and of class $C^1
(\mathbb{R}_+)$ satisfying
$$
-L'(s)=\Psi (E (s))\geq \Psi (L(s)),\quad \forall s\geq 0.
$$
The function $\psi$ is decreasing,
then
$$
\Bigl(\psi (L(s))\Bigr)'={{- L'(s)}\over{\Psi ( L (s))}}\geq 1,\quad
\forall s\geq 0.
$$
Integration on $[0,t]$, we obtain
\begin{equation}
\psi(L(t))\geq t+\psi (E(0)),\quad \forall t\geq 0. \label{x7}
\end{equation}
Since $\Psi$ is convex and $\Psi (0)=0$, we have
$$
\Psi (s)\leq \Psi (1)s,\,\,\forall s\in [0,1]\quad\text{and}\quad
\Psi (s)\geq \Psi (1)s,\,\,\forall s\geq 1,
$$
then $\lim_{t\to 0}\psi(t)=+\infty$ and $[\psi
(E(0)),+\infty[\subset\,\text{Image}\,(\psi)$. Then (\ref{x7})
implies that
\begin{equation}
L (t)\leq \psi^{-1}\Bigl(t+\psi (E(0))\Bigr),\quad \forall t\geq 0.
\label{x8}
\end{equation}
Now, for $s\geq 0$, let
$$
f_s(t)=e^{-\lambda t}\int_{s}^{t} e^{{\lambda}\tau}\,d\tau,\quad
\forall t\geq s.
$$
The function $f_s$ is increasing on $[s,+\infty[$ and strictly
positive on $]s,+\infty[$ such that
$$
f_s (s)=0 \quad\text{and}\quad f'_s(t)+\lambda f_s
(t)=1,\quad\forall t\geq s\geq 0,
$$
and the function $d$ is well defined, positive and increasing such
that
$$
d(t)\leq \Psi(t)\quad\text{and}\quad \lambda td'(t)=\lambda
\Psi(t),\quad\forall t\geq 0,
$$
then
\begin{align*}
\partial_{\tau} \Bigl(f_s(\tau) d(E(\tau))\Bigr)
&=f'_s (\tau) d(E(\tau))+f_s (\tau) E'(\tau)d'(E(\tau)) \\
&\leq \Bigl(1-\lambda f_s (\tau)\Bigr)\Psi(E(\tau)) +\lambda f_s
(\tau)\Psi (E(\tau))\\
&=\Psi (E(\tau)),\quad \forall \tau\geq s\geq 0.
\end{align*}
Integrating on $[s, t]$, we obtain
\begin{equation}
L (s)\geq \int_{s}^{t} \Psi (E(\tau)) \,d\tau\geq f_{s} (t)d (E
(t)),\quad\forall t\geq s\geq 0. \label{x9}
\end{equation}
Since $\lim_{t\to +\infty} d(s)=+\infty$, $d(0)=0$ and $d$ is
increasing, then (\ref{x8}) and (\ref{x9}) imply
\begin{equation}
E (t)\leq d^{-1}\Bigl(\inf_{s\in[0,t[}{{\psi^{-1}\Bigl(s+\psi
(E(0))\Bigr)}\over{f_{s} (t)}}\Bigr),\quad\forall t>  0. \label{x10}
\end{equation}
Now, let $t> T_0$ and
$$
J(s)={{\psi^{-1}\Bigl(s+\psi (E(0))\Bigr)}\over{f_{s} (t)}},\quad
\forall s\in [0,t[.
$$
The function $J$ is differentiable and we have
$$
J'(s)=f^{-2}_{s} (t) \Bigl[e^{-\lambda (t-s)}\psi^{-1}\Bigl(s+\psi
(E(0))\Bigr)-f_{s}(t) \Psi\Bigl(\psi^{-1}\Bigl(s+\psi
(E(0))\Bigr)\Bigr)\Bigr].
$$
Then
$$
J'(s)=0\,\,\Leftrightarrow\,\, K(s)=D(t)\quad\text{and}\quad J'(s)<
0\,\,\Leftrightarrow\,\, K(s)<D(T).
$$
Since $K(0)={{E(0)}\over{\Psi (E(0))}}$, $D(0)=0$ and $K$ and $D$
are increasing (because $\psi^{-1}$ is decreasing and $s\mapsto
{{s}\over{\Psi(s)}}$, $s>0$, is non increasing thanks to the fact
that $\Psi$ is convex). Then, for $t>T_0$,
$$
\inf_{s\in[0,t[}J(s)=J\Bigl(K^{-1}(D(t))\Bigr)=J(h(t)).
$$
Since $h$ satisfies $J'(h(t))=0$, we conclude from (\ref{x10})
our desired estimate for $t>T_0$.

For $t\in [0,T_0]$, we have just to note that $E'(t)\leq \lambda
E(t)$ and the fact that $d\leq \Psi$ imply
$$
E(t)\leq e^{\lambda t} E(0)\leq  e^{\lambda T_0} E(0)\leq e^{\lambda
T_0}\Psi^{-1} \Bigl(e^{\lambda t}\Psi (E(0))\Bigr)\leq e^{\lambda
T_0}d^{-1} \Bigl(e^{\lambda t}\Psi (E(0))\Bigr).
$$
\end{proof}

\begin{remark} \rm
Under the hypotheses of Lemma \ref{ll1}, we have
$\lim_{t\to +\infty} E(t)=0$.
Indeed, we have just to choose $s={1\over 2}t$ in (\ref{x10})
instead of $h(t)$ and note that $d^{-1} (0)=0$, $\lim_{t\to +\infty}
\psi^{-1} (t)=0$ and $\lim_{t\to +\infty} f_{{1\over 2}t} (t)> 0$.
\label{rr1}
\end{remark}


Before stating the global existence theorem, we will give some
notions of the theory of Orlitz spaces (see \cite{adam} and
\cite{krru}) which is suitable for a large class of
quasilinear equations.

\begin{definition} \label{def2.1} \rm
A function $\Phi: \mathbb{R}_{+}\to \mathbb{R}_{+}$ is called an N-function
if it is continuous, convex, strictly increasing and such that
$$
\lim_{s\to 0} \frac{\Phi(s)}{s}=0 \text{ and }
\lim_{s\to 0} \frac{\Phi(s)}{s}=+\infty.
$$
\end{definition}

The N-function complementary to $\Phi$ is defined by
${\tilde \Phi}(s)=\max_{\sigma\geq 0}(s\sigma-\Phi(\sigma))$.
The Simonenko indices $p(\Phi)$ and $q(\Phi)$ are defined by
$$
p(\Phi)=\inf_{t> 0} \frac{t \Phi'(t)}{\Phi(t)}, \quad
q(\Phi)=\sup_{t> 0} \frac{t \Phi'(t)}{\Phi(t)}.
$$
Clearly, $1\leq p(\Phi)\leq q(\Phi)\leq \infty$, and if $q(\Phi)<
\infty$, then
\begin{equation}
p(\Phi)\frac{\Phi(t)}{t}\leq \Phi'(t)\leq q(\Phi)\frac{\Phi(t)}{t}
\text{ for all } t> 0.
\label{o6}
\end{equation}
Integrating these inequalities, one sees that (\ref{o6}) is
equivalent to
\begin{equation}
\frac{\Phi(t)}{t^{p(\Phi)}} \text{ is increasing,  and }
\frac{\Phi(t)}{t^{q(\Phi)}} \hbox{ is decreasing for all } t> 0.
\label{o7}
\end{equation}
We say that two N-function $\Phi$ and $\Phi_{1}$ are equivalent if
there exists two constants $C_{1}$ and $C_{2}$ such that
$$
C_{1} \Phi_{1}(t)\leq \Phi(t)\leq C_{2}\Phi_{1}(t) \quad \text{ for all }t\geq 0.
$$
We denote by $i(\Phi)$ and $I(\phi)$ the reciprocal Boyd indices of
$\Phi$. Sometimes, $i(\Phi)$ is called the lower index and $I(\Phi)$
the upper index of $\Phi$. We have the following characterisations:
$$
i(\Phi)=\sup_{\Phi_{1}\sim \Phi} p(\Phi_{1}) \quad\text{and}\quad
I(\Phi)=\inf_{\Phi_{1}\sim \Phi} q(\Phi_{1}).
$$
Let $\Phi$ be an N-function satisfying $I(\Phi)< \infty$. The Orlitz
space $L^{\Phi}=L^{\Phi}(\Omega)$ is the space of all measurable
functions $f$ defined on $\Omega$ such that $\int_{\Omega}
\Phi(|f|)\, dx< +\infty$. It is endowed with the norm
$$
\|f\|_{\Phi}=\inf\big\{\lambda> 0;
\int_{\Omega}\Phi\big(\frac{|f|}{\lambda}\big)\, dx
\leq 1\big\}
$$
For every $f\in L^{\Phi}$ and every $g\in L^{{\tilde \Phi}}$ the
following H\"{o}lder type inequality holds:
$$
\int_{\Omega}|fg|\, dx \leq 2 \|f\|_{\Phi} \|g\|_{\tilde\Phi}.
$$
Let us denote by $\mathcal{W}^{1, \Phi}=\mathcal{W}^{1, \Phi}(\Omega)$
the space of all functions in $L^{\Phi}$ such that the
distributional partial derivatives belong to $L^{\Phi}$, and by
$\mathcal{W}_{0}^{1, \Phi}(\Omega)$ the closure of the test functions
in this space. Such spaces are well known in the literature as
Orlitz-Sobolev spaces (see \cite{adam}). We have Poincar\'e's
inequality for Orlitz-Sobolev spaces
\begin{equation}
\|u\|_{\Phi}\leq C \|\nabla_{x} u\|_{\Phi}, \quad u\in \mathcal{W}_{0}^{1, \Phi}(\Omega), \label{o1}
\end{equation}
so that $\|\nabla_{x} u\|_{\Phi}$ defines an equivalent norm in
$\mathcal{W}_{0}^{1, \Phi}(\Omega)$. By $\mathcal{W}_{0}^{-1, {\tilde
\Phi}}=\mathcal{W}_{0}^{-1, {\tilde \Phi}}(\Omega)$ we denote the dual
space of $\mathcal{W}_{0}^{1, \Phi}(\Omega)$.

The classical Sobolev embedding theorem has been extended into
Orlitz setting. In the following we only need that if $\Phi$ is an
N-function such that for $n'=\frac{n}{n-1}$\ \ $(n>1)$
\begin{equation}
 \int_{1}^{+\infty}\frac{\tilde \Phi(s)}{s^{n'+1}}\, ds=+\infty,
\label{o2}
\end{equation}
then it is possible to define an optimal N-function $\Phi^{*}$ such
that the embedding
\begin{equation}
\mathcal{W}_{0}^{1, \Phi}(\Omega)\hookrightarrow L^{\Phi^{*}}
\label{o3}
\end{equation}
holds; optimality means that $L^{\Phi^{*}}$ is the smallest Orlitz
space for which (\ref{o3}) holds. If the integral in (\ref{o2}) is
finite or $n=1$, then
\begin{equation}
\mathcal{W}_{0}^{1, \Phi}(\Omega)\hookrightarrow L^{\infty}(\Omega).
\label{o4}
\end{equation}
We assume that $\Phi$ is N-function satisfying
\begin{equation}
i(\Phi)\in \left]\frac{2n}{n+2}, +\infty\right[\cap ]1,
+\infty[\quad \text{and}\quad  I(\Phi)<+\infty. \label{o5}
\end{equation}
and where  $\Phi^{*}$ is the N-function from (\ref{o3}), and
identifying $L^{2}$ with its dual, we have
$$
W_{0}^{1, \Phi}\hookrightarrow L^{2}\hookrightarrow W^{-1,
{\tilde\Phi}}.
$$
These dense inclusions also hold by the Sobolev embedding (\ref{o4})
in the case that the integral in (\ref{o2}) is finite. Set
$$
\Phi(s)=\frac{1}{2}\int_{0}^{s^{2}}\phi(t)\, dt.
$$

\begin{theorem}
Assume that $(u_{0}, u_{1})\in \mathcal{W}_{0}^{1, \Phi}(\Omega)\times
L^{2}(\Omega)$. Then problem \eqref{eP} admits a unique strong
solution on $\Omega\times [0, \infty[$ in the class
$$
C([0, \infty[, \mathcal{W}_{0}^{1, \Phi}(\Omega))\cap C^{1}([0,
\infty[, L^{2}(\Omega))
$$
\label{the1}
\end{theorem}

\begin{proof} The theory of maximal monotone operators associated
with subdifferentials (see \cite{gues11}, \cite{gues12},
\cite{brez} and \cite{akot1}) imply that,
for every $(u_{0}, u_{1})\in \mathcal{W}_{0}^{1, \phi}(\Omega)\times
L^{2}(\Omega)$, the problem \eqref{eP} admits
a unique global strong solution.
\end{proof}

Our main result on stabilization is the following.

\begin{theorem} \label{thh1}
Assume that {\rm (H1)-(H3)} hold. Let
${\tilde{\sigma}}(t)=\int_0^t \sigma(\tau)d\tau$. Then there exist
$\omega,\,\epsilon_0> 0$ such that the energy $E$ satisfies
\begin{equation}
E(t)\leq {\varphi_1}\Bigl(\psi^{-1}\Bigl(h({\tilde{\sigma}}(t))+\psi
({\varphi_1}^{-1} (E(0)))\Bigr)\Bigr),\quad\forall t\geq 0 \label{x11}
\end{equation}
where
\begin{gather*}
\psi (t)=\int_t^1 {1\over {\omega\varphi(\tau)}}d\tau\quad\text{for }
t>0; \quad h(t)=0\quad\text{ for } 0\leq t\leq
{{E(0)}\over{\omega\varphi(E(0))}};
\\
h^{-1}(t)=
t+{{\psi^{-1}\Bigl(t+\psi(E(0))\Bigr)}\over{\omega\varphi
\Bigl(\psi^{-1}\Bigl(t+\psi(E(0))\Bigr)\Bigr)}},\quad\text{for }
t> 0;
\\
\varphi (s)=\begin{cases}
{\tilde{\phi}} (s) &\text{if $r=1$ and $G$ is linear on }
 [0, \epsilon_1],\\
{{({\tilde{\phi}} (s))^{1+{1\over r}}}\over{s^{1/r}}} &\text{if $r\ne 1$ and $G$ is linear on }[0, \epsilon_1],\\
{{2\epsilon_{0}s^2}\over{{{\tilde{\phi}}^{-1}(s)}}}
G'\left({{\epsilon^{2}_{0}s^2}\over{{{\tilde{\phi}}^{-1}(s)}}}\right)
 &\text{if }G'(0)=0;
\end{cases}
\\
\varphi_1 (s)=\begin{cases}
{\tilde{\phi}} (s)&\text{if $G$ is linear on } [0, \epsilon_1],\\
s &\text{if } G'(0)=0.
\end{cases}
\end{gather*}
\end{theorem}

\begin{remark} \rm
1. Under the hypotheses of Theorem \ref{thh1} and thanks to Remark
\ref{rr1}, we have  strong stability of \eqref{eP}; that is,
\begin{equation}
\lim_{t\to +\infty} E(t)=0. \label{x12}
\end{equation}

2. Thanks to (H2) and (H3), the function $\varphi$ (defined in
Theorem \ref{thh1}) is of class $C^1 (\mathbb{R}_+)$ and satisfies
the same hypotheses as the function $\Psi$ in Lemma \ref{ll1}.
Then we can apply Lemma \ref{ll1} for $\Psi=\omega\varphi$.
%\label{rdl1}

3. We obtain same results for the  problem
$$
u''-\Delta_{\phi}u-\sigma(t) \mathop{\rm div}(\psi(|\nabla_{x} u'|^{2})\nabla_{x}
u')=0
$$
such that
\begin{gather}
c_1 \leq \psi(s^{2}) \leq c_2 \quad \text{if }|s|\geq \epsilon_1 , \\
|s|^2 +\psi^2 (s)\leq G^{-1} (sg(s))\quad\text{if } |s|\leq
\epsilon_1.
\end{gather}
\end{remark}

Using (\ref{x11}), we give several
 significant examples of growth at the origine of $g$ and
 $\phi$, and
 the corresponding decay estimates. Some of these examples
 were given (in less general form) by Liu and Zuazua
 {\bf\cite{lizu}} and Alabau-Boussouira
 {\bf\cite{alab}} for the wave equation,
 and Eller, Lagnese and Nicaise {\bf\cite{elle}} for
 Maxwell system.
\subsection*{Polynomial or logarithmic growth for $\phi$ and polynomial growth
for $g$} If $\phi (t)=ct^{m}\Bigl(\ln (t+1)\Bigr)^q$\ (degeneracy of
finite order) and $G(t)=c't^{{{p+1}\over 2}}$ for $c,\,c'> 0$ (that
is $c_1'|s|^{{{l(p+1)-2}\over{2}}}\leq |g(s)|\leq c_2'|s|^{1\over p}$
on
$[-\epsilon_1,\epsilon_1]$ for some $c_1',\,c_2'> 0$),
$m\geq 0$, $q\geq -m$ and $p\geq 1$ (note that $c_3 s^{m+q+1}\leq
{\tilde{\phi}}(s)\leq c'_3 s^{m+q+1}$ for $c_3,\,c'_3> 0$ when $s$
is near 0), then there exists $\alpha> 0$ such
that for all $t\geq 0$
$$
E(t)\leq \begin{cases}
\alpha e^{-\omega {\tilde{\sigma}}(t)}&\text{if }(m+q,p)=(0,1), \\
\alpha\Bigl({\tilde{\sigma}}(t)+1\Bigr)^{-{{{r(m+q+1)}\over{(r+1)(m+q)}}}}
&\text{if }m+q>0, \; p=1 ,\; r>1, \\
\alpha\Bigl({\tilde{\sigma}}(t)+1\Bigr)^{-{{{2(m+q+1)}\over{2p(m+q)+p-1}}}}
 &\text{otherwise}.
\end{cases}
$$
Moreover, we can obtain more precise  rate of decay, in the case
$\phi(s)=s^{m}$ with $m\geq 0$ and
$$
c_{1}|s|^{p}\leq |g(s)|\leq c_{2} |s|^{\theta}\quad\text{if }
|s|\leq \epsilon_1
$$
where $\frac{1}{p}\leq \theta\leq p$.
We have the following estimates:
If $l\geq p+1$, then for all $t\geq 0$,
$$
E(t)\leq \begin{cases}
\alpha e^{-\omega {\tilde{\sigma}}(t)}
&\text{if } 2m+1\leq \theta ,\\
 \alpha ({\tilde{\sigma}}(t)+1)^{-\frac{2\theta(m+1)}{2m+1-\theta}}
&\text{if } 2m+1> \theta .
\end{cases}
$$
If $l< p+1$, then for all $t\geq 0$,
$$
E(t)\leq \begin{cases}
\alpha ({\tilde{\sigma}}(t)+1)^{-\frac{2\theta(m+1)}{2m+1-\theta}}
&\text{if } l\geq \frac{2\theta (m+1)(p+1)}{(\theta+1)(2m+1)}, \\
\alpha ({\tilde{\sigma}}(t)+1)^{-\frac{l}{p+1-l}}
&\text{if } l< \frac{2\theta (m+1)(p+1)}{(\theta+1)(2m+1)}.
\end{cases}
$$

\subsection*{Polynomial or logarithmic growth for $\phi$ and exponential growth
for $g$} If $\phi (t)=ct^{m}\Bigl(\ln (t+1)\Bigr)^q$  (degeneracy of
finite order) and $H(|s|)\leq |g(s)|\leq H^{-1} (|s|)$ on
$[-\epsilon_1,\epsilon_1]$ where $H(s)={1\over
s}e^{-{s^{-\gamma}}}$, $m\geq 0$, $q\geq -m$ and $c,\gamma> 0$ (note
that $c_3 s^{m+q+1}\leq
{\tilde{\phi}}(s)\leq c'_3 s^{m+q+1}$ and $G(s)=e^{-2^{{{\gamma}\over
{2}}}s^{-{{\gamma}\over {2}}}}$ for $c_3,\,c'_3> 0$ when $s$
is near 0, and $\psi (s)\leq c'_1 e^{c'_2 s^{-{{\gamma}\over {2}}}}$ on
$]0,1]$ for $c'_1,\,c'_2> 0$),
then there exist $\alpha,\beta> 0$
such that
$$
E(t)\leq \beta\Bigl(\ln
\Bigl(\alpha{h(\tilde{\sigma}(t))}+2\Bigr)\Bigr)^{{{-2(m+q+1)}\over{\gamma(2(m+q)+1)}}},\quad\forall
t\geq 0.
$$

\subsection*{Polynomial or logarithmic growth for $\phi$ and faster than
exponential growth for $g$} If $\phi (t)=ct^{m}\Bigl(\ln
(t+1)\Bigr)^q$  (degeneracy of finite order) and $H(|s|)\leq
|g(s)|\leq H^{-1} (|s|)$ on $[-\epsilon_1,\epsilon_1]$ where
$H(s)={1\over s}H_n(s)$, $m\geq 0$, $q\geq -m,
c,\,\gamma> 0$ and
$$
H_1 (s)=
e^{-s^{-\gamma}}\quad\hbox{and}\quad H_n
(s)=e^{-{{1}\over{H_{n-1}(s)}}},\,\,n=2,\,3,\cdots,
$$
then (as in the example 2) there exist
$\alpha,\beta, \delta> 0$ such that
$$
E(t)\leq \beta\Bigl({\bar H}_n(h({\tilde\sigma}(t)))\Bigr)
^{{{-2(m+q+1)}\over{\gamma(2(m+q)+1)}}},\quad\forall
t\geq 0.
$$
where
$$
{\bar H}_1 (t)=\ln (\alpha t
+ \delta)\quad\hbox{and}\quad {\bar H}_n
(t)=\ln ({\bar H}_{n-1}
(t)),\,\,n=2,\,3,\cdots .
$$

\subsection*{Polynomial or logarithmic growth for $\phi$ and between polynomial
and exponential growth for $g$} If $\phi (t)=ct^{m}\Bigl(\ln
(t+1)\Bigr)^q$ \ (degeneracy of finite order) and $H(|s|)\leq
|g(s)|\leq H^{-1} (|s|)$ on $[-\epsilon_1,\epsilon_1]$ where
$H(s)={1\over s}e^{-(H_n (s))^{\gamma}}$, $\gamma> 1$, $m\geq 0$,
$q\geq -m, c> 0$ and
$$
 H_1 (s) =-\ln s\quad\hbox{and}\quad H_n
 (s)=\ln (H_{n-1}(s)),\,\,n=2,\,3,\cdots
 $$
(then $G(s)=e^{-({{-1}\over{2}}\ln
{{s}\over{2}})^{\gamma}}$ when $s$ is near $0$), then there exist
$\alpha,\,\beta, \delta> 0$ such that
$$
E(t)\leq \beta e^{-{{2(m+q+1)}\over{2(m+q)+1}}{\bar H}_n (h({\tilde\sigma}(t)))},\quad\forall
t\geq 0.
$$
where
$$
{\bar H}_1 (t)=\ln (\alpha t
+ \delta)^{1\over{\gamma}}\quad\hbox{and}\quad {\bar H}_n
(t)=e^{{\bar H}_{n-1}
(t)},\,\,n=2,\,3,\cdots .
$$

\subsection*{Exponential growth for $\phi$  (degeneracy of infinite order) and
linear growth for $g$} If $\phi (t)=e^{-t^{-\gamma}}, \gamma> 0$,
(note that $c'_{1} t^{\gamma+1}e^{-t^{-\gamma}}\leq
{\tilde{\phi}}(t)\leq c'_{2} t^{\gamma+1}e^{-t^{-\gamma}}$ for
$c'_1,\,c'_2> 0$ when $s$ is near 0) then there exist
$\alpha,\,\beta>  0$ such that for all $t\geq 0$
$$
E(t)\leq \begin{cases}
\beta (h({\tilde\sigma} (t)))^{{{-r}\over{r+1}}}\Bigl(\ln
(\alpha h({\tilde{\sigma}}(t))+2)\Bigr)^{-\frac{\gamma+1}{\gamma}},
&\text{if }r>1, \\
 \beta (h({\tilde\sigma} (t)))^{-1}\Bigl(\ln
(\alpha h({\tilde{\sigma}}(t))+2)\Bigr)^{-\frac{\gamma+1}{\gamma}},
&\text{if }r=1.
\end{cases}
$$

\subsection*{Faster than exponential growth for $\phi$ (degeneracy of infinite
order) and linear growth for $g$} If
$\phi (t)=e^{-e^{t^{-\gamma}}}, \gamma> 0$,
(note that $c'_{1} s^{\gamma+1}e^{-e^{t^{-\gamma}}}
e^{-t^{-\gamma}}\leq {\tilde{\phi}}(t)\leq c'_{2}
t^{\gamma+1}e^{-e^{t^{-\gamma}}} e^{-t^{-\gamma}}$ for $c'_1,\,c'_2>
0$ when $t$ is near 0) then there exist $\alpha,\,\beta>  0$ such
that for all $t\geq 0$,
$$
E(t)\leq \begin{cases}
\beta (h({\tilde\sigma}(t)))^{{{-r}\over{r+1}}} \Bigl(\ln
(\alpha{h(\tilde{\sigma}}(t))+2)\Bigr)^{-1}\Bigl(\ln\Bigl({{r}\over{r+1}}\ln
(\alpha h({\tilde{\sigma}}(t))+3)\Bigr)\Bigr)^{-\frac{\gamma+1}{\gamma}},
&\text{if }r>1, \\
 \beta (h({\tilde\sigma}(t)))^{-1} \Bigl(\ln
(\alpha h({\tilde{\sigma}}(t))+2)\Bigr)^{-1}\Bigl(\ln\Bigl(\ln
(\alpha h({\tilde{\sigma}}(t))+3)\Bigr)\Bigr)^{-\frac{\gamma+1}{\gamma}},
&\text{if }r=1.
\end{cases}
$$

\subsection*{Faster than polynomials, less than exponential growth for
$\phi$ (degeneracy of infinite order) and linear growth for $g$}
If $\phi
(t)=e^{-(-\ln t)^{\gamma}}, \gamma\geq 1$, (note that $c'_{1} t
e^{-(-\ln t)^{\gamma}}(-\ln t)^{1-\gamma}\leq {\tilde{\phi}}(t)\leq
c'_{2} t e^{-(-\ln t)^{\gamma}}(-\ln t)^{1-\gamma}$ for
$c'_1,\,c'_2> 0$ when $t$ is near 0) then there exist $\alpha,\beta>
0$ such for all $t\geq 0$,
that
$$
E(t)\leq \begin{cases}
\beta (h({\tilde\sigma}(t)))^{{{-r}\over{r+1}}}
e^{-\left({{r}\over{r+1}}\ln
(\alpha h({\tilde\sigma}(t))+2)\right)^{\frac{1}{\gamma}}} \Bigl(\ln
(\alpha h({\tilde\sigma}(t))+2)\Bigr)^{-\frac{\gamma-1}{\gamma}},
&\text{if } r>1, \\
 \beta (h({\tilde\sigma}(t)))^{-1} e^{-\left(\ln
(\alpha h({\tilde\sigma}(t))+2)\right)^{\frac{1}{\gamma}}} \Bigl(\ln
(\alpha h({\tilde\sigma}(t))+2)\Bigr)^{-\frac{\gamma-1}{\gamma}},
&\text{if }r=1.
\end{cases}
$$

\subsection*{Slow than polynomials for $\phi$ (slow degeneracy) and linear
growth for $g$} If $\phi (t)=|\ln t|^{-\gamma}$ near of $0$ where
$\gamma> 0$, (note that $c'_{1} s (-\ln s)^{-\gamma}\leq
{\tilde{\phi}}(s)\leq c'_{2} s (-\ln s)^{-\gamma}$ for
$c'_1,\,c'_2>0$ when $s$ is near 0) then there exists $\alpha> 0$ such
that for all $t\geq 0$,
$$
E(t)\leq \begin{cases}
\alpha (h({\tilde\sigma}(t)))^{-\frac{\gamma}{\gamma (1+{1\over r})+1}}
e^{-(h({\tilde\sigma}(t)))^{\frac{1}{\gamma (1+{1\over r})+1}}},
&\text{if } r>1,\\
\alpha (h({\tilde\sigma}(t)))^{-\frac{\gamma}{\gamma+1}}
e^{-(h({\tilde\sigma}(t)))^{\frac{1}{\gamma+1}}}, &\text{if }r=1.
\end{cases}
$$

\section{Proof of Theorem \ref{thh1}}

For the rest of this article, we denote by $c$ various positive constants
which may be different at different occurrences.

If $E(t_0)=0$ for some $t_0 \geq 0$, then $E(t)=0$ for all $t\geq
t_0$, and then we have nothing to prove in this case. So we assume
that $E(t)> 0$ for all $t\geq 0$ without loss of generality.

We multiply the first equation of \eqref{eP} by
$\sigma (t){{{\tilde{\varphi}} (E)}\over{E}}u$ where
${\tilde{\varphi}}:\mathbb{R}_+\to \mathbb{R}_+$ is convex,
increasing and of class $C^1 (]0,+\infty[)$ such that
${\tilde{\varphi}}(0)=0$, and we integrate by parts, we have, for
all $0\leq S\leq T$,
\begin{align*}
0&=\int_{S}^{T} \sigma (t){{{\tilde{\varphi}} (E)}\over{E}}
\int_{\Omega} u\Bigl((|u'|^{l-2}u')'- \Delta_{\phi} u + \sigma(t)
g(u')\Bigr)\,dx\, dt \\
&=\Bigl[\sigma (t){{{\tilde{\varphi}} (E)}\over{E}}\int_{\Omega}uu'
|u'|^{l-2} dx\Bigr]_S^T \\
&\quad- \int_S^T\int_{\Omega}u'
|u'|^{l-2}\Bigl(\sigma'(t){{{\tilde{\varphi}} (E)}\over{E}}u+\sigma
(t){{{\tilde{\varphi}} (E)}\over{E}}u'+\sigma
(t)({{{\tilde{\varphi}} (E)}\over{E}})'u\Bigr)\,dx\,dt
\\
&\quad +\int_{S}^{T}\sigma (t){{{\tilde{\varphi}}
(E)}\over{E}}\int_{\Omega}\sum_{i=1}^n \phi
(|\partial_{x_i} u|^2)|\partial_{x_i} u|^2 \,dx\,dt +\int_S^T \sigma^2
(t){{{\tilde{\varphi}} (E)}\over{E}}\int_{\Omega} ug(u')\,dx\,dt.
\end{align*}
Using Lemma \ref{dl1} for $p=2$ and $q=l$ and the definition of
$E$, we have (note also that ${\tilde{\phi}}$ is convex and
defines a bijection from $\mathbb{R}_+$ to $\mathbb{R}_+$)
\begin{equation}
\begin{aligned}
\Bigl|\int_{\Omega} uu'|u'|^{l-2}dx\Bigr|
&\leq \Bigl(\int_{\Omega} |u|^l dx\Bigr)^{1/ l}
 \Bigl(\int_{\Omega}|u'|^{l} dx\Bigr)^{{l-1}\over l} \\
&\leq c\Bigl(\int_{\Omega}|\nabla u|^2 dx \Bigr)^{1/2}E^{{{l-1}\over l}}\\
&\leq cE^{{{l-1}\over l}}\Bigl(\sum_{i=1}^n{\tilde{\phi}}^{-1}
 \Bigl(\int_{\Omega}\sum_{i=1}^n
 {\tilde{\phi}}(|\partial_{x_i} u|^2)dx\Bigr)\Bigr)^{1/2} \\
&\leq cE^{{{l-1}\over l}}{\sqrt{{\tilde{\phi}}^{-1}(E)}}.
\end{aligned} \label{x13}
\end{equation}
In the other hand, we have $s\phi (s)\geq 2{\tilde{\phi}} (s)$,
$l\geq 2$, ${\tilde{\phi}}^{-1}$ is non decreasing and ${\tilde{\varphi}}$
is convex, increasing and of class $C^1 (]0,+\infty[)$ such that
${\tilde{\varphi}} (0)=0$ (then $s\mapsto s^{{{l-1}\over {l}}}$,
$s\mapsto {\tilde{\phi}}^{-1}(s)$
and $s\mapsto {{{\tilde{\varphi}} (s)}\over{s}}$ are non
decreasing). Then we deduce that
\begin{equation}
\begin{aligned}
&\int_S^T \sigma (t){\tilde{\varphi}}(E(t))dt\\
&\leq cE^{{{l-1}\over l}} (S){\sqrt{{\tilde{\phi}}^{-1}
(E(S))}}{{{\tilde{\varphi}} (E(S))}\over{E(S)}}
+c\int_{S}^{T} \sigma (t){{{\tilde{\varphi}} (E)}\over{E}}
\int_{\Omega} (|u'|^l +|ug(u')|)\,dx\,dt.
\end{aligned} \label{x14}
\end{equation}
To estimate the last integral above, we distinguish three
cases:

\noindent\textbf{Case 1: $r=1$ and $G$ is linear on $[0,\epsilon_1]$:}
 We choose ${\tilde{\varphi}}(s)=s$.
For all $t\geq 0$, we denote
$$
\Omega^+_t=\{x\in\Omega:uu'\geq 0\},\quad
\Omega^-_t=\{x\in\Omega:uu'\leq 0\}.
$$
We have $C_1 |s|\leq |g(s)|\leq C_2 |s|$ for all $s\in \mathbb{R}$
(because $2\leq l\leq r+1=2$), and then (using (\ref{x5}))
$$
\int_{S}^{T} \sigma (t){{{\tilde{\varphi}} (E)}\over{E}}
\int_{\Omega} |u'|^l \,dx\,dt\leq c\int_{S}^{T} \sigma (t)\int_{\Omega}
u'g(u')\,dx\,dt\leq cE(S)
$$
and (note that $\sigma'\leq 0$)
\begin{align*}
&\int_{S}^{T} \sigma (t){{{\tilde{\varphi}} (E)}\over{E}}
\int_{\Omega} |ug(u')|\,dx\,dt\\
& \leq c\int_{S}^{T} \sigma (t) \int_{\Omega} |uu'|\,dx\,dt\\
&\leq c\Bigl[\sigma (t)\int_{\Omega^+_t}u^2 dx-\sigma
(t)\int_{\Omega^-_t}u^2 dx\Bigr]_S^T +c \int_{S}^{T} \sigma'
(t)\Bigl(-\int_{\Omega^+_t}u^2 dx+\int_{\Omega^-_t}u^2 dx\Bigr)dt\\
&\leq c{\tilde{\phi}}^{-1}(E(S))+c{\tilde{\phi}}^{-1}(E(S))\int_S^T
(-\sigma' (t))dt\leq c{\tilde{\phi}}^{-1}(E(S)).
\end{align*}
Then
$$
\int_S^T \sigma (t)E(t)dt\leq
c\Bigl(1+{{E(S)}\over{{\tilde{\phi}}^{-1}(E(S))}}
+{\sqrt{{{E(S)}\over{{\tilde{\phi}}^{-1}(E(S))}}}}\Bigr)
{\tilde{\phi}}^{-1}(E(S)).
$$
Using the fact that ${\tilde{\phi}}$ is convex,
increasing and ${\tilde{\phi}}(0)=0$ (then $s\mapsto
{{s}\over{{\tilde{\phi}}^{-1}(s)}}$ is non
decreasing) we obtain from (\ref{x14}) that
$$
\int_S^{+\infty} \sigma (t)E(t)dt\leq c{\tilde{\phi}}^{-1}(E(S)).
$$
Let ${\tilde{E}}={\tilde{\phi}}^{-1}\circ E\circ
{\tilde{\sigma}}^{-1}$ (note that ${\tilde{\sigma}}$ is a
bijection from $\mathbb{R}_+$ to $\mathbb{R}_+$). Then, for
$\omega> 0$,
$$
\int_S^{+\infty} {\tilde{\phi}}({\tilde{E}}(t))dt\leq {1\over
{\omega}}{\tilde{E}}(S).
$$
Using Lemma \ref{ll1} for ${\tilde{E}}$ in the particular case
$\Psi (s)=\omega {\tilde{\phi}}(s)$ and $\lambda=0$, we deduce
from (\ref{x6}) that
$$
{\tilde{E}}(t)\leq
\psi^{-1}\left(h(t)+\psi({\tilde{\phi}}^{-1}(E(0)))\right),\quad\forall
t\geq 0.
$$
Then, using the definition of ${\tilde{E}}$, we obtain (\ref{x11})
in the case where $r=1$ and $G$ is linear on $[0,\epsilon_1]$.

\noindent\textbf{Case 2: $r>1$ and $G$ is linear on $[0,\epsilon_1]$.}
We choose
${\tilde{\varphi}}(s)={{s^{1+{1\over r}}}
\over{({\tilde{\phi}}^{-1}(s))^{1/r}}}$.
For all $t\geq 0$, we denote
$$
\Omega^1_t=\{x\in\Omega:|u'|\geq \epsilon_1\},\quad
\Omega^2_t=\{x\in\Omega:|u'|\leq \epsilon_1\}.
$$
Using Young's and Lemma \ref{dl1} (for $q=r+1$ and $p=2$) and condition
(\ref{x2}) we have, for all $\epsilon> 0$ (using also the fact that
$s\mapsto
{{{\tilde{\varphi}} (s)}\over{s}}$ is non decreasing),
\begin{align*}
&\int_{S}^{T} \sigma (t){{{\tilde{\varphi}} (E)}\over{E}}\int_{\Omega^1_t}
(|u'|^l +|ug(u')|)\,dx\,dt \\
&\leq \int_{S}^{T} \sigma (t)
{{{\tilde{\varphi}}(E)}\over{E}}\Bigl(\int_{\Omega^1_t} |u|^{r+1}
dx\Bigr)^{1\over{r+1}}\Bigl(\int_{\Omega^1_t}
|g(u')|^{{r+1}\over{r}} dx\Bigr)^{{r}\over{r+1}}dt \\
&\quad +c\int_{S}^{T} \sigma (t) \int_{\Omega^1_t} u'g (u')\,dx\,dt \\
&\leq \epsilon\int_{S}^{T} \sigma (t) {{{\tilde{\varphi}}^{r+1}
(E)}\over{E^{r+1}}}\int_{\Omega^1_t} |\nabla u|^{2} \,dx\,dt
+c\int_{S}^{T} \sigma (t) \int_{\Omega^1_t} (|g (u')|^{1+{1\over
r}}+u'g (u'))\,dx\,dt \\
&\leq \epsilon\int_{S}^{T} \sigma (t) {{{\tilde{\varphi}}^{r+1}
(E){\tilde{\phi}}^{-1} (E)}\over{E^{r+1}}}dt
+c\int_{S}^{T} \sigma (t) \int_{\Omega^1_t} u'g (u')\,dx\,dt \\
&\leq \epsilon\int_{S}^{T} \sigma (t) {\tilde{\varphi}}(E)dt+cE(S).
\end{align*}
Choosing $\epsilon$ small enough, we obtain from (\ref{x14}) that
\begin{align*}
\int_S^T \sigma (t){\tilde{\varphi}} (E(t))dt
&\leq c\Big(E(S)+E^{{l-1}\over l}
(S){\sqrt{{\tilde{\phi}}^{-1}(E(S))}}  {{{\tilde{\varphi}}
(E(S))}\over{E(S)}}\Big)\\
&\quad +c\int_{S}^{T} \sigma (t){{{\tilde{\varphi}} (E)}\over{E}}
\int_{\Omega^2_t}
(|u'|^l +|ug(u')|)\,dx\,dt.
\end{align*}
On the other hand, we have $C_1 |s|^{l-1}\leq |g(s)|\leq C_2 |s|$ for
all $s\in [-\epsilon_1,\epsilon_1]$
and then (note that $s\mapsto {{{\tilde{\varphi}} (s)}\over{s}}$ is non
decreasing and follow the proof in the case 1)
\begin{align*}
c\int_{S}^{T} \sigma (t){{{\tilde{\varphi}} (E)}\over{E}}
\int_{\Omega^2_t} (|u'|^l +|ug(u')|)\,dx\,dt
&\leq c\int_{S}^{T} \sigma (t)\int_{\Omega}(u'g(u') +|uu'|)\,dx\,dt\\
&\leq c\Bigl (E(S)+{\tilde{\phi}}^{-1} (E(S))\Bigr).
\end{align*}
Then from \eqref{x14} we deduce that
\begin{align*}
&\int_S^T \sigma (t) {\tilde{\varphi}} (E(t))dt\\
&\leq c\Bigl(1+{{E(S)}\over{{\tilde{\phi}}^{-1}(E(S))}}
+E^{{{l-2}\over{2l}}}(S){{{\tilde{\varphi}}(E(S)))}\over{E(S))}}
{\sqrt{{{E(S)}\over{{\tilde{\phi}}^{-1}(E(S))}}}}\Bigr)
{\tilde{\phi}}^{-1}(E(S)).
\end{align*}
Finally (note that $s\mapsto
s^{{l-2}\over{2l}}$, $s\mapsto {{{\tilde{\varphi}} (s)}\over{s}}$ and
$s\mapsto
{{s}\over{ {\tilde{\phi}}^{-1}(s)}}$ are non decreasing), we
obtain
$$
\int_S^{+\infty} \sigma (t){\tilde{\varphi}}(E(t))dt\leq
c{\tilde{\phi}}^{-1}(E(S)).
$$
Let ${\tilde{E}}={\tilde{\phi}}^{-1} \circ E\circ
{\tilde{\sigma}}^{-1}$. Then we deduce from
this inequality that, for $\omega> 0$,
$$
\int_S^{+\infty} {\tilde{\varphi}}\Bigl({\tilde{\phi}}
({\tilde{E}}(t))\Bigr)dt\leq
{1\over{\omega}}{\tilde{E}}(S).
$$
Using Lemma \ref{ll1} for ${\tilde{E}}$ in the particular case
$\Psi (s)=\omega {\tilde{\varphi}}({\tilde{\phi}} (s))=\omega
{{{\tilde{\phi}} (s)^{1+{1\over r}}}\over{s^{1/r}}}$ and
$\lambda=0$, we deduce from (\ref{x6}) our estimate (\ref{x11}).

\noindent\textbf{Case 3: $G'(0)=0$.} We choose
${\tilde{\varphi}}(s)={{2\epsilon_0 s^2}\over{{\tilde{\phi}}^{-1}(s)}}G'\Bigl({{\epsilon_0^2
s^2}\over{{\tilde{\phi}}^{-1}(s)}}\Bigr)$.
Using the fact that $s\mapsto G'(s)$,
$s\mapsto{{s^2}\over{{\tilde{\phi}}^{-1}(s)}}$ and $s\mapsto
{{{\tilde{\varphi}}^{r-1}(s)}\over{s^{r-1}}}$
are non decreasing, we obtain (as in case 2)
\begin{align*}
\int_{S}^{T}\! \sigma (t){{{\tilde{\varphi}} (E)}\over{E}}
\int_{\Omega^1_t} (|u'|^l +|ug(u')|)\,dx\,dt
&\leq \epsilon\int_{S}^{T} \sigma (t) {{{\tilde{\varphi}}^{r+1}
(E){\tilde{\phi}}^{-1} (E)}\over{E^{r+1}}}dt
+cE(S)
\\
&\leq \epsilon\int_{S}^{T} \sigma (t) {{{\tilde{\varphi}}^{2}
(E){\tilde{\phi}}^{-1} (E)}\over{E^{2}}}dt+cE(S)
\\
&=2\epsilon\epsilon_0\int_{S}^{T} \sigma (t) {\tilde{\varphi}}
(E)G'({{\epsilon_0^2 E^{2}}\over{{\tilde{\phi}}^{-1} (E)}})dt+cE(S)
\\
&\leq 2\epsilon\epsilon_0\int_{S}^{T} \sigma (t) {\tilde{\varphi}}
(E)dt+cE(S)
\end{align*}
Choosing $\epsilon$ small enough, we obtain from (\ref{x14}) that
\begin{align*}
\int_S^T \sigma (t){{\tilde{\varphi}}} (E(t))dt
&\leq c\Big(E(S)+E^{{l-1}\over l}(S)
{\sqrt{{\tilde{\phi}}^{-1}(E(S))}} {{{\tilde{\varphi}}
(E(S))}\over{E(S)}}\Big)\\
&\quad +c\int_{S}^{T} \sigma (t){{{\tilde{\varphi}} (E)}\over{E}}
\int_{\Omega^2_t}
(|u'|^l +|ug(u')|)\,dx\,dt.
\end{align*}
Let now  $G_1 (s)=G(s^2)$ (note that
$G_1$ satisfies the same hypotheses as $G$) and let $G^{*}$ and
$G^{*}_1$ denote the dual functions of the convex functions $G$
and $G_1$ respectively in the sense of Young (see  Arnold
\cite[page 64]{arno}, for the definition). Because $G$ is convex and
$G$ is not linear near $0$, then there exists $\epsilon_1'>0$ such
that $G''>0$ on $]0,\epsilon_1']$. Since, because $G'(0)=0$ and
$(2)-(3)$ are still satisfied for $\epsilon''=\min\{\epsilon_1,
\epsilon_1'\}$ instead of $\epsilon_1$, we can assume, without
lose of generality, that $G'$ defines a bijection from
$\mathbb{R}_+$ to $\mathbb{R}_+$. Then $G^*$ and $G^{*}_1$ are the
Legendre transform of $G$ and $G_1$ respectively, which are given
by (see  Arnold \cite[pp. 61-62]{arno})
$$
G^{*} (s)=s(G')^{-1} (s)-G [(G')^{-1} (s)],\quad
G^{*}_1 (s)=s(G_1')^{-1} (s)-G_1 [(G_1')^{-1} (s)].
$$
Thanks to our choice
$$
{\tilde{\varphi}} (s)={{2\epsilon_0
s^2}\over{{{{\tilde{\phi}}^{-1} (s)}}}}G'\Big({{\epsilon^2_0
s^2}\over{{{{\tilde{\phi}}^{-1} (s)}}}}\Big)
= {s\over{{\sqrt{{\tilde{\phi}}^{-1} (s)}}}}G_1'\Big({{\epsilon_0
s}\over{{\sqrt{{\tilde{\phi}}^{-1} (s)}}}}\Big),
$$
 we have
\begin{gather*}
G^{*} \Bigl({{{\tilde{\varphi}} (s)}\over {s}}\Bigr)\leq
{\tilde{\varphi}}
(s){{(G')^{-1}({{{\tilde{\varphi}} (s)}\over {s}})}\over {s}},\\
G^{*}_1 \Big({{{\tilde{\varphi}} (s)}\over
{s}}{\sqrt{{\tilde{\phi}}^{-1}
(s)}}\Big)\leq {{\epsilon_0 s}\over{{\sqrt{{\tilde{\phi}}^{-1}
(s)}}}}G_1'\Big({{\epsilon_0 s}\over{{\sqrt{{\tilde{\phi}}^{-1}
(s)}}}}\Big)
=\epsilon_0 {\tilde{\varphi}} (s).
\end{gather*}
Then, by Poincar\'e's inequality, Young's inequality (see
Arnold \cite[p. 64]{arno}) and Jensen's inequality (see  Rudin
\cite{rudi}), we deduce ($|\Omega|$ is the measure of $\Omega$ in
$\mathbb{R}^n$)
\begin{align*}
&\int_{S}^{T} \sigma (t){{{\tilde{\varphi}} (E)}\over{E}}
 \int_{\Omega^2_t}(|u'|^l+|ug (u')|) \,dx\,dt \\
&\leq \int_{S}^{T} \sigma (t){{{\tilde{\varphi}} (E)}\over{E}}
 \Bigl(\int_{\Omega^2_t} G^{-1}(u'g(u'))dx
\\
&\quad +\Bigl(\int_{\Omega}|\nabla u|^2 dx\Bigr)^{1/2}
 \Bigl(\int_{\Omega^2_t} G^{-1}(u'g(u')) dx\Bigr)^{1/2}\Bigr)dt \\
&\leq \int_{S}^{T} \sigma (t){{{\tilde{\varphi}} (E)}\over{E}}
\Bigl({\sqrt{{\tilde{\phi}}^{-1}
(E)}}{\sqrt{|\Omega|G^{-1}\Bigl({1\over{|\Omega|}}\int_{\Omega}u'g(u')
 dx\Bigr)}}\\
&\quad +|\Omega|G^{-1}\Bigl({1\over{|\Omega|}}
 \int_{\Omega}u'g(u')dx\Bigr)dt\Bigr)
\\
&\leq c\int_{S}^{T}\sigma (t)\Bigl(G^*_1 \Bigl({{{\tilde{\varphi}}
(E)}\over{E}}{\sqrt{{\tilde{\phi}}^{-1}
(E)}}\Bigr)+G^{*}({{{\tilde{\varphi}}
(E)}\over{E}})\Bigr)dt +c\int_{S}^{T}\sigma (t)\int_{\Omega}u'g(u')\,dx\,dt
\\
&\leq c\int_{S}^{T}\sigma (t)\Bigl(\epsilon_0 +{{(G')^{-1}({{\varphi
(E)}\over {E}})}\over {E}}\Bigr){\tilde{\varphi}} (E)dt+cE(S).
\end{align*}
Using the fact that $s\mapsto (G')^{-1} (s)$ and $s\mapsto
{{s}\over{{\tilde{\phi}}^{-1}(s)}}$ are non decreasing, we deduce
that, for $0< \epsilon_0\leq
{{{\tilde{\phi}}^{-1}(E(0))}\over{2E(0)}}$,
$$
\int_{S}^{T} \sigma(t){{{\tilde{\varphi}} (E)}\over{E}}
\int_{\Omega^2_t}(|u'|^l+|ug (u')|) \,dx\,dt\leq c\epsilon_0
\int_{S}^{T}\sigma (t) {\tilde{\varphi}} (E)dt+cE(S).
$$
Then, choosing $\epsilon_0$ small enough, we deduce from (\ref{x14})
that
$$
\int_S^T \sigma (t) {\tilde{\varphi}} (E(t))dt\leq
c\Bigl(1+E^{{l-2}\over {2l}}
(S){\sqrt{{{\tilde{\phi}}^{-1}(E(S))}\over {E(S)}}}{{{\tilde{\varphi}}
(E(S))}\over{E(S)}}\Bigr)E(S)
$$
Finally (note that $s\mapsto
s^{{l-2}\over{2l}}$ and $s\mapsto
{\sqrt{{{{\tilde{\phi}}^{-1}(s)}\over{s}}}}{{{\tilde{\varphi}}
(s)}\over{s}}=2\epsilon_0{\sqrt{{{s}\over{{\tilde{\phi}}^{-1}(s)}}}}G'\Bigl({{\epsilon^2_0
s^2}\over{ {\tilde{\phi}}^{-1}(s)}}\Bigr)$ are non decreasing), we
obtain
$$
\int_S^{+\infty} \sigma (t){\tilde{\varphi}} (E(t))dt\leq cE(S).
$$
Let ${\tilde{E}}=E\circ{\tilde{\sigma}}^{-1}$. Then we deduce from
this inequality that, for $\omega> 0$,
$$
\int_S^{+\infty}{\tilde{\varphi}} ({\tilde{E}}(t))dt\leq
{1\over{\omega}}{\tilde{E}}(S).
$$
Using Lemma \ref{ll1} for ${\tilde{E}}$
in the particular case $\Psi (s)=\omega {\tilde{\varphi}} (s)$ and
$\lambda=0$, we deduce from (\ref{x6}) our estimate (\ref{x11}).
This is completes the proof.

\section{An application to wave equations of $\phi$-Laplacian with
source term}

In this section we shall propose some applications of Theorem
\ref{thh1}.

\noindent\textbf{Example 1.} Let us consider the Cauchy problem for the wave
equation, in $\Omega\times\mathbb{R}_+$,
\begin{equation} \label{eP1b}
\begin{gathered}
(|u'|^{l-2}u')'-e^{-\lambda (x)}\sum_{i=1}^n
\partial_{x_i}\Bigl(e^{\lambda (x)}\phi (|\partial_{x_i}
u|^2)\partial_{x_i} u\Bigr) +\sigma(t) g(u')+f(u)=0,\\
 u=0 \quad \text{on } \Gamma \times \mathbb{R}_+\\
 u(x, 0)=u_{0}(x),\quad u'(x, 0)=u_{1}(x) \quad \text{on }\Omega.
\end{gathered}
\end{equation}
We define the energy associated to the solution
as
\begin{align*}
E(t) &= \frac{l-1}{l}\int_{\Omega}e^{\lambda (x)} |u'|^{l}
dx+\int_{\Omega}e^{\lambda (x)}\sum_{i=1}^n
\tilde{\phi}(|\partial_{x_i} u|^{2})dx
+\int_{\Omega}e^{\lambda (x)} F(u)\, dx \\
&= \frac{l-1}{l}\int_{\Omega}e^{\lambda (x)} |u'|^{l}
dx+ J(u)
\end{align*}
where $F(u)=\int_{0}^{u}f(s)\, ds$.
For the function $f\in C(\mathbb{R})$ we assume that there exists an N-function
$\psi$ satisfying
$i(\psi), I(\psi)\in ]1, +\infty[$ and
\begin{equation}
|f(t)|\leq \psi'(|t|) \quad \text{for every } t\in \mathbb{R}. \label{el1}
\end{equation}
If condition (\ref{o2}) is satisfied (so that $\Phi^{*}$ exists), then
we assume in addition that
\begin{equation}
\psi(t)\leq \Phi^{*}(C t) \quad \text{for all large }  t> 0. \label{el2}
\end{equation}
Thus, we can verify that for all $u\in W_{0}^{1, \Phi}$ with norm
small enough that
\begin{equation}
\frac{1}{C}\int_{\Omega} \Phi(|\nabla_{x} u|)\, dx\leq |J(u)|\leq
C\int_{\Omega} \Phi(|\nabla_{x} u|)\, dx.
\label{el3}
\end{equation}
So, we obtain same results as in the theorem \ref{the1}.

\noindent\textbf{Proof of the example 1.}
We prove only the second part. The proof of the first part is a direct
application of the theorem \ref{thh1}.
We make an additional assumption on $g(v)$:
\begin{itemize}
\item[(H3')]  Suppose that there exist $c_{i}> 0$; $i=1, 2, 3, 4$ such
that
\begin{gather}
c_{1}|v|^{p}\leq |g(v)|\leq c_{2} |v|^{\theta} \text{if } |v|\leq
1, \label{ee3} \\
c_{3}|v|^{s}\leq |g(v)|\leq c_{4}|v|^{r} \text{for all } |v|\geq
1, \label{ee4}
\end{gather}
where $1\leq m\leq r$, $\theta\leq p$, $l-1\leq s\leq r\leq \frac{n +2}{n-2}$.
\end{itemize}

\noindent\textbf{Proof of the energy decay.}
We denote by $c$ various positive constants which may
be different at different occurrences. We multiply the first
equation of \eqref{eP} by $E^{q}{\tilde\sigma}' u$, where
${\tilde\sigma}$ is a function
satisfying all the hypotheses of lemma \ref{ll1}, we obtain
\begin{align*}
0&=\int_{S}^{T} E^{q}{\tilde\sigma}'\int_{\Omega} u((|u'|^{l-2}u')_{t}-
\Delta_{\phi} u+ \sigma(t) g(u'))\,dx\,dt\\
&=\big[E^{q}{\tilde\sigma}'\int_{\Omega}
uu'|u'|^{l-2}\,dx\big]_S^T-\int_S^T(q E'E^{q-1}{\tilde\sigma}' +E^{q}{\tilde\sigma}'')
\int_{\Omega} uu'|u'|^{l-2}\,dx\,dt \\
&\quad -\int_S^T E^{q}{\tilde\sigma}' \int_{\Omega} |u'|^{l}\,dx\,dt
 + \int_S^T E^{q}{\tilde\sigma}'
\|\nabla u\|_{2}^{2(\gamma+1)} \,dx\,dt\\
&\quad +\int_S^T
E^{q}{\tilde\sigma}' \int_{\Omega} \sigma(t) u g(u')\,dx\,dt.
\end{align*}
we deduce that
\begin{equation}
\begin{aligned}
&2(m+1)\int_S^TE^{q+1}{\tilde\sigma}' \,dt\\
&\leq -\big[E^{q}{\tilde\sigma}' \int_{\Omega} uu'|u'|^{l-2}\,dx\big]_S^T
+\int_S^T(q E'E^{q-1}{\tilde\sigma}' + E^{q}{\tilde\sigma}'')\int_{\Omega}
uu'|u'|^{l-2}\,dx\, dt \\
&\quad +\frac{2(l - 1)(m+1) +l}{l}\int_S^T E^{q}{\tilde\sigma}'
 \int_{\Omega} |u'|^{l}\,dx\,dt -\int_S^T
E^{q} {\tilde\sigma}' \int_{\Omega} \sigma(t) ug(u')\,dx\, dt.
\end{aligned} \label{e5}
\end{equation}
Since $E$ is non-increasing, ${\tilde\sigma}' $ is a bounded nonnegative
function on $\mathbb{R}_{+}$ (and we denote by $\mu$ its maximum) and using
H\"older inequality, we have
$$
\big|E(t)^{q}{\tilde\sigma}' \int_{\Omega}uu'|u'|^{l-2}\,dx\,dt \big|
\leq c\mu E(S)^{q+\frac{l - 1}{l}+\frac{1}{2(m+1)}}\qquad \forall t\geq S.
$$
and
\begin{align*}
&\int_{S}^{T}(q E'E^{q-1}{\tilde\sigma}' +E^{q}{\tilde\sigma}'')\int_{\Omega}
uu'|u'|^{l-2}\,dx\,dt,\,dx\,dt \\
&\leq  c\mu \int_{S}^{T} -E'(t)E(t)^{q-\frac{1}{l}+\frac{1}{2(m+1)}}\, dt
+ c \int_{S}^{T}E(t)^{q+\frac{l -1}{l}
 +\frac{1}{2(m+1)}}(-{\tilde\sigma}''(t))\, dt\\
&\leq  c\mu E(S)^{q+\frac{l - 1}{l}+\frac{1}{2(m+1)}}.
\end{align*}
Using these estimates we conclude from the above inequality that
\begin{align*}
&2(m+1)\int_{S}^{T}E(t)^{1+q}{\tilde\sigma}' (t)\, dt \\
& \leq c E(S)^{q+\frac{l - 1}{l}+\frac{1}{2(m+1)}}
+ \frac{2(l -1)(m+1) +l}{l}\int_S^T E^{q}{\tilde\sigma}'
 \int_{\Omega} |u'|^{l}\,dx\,dt\\
& \quad -\int_S^T E^{q}{\tilde\sigma}' \int_{\Omega} \sigma(t)
 ug(u')\,dx\,dt  \\
&\leq c E(S)^{q+\frac{l-1}{l}+\frac{1}{2(m+1)}}+ \frac{2(l - 1)(m+1) +l}{l}
\int_S^T E^{q}{\tilde\sigma}' \int_{\Omega} |u'|^{l}\,dx\,dt \\
&\quad -\int_S^T E^{q}{\tilde\sigma}' \int_{|u'|\leq 1} \sigma(t)
 ug(u')\,dx\,dt
 -\int_S^T E^{q}{\tilde\sigma}' \int_{|u'|> 1} \sigma(t) ug(u')\,dx\,dt.
\end{align*} \label{e6}
Now, we  estimate each terms on the right-hand side of
the above inequality, to apply Lemma \ref{ll1}.
Using H\"older inequality, we obtain
\begin{align*}
&\int_S^T E^{q}{\tilde\sigma}' \int_{\Omega} |u'|^{l}\,dx\,dt \\
&\leq C\int_{S}^{T}E^{q}{\tilde\sigma}' \int_{\Omega}\frac{1}{\sigma(t)}
 u'\rho(t,u')\, dx\, dt +C'\int_{S}^{T}E^{q}{\tilde\sigma}' \int_{\Omega}
\Big(\frac{1}{\sigma(t)}u'\rho(t, u')\Big)^{\frac{l}{(p+1)}}\,
dx\, dt\\
&\leq  C \int_{S}^{T}E^{q}\frac{{\tilde\sigma}'}{\sigma(t)}(-E')\, dt
 +C'(\Omega) \int_{S}^{T}E^{q}\frac{{\tilde\sigma}'}
 {\sigma^{\frac{l}{p+1}}(t)}
(-E')^{\frac{l}{p+1}}\, dt\\
& \leq  C E^{q+1}(S)+C'(\Omega)
\int_{S}^{T}E^{q} {\tilde\sigma}'^{\frac{p+1-l}{p+1}}
\Big(\frac{{\tilde\sigma}'}{\sigma(t)}\Big)^{\frac{l}{p+1}}
(-E')^{\frac{l}{p+1}}\, dt.
\end{align*}
Now, fix an arbitrarily small $\varepsilon> 0$ (to be chosen later),
by applying Young's inequality, we obtain
\begin{equation}
\begin{aligned}
&\int_S^T E^{q}{\tilde\sigma}' \int_{\Omega} |u'|^{l}\,dx\,dt \\
&\leq  C E^{q+1}(S)+ C'(\Omega) \frac{p
+1-l}{p+1}\varepsilon^{\frac{(p+1)}{(p+1-l)}}
\int_{S}^{T}E^{q\frac{p+1}{p+1-l}} {\tilde\sigma}' \, dt+
C'(\Omega)\frac{l}{p+1}\frac{1}{\varepsilon^{\frac{(p+1)}{l}}}E(S).
\end{aligned}\label{e8}
\end{equation}
If $l\geq p+1$,  from (\ref{ee3}) and (\ref{ee4}) we obtain easily that
\begin{equation}
\int_S^T E^{q}{\tilde\sigma}' \int_{\Omega} |u'|^{l}\,dx\,dt \leq C E^{q+1}(S).
\label{e9}
\end{equation}
Thanks to Young's inequality,
%\begin{equation}
\begin{align*}
&\int_S^T E^{q}{\tilde\sigma}'
\int_{|u'|\leq 1} \sigma(t) u g(u')\,dx\,dt
\int_S^T E^{q}{\tilde\sigma}' \int_{|u'|\leq 1} \sigma(t) \|u\|_{2}
\Big(\int_{|u'|\leq 1}|g(u')|^{2}\, dx\Big)^{\frac12}\, dt \\
&\times \int_S^T E^{q}{\tilde\sigma}' \int_{|u'|\leq 1} \sigma(t)
\|\nabla_{x} u\|_{2m+2} \Big(\int_{|u'|\leq 1}
(u'g(u'))^{\frac{2\theta}{\theta+1}}\, dx\Big)^{1/2}\, dt\\
&\leq c \int_S^T
E^{q+\frac{1}{2(m+1)}}{\tilde\sigma}' \sigma^{\frac{1}{(\theta+1)}}(t)
\Big(\int_{|u'|< 1}\sigma  u'g(u') \,
dx\Big)^{\theta/(\theta+1)}\,dt\\
&\leq c \int_S^T E^{q+\frac{1}{2(m+1)}}{\tilde\sigma}'
\sigma^{\frac{1}{(\theta+1)}}(t)
(-E')^{\frac{\theta}{\theta+1}}\, dt.
\end{align*}% \label{ee12}
%\end{equation}
Applying Young's inequality, we obtain
\begin{equation} \label{eee1}
\begin{aligned}
\int_S^T E^{q}{\tilde\sigma}' \int_{|u'|\leq 1} \sigma(t) u
g(u')\,dx\,dt
&\leq  C(\Omega)\varepsilon_{2}^{\theta+1} \int_S^T
\left(E^{q+\frac{1}{2(m+1)}}{\tilde\sigma}'
\sigma^{\frac{1}{(\theta+1)}}(t)\right)^{\theta+1}\,
dt \\
&\quad + C(\Omega) \frac{1}{\varepsilon_{2}^{\frac{\theta+1}{\theta}}}\int_S^T
(-E')\, dt
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
&\int_S^T E^{q}{\tilde\sigma}' \int_{|u'|\geq 1} \sigma(t) u g(u')\,dx\,dt\\
& \leq C(\Omega)\frac{1}{(r+1)}\varepsilon_{1}^{(r+1)} \int_S^T
E^{\left(q+\frac{1}{2(\gamma+1)}\right)(r+1)}{\tilde\sigma}'\sigma(t)^{r+1}\,
dt
+ \frac{C(\Omega)r}{(r+1)}\frac{1}{\varepsilon_{1}^{\frac{r+1}{r}}}E(S),
\end{aligned}
\label{ee13}
\end{equation}

\subsection*{The case  $l \geq p + 1$}
We consider two subcases
\begin{itemize}
\item $\theta \geq 2m + 1$.
Choose $q=0$ and we have
$\big(\frac{1}{2(\gamma +1)}\big)(\theta + 1)=1+ \alpha$,
where $\alpha=\frac{\theta-(2m+1)}{2(m+1)}\geq 0$.

\item $\theta < 2m + 1$.
Choose $q$ such that
$\big(q+\frac{1}{2(m +1)}\big)(\theta + 1)=q+1$.
Thus, $q=\frac{2m - \theta + 1}{2\theta(m + 1)}$.
\end{itemize}

\subsection*{The case $l < p + 1$.}
\begin{itemize}
\item $2m+1> \theta$
If $l \geq \frac{2\theta(m+ 1)(p+1)}{(\theta+1)(2m + 1)}$,
we choose $q$ such that
$\big(q+\frac{1}{2(m +1)}\big)(\theta + 1)=q+1$.
Thus,  $q=\frac{2m - \theta + 1}{2\theta(m + 1)}$ and
$q\frac{p + 1}{p + 1 - l}=q+1+\alpha$ with
$$
\alpha=\frac{l(2m + 1)(\theta+1)-2\theta(m + 1)(p+1)}{2`\theta(m +
1)(p + 1 - l)}\geq 0.
$$
If $l< \frac{2\theta(m+ 1)(p+1)}{(\theta+1)(2m + 1)}$, we choose $q$
such that $q\frac{p + 1}{p + 1 - l}=q+1$.
Thus
$q=\frac{p + 1-l}{l}$
and
$\big(q+\frac{1}{2(m +1)}\big)(\theta + 1)=q+1+\alpha$,
where
$$
\alpha=\frac{2\theta(m+1)(p+1)-l(2m+1)(\theta+1)}{2l(m+1)}> 0.
$$

\item $2m+ 1\leq \theta$, we choose $q$ such that
$q\big(\frac{p+1}{p+1-l}\big)=q+1$, thus
$q=\frac{p+1-l}{l}$
and  $(q+\frac{1}{2(\gamma +1)})(m + 1)=q+1+\alpha$ with
$$
\alpha=m \frac{m+1-l}{l}+ \frac{m-(2\gamma+1)}{2(\gamma+1)}> 0.
$$
\end{itemize}
We may thus complete the proof by applying Lemma \ref{ll1} with
${\tilde E}=E\circ {\tilde\sigma}^{-1}$ instead of $E$ and
$\Psi(s)=s^{q}$.

\section{Comments and open questions}

1. It is interesting to
study the asymptotic behavior of solutions for Klein-Gordon nonlocal
equation
\begin{gather*}
(|u'|^{l-2}u')'-\phi_{1}(\|\nabla u\|_{2}^{2},
\|u\|_{2}^{2})\Delta u+ \phi_{2}(\|\nabla u\|_{2}^{2},
\|u\|_{2}^{2})u+\sigma(t) g(u')=0\quad \text{in } \Omega\times
\mathbb{R}_+\\
u=0  \quad \text{on } \Gamma_{0}\times \mathbb{R}_+\\
 u(x, 0)=u_{0}(x),\quad u'(x, 0)=u_{1}(x) \quad \text{on }\Omega,
\end{gather*}
in particular when there exists a continuous function
$E(w, r,s) =\frac{l-1}{l}w+ \frac{1}{2}L(r, s)$ defined for
$w, r, s\geq 0$
such that for all solutions,
$$
E(\|u'\|_{l}^{l}, \|\nabla u\|_{2}^{2}, \|u\|_{2}^{2})
+\int_{0}^{t}\sigma(t)\int_{\Omega} u'g(u')\, dx\, ds
=E(\|u_{1}\|_{l}^{l}, \|\nabla u_{0}\|_{2}^{2}, \|u_{0}\|_{2}^{2}).
$$
For example when $\phi_{1}(r, s)=m(r)$ and $\phi_{2}(r, s)=n(s)$
where $m$ and $n$ are two continuous positives functions. We can
take
$$
E(w, r, s)=\frac{l-1}{l}w+\frac{1}{2}\int_{0}^{r}m(\tau)\, d\tau
+\frac{1}{2}\int_{0}^{s} n(\tau)\, d\tau.
$$
So,
$E'(t)=-\sigma(t)\int_{\Omega} u'g(u')\, dx\, ds$.

As another example, when $\phi_{1}(r, s)=\frac{r}{1+s^{2}}$ and
$\phi_{2}(r, s)=-\frac{r^{2} s}{(1+s^{2})^{2}}$ we can take
$$
E(w, r, s)=\frac{l-1}{l}w+\frac{r^{2}}{4(1+ s^{2})}.
$$
So,
$E'(t)=-\sigma(t)\int_{\Omega} u'g(u')\, dx$.

As another example when $\phi_{1}(r, s)=\frac{s}{1+r}$ and
$\phi_{2}(r, s)=\arctan(r)$ we can take
$$
E(w, r, s)=\frac{l-1}{l}w+ \frac{1}{2} \arctan(r)\ s.
$$
So, $E'(t)=-\sigma(t)\int_{\Omega} u'g(u')\, dx$.

2. An interesting problem is to study the asymptotic behavior
of solutions for Kirchhoff type systems,
\begin{gather*}
(|v|^{l-2}v)'=\psi_{1}(\|v(t)\|_{2}^{2},
\|w(t)\|_{2}^{2}) v_{x}
+\phi_{1}(\|v(t)\|_{2}^{2}, \|w(t)\|_{2}^{2}) w_{x}- \rho_{1}(t) g(v)
\\
(|w|^{r-2}w)'=\phi_{2}(\|v(t)\|_{2}^{2}, \|w(t)\|_{2}^{2}) v_{x}
+\psi_{2}(\|v(t)\|_{2}^{2}, \|w(t)\|_{2}^{2}) w_{x}- \mu_{2}(t) h(w)
\end{gather*}
where $\phi_{1}, \phi_{2}, \psi_{1}$ and $\psi_{2}$ are real and
continuous functions on $\mathbb{R}_+^{2}$,
$\phi_{1}\phi_{2}\geq 0$, $\rho_{1}$ and $\mu_{2}$ are two  positives
and decreasing functions, $g, h:\mathbb{R}\to \mathbb{R}$ are
non-decreasing functions of class $C (\mathbb{R})$.

If there is a $C^{1}$ function $L(r, s)$ defined on $\mathbb{R}_{+}^{2}$,
with
$$
\frac{l}{l-1}\frac{\partial L}{\partial
r}\phi_{1}=\frac{r}{r-1}\frac{\partial L}{\partial r}\phi_{2},\quad
\frac{\partial L}{\partial r}\geq 0,\quad
\frac{\partial L}{\partial s}\geq 0.
$$
We define the energy function
$E(t)=L(\|v(t)\|_{l}^{l}, \|w(t)\|_{r}^{r})$.
So that
$$
E'(t)=-\frac{l}{l-1}\rho_{1}\int_{0}^{2\pi}g(v) v\, dx
-\frac{r}{r-1}\mu_{2}\int_{0}^{2\pi}h(v) v\, dx\leq 0.
$$

3. Another interesting problem is to study the asymptotic behavior
of solutions for Kirchhoff equation with memory,
\begin{gather*}
(|u'|^{l-2}u')'-\phi_{1}(\|\nabla u\|^2)\Delta u-
\int^t_0 a(t-s)\phi_{2}(\|\nabla u\|^2)\Delta uds=0\quad\text{in }
\Omega\times \mathbb{R}_+\\
 u=0 \quad \text{on } \Gamma_{0}\times \mathbb{R}_+\\
u(x, 0)=u_{0}(x),\quad  u'(x, 0)=u_{1}(x) \quad\text{on }\Omega.
\end{gather*}
In the non-degenerate case, the global existence in
$H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$ was treated by  Abdelli
and Benaissa \cite{abbe} when $\phi_{2}\equiv$const and the
function $a$ is a polynomial. The asymptotic behaviour of
the energy play an important role to prove global existence.

In the degenerate case, when $\phi_{1}\geq \phi_{2}\geq 0$,
Dix and  Torrejon \cite{dito} proved a global existence of
the $(-\Delta)$-analytic solution. It is an interesting question
to study the decay rate of the energy (the energy is a decreasing
function). It is clear that the energy decay rate depends on the
order of degeneracy of $\phi_{1}, \phi_{2}$ and the form of $a$.

4. Another interesting problem is to study global existence and
asymptotic behaviour for the following Kirchhoff equation with
dissipation and source term with initial data less regular than as
in the classical case
(i.e $(u_{0}, u_{1})\in H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$),
\begin{gather*}
(|u'|^{l-2}u')'-\phi_{1}(\|\nabla u\|^2)\Delta u+
\sigma(t) g(u')+f(u)=0\quad \text{in } \Omega\times
\mathbb{R}_+\\
u=0  \quad \text{on } \Gamma_{0}\times \mathbb{R}_+\\
u(x, 0)=u_{0}(x),\quad u'(x, 0)=u_{1}(x) \quad \text{on }\Omega.
\end{gather*}
This study makes possible to consider the case when $g$ and $f$
are not Lipschitz functions (see  Serrin,  Todorova and
Vitillaro \cite{serr} and  Panizzi \cite{pani}). A
convenient space is
$D((-\Delta)^{\kappa/ 2})\cap D((-\Delta)^{{{\kappa-1}\over 2}})$
where $\kappa\geq 3/ 2$,
in particular when $\kappa=2$, we find
$D((-\Delta)^{\kappa/2})\cap
D((-\Delta)^{{{\kappa-1}\over 2}})=H^{2}(\Omega)\cap
H_{0}^{1}(\Omega)$. When $1\leq  \kappa<  3/2$, the problem
of local existence is open for the non-degenerate Kirchhoff equation
without dissipation and source term.

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\end{document}