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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 222, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/222\hfil Energy decay]
{Energy decay for degenerate Kirchhoff equations with weakly nonlinear
dissipation}

\author[M. Abdelli, S. A. Messaoudi \hfil EJDE-2013/222\hfilneg]
{Mama Abdelli, Salim A. Messaoudi}  % in alphabetical order

\address{Mama Abdelli \newline
Universit\'e Djillali Liab\'es, Laboratoire de Math\'ematique,
B.P. 89. Sidi Bel Abb\'es 22000, Algeria}
\email{abdelli\_mama@yahoo.fr}

\address{Salim A. Messaoudi \newline
King Fahd University of Petroleum and Minerals\\
Department of Mathematics and Statistics \\
Dhahran 31261, Saudi Arabia}
\email{messaoud@kfupm.edu.sa}

\thanks{Submitted September 18, 2013. Published October 11, 2013.}
\subjclass[2000]{35B37, 35L55, 74D05, 93D15, 93D20}
\keywords{Decay of solutions; nonlinear; degenerate; Kirchhoff equation}

\begin{abstract}
 In this article we consider a degenerate Kirchhoff equation wave
 equation with a weak frictional damping,
 \[
 (|u_t|^{l-2}u_t)_t-\Big( \int_{\Omega }|\nabla
 _xu|^{2}\,dx\Big)^{\gamma }\Delta _xu+\alpha (t)g(u_t)=0.
 \]
 We prove general stability estimates using some properties of convex
 functions, without imposing any growth condition at the frictional damping
 term.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

In this article, we consider the initial-boundary value problem for the
nonlinear Kirchhoff equation
\begin{gather}
(|u_t|^{l-2}u_t)_t-\Big( \int_{\Omega }|\nabla u|^{2}\,dx
\Big)^{\gamma }\Delta u+\alpha (t)g(u_t)=0,\quad \text{in }\Omega
\times (0,\infty )  \label{p1}
\\
u=0,\quad \text{on } \partial \Omega \times (0,\infty )  \label{p2}
\\
u(x,0)=u^{0}(x),\quad u_t(x,0)=u^{1}(x),\quad x\in \Omega ,  \label{p3}
\end{gather}
where $l\geq 2$, $\gamma \geq 0$ are given constants,  $\Omega $ is
a bounded domain in $\mathbb{R}^{n}$ with a smooth
boundary $\partial \Omega $, and $g$ and $\alpha $ are one-variable functions
satisfying some conditions to be specified later. This problem has been
studied by many authors and several existence, nonexistence, and decay results
have appeared. For instance, when $l=2$ and $\gamma =0$, the problem was
treated by Mustafa and Massaoudi \cite{MM}. By using some
properties of convex functions, they established a general decay result
without imposing any growth condition on $g$ at the origin. Abdelli and
Benaissa \cite{AB} treated system \eqref{p1}-\eqref{p3} for $g$
having a polynomial growth near the origin and established energy decay
results depending on $\alpha $ and $g$ under appropriate relations between 
$l $ and $\gamma $. In a realted work, Amroun and Benaissa \cite{AM}
constructed an exact solution of \eqref{p1}-\eqref{p3} in the presence of a
nonlinear source term and for $\alpha \equiv 0.$ They also proved a
finite-time blow-up result for some specific initial data. 
Benaissa and Guesmia \cite{BG} proved the existence of global solution, as well
as, a general stability result for the following equation
\[
(|u'|^{l-2}u')'-\Delta _{\phi }u+\alpha
(t)g(u')=0,\quad \text{in } \Omega \times  \mathbb{R}_{+},
\]
where $\Delta _{\phi }=\sum_{i=1}^{n}\partial _{x_i}(\phi (|\partial
_{x_i}|^{2})\partial _{x_i})$.

In this article, we use some technique from \cite{MM} to establish an
explicit and general decay result, depending on $g$ and $\alpha $. The proof
is based on the multiplier method and makes use of some properties of convex
functions, the general Young inequality and Jensen's inequality. These
convexity arguments were introduced and developed by Lasiecka and co-workers
(\cite{L,LT1,LT2}) and used, with
appropriate modifications, by Liu and Zuazua \cite{LZ},
Alabau-Boussouira \cite{FA} and others. 

The paper is organized as follows: in section $2$, we give our hypotheses and
establish a useful lemma. In section $3$, we state and prove our main result.

\section{Preliminaries}

To state and prove our result, we need the
following hypotheses:
\begin{itemize}
\item[(H1)] $\alpha :{\mathbb{R}_{+}}\to {\mathbb{R}_{+}}$ is a nonincreasing
differentiable function.

\item[(H2)] $g:\mathbb{R}\to \mathbb{R}$ is a nondecreasing $C^{0}$
 function such that there exist $\varepsilon ,c_1,c_2>0$, and a
convex and increasing function $G:{\mathbb{R}_{+}}\to {\mathbb{R}_{+}}$
of class $C^{1}(\mathbb{R}_{+})\cap C^{2}(]0,+\infty[)$ 
satisfying $G(0)=G'(0)=0$ or $G$ is linear on $[0,\varepsilon ]$ such that
\begin{gather*}
c_1|s|^{l-1}\leq |g(s)|\leq c_2|s|^{p},\quad \text{if }|s|\geq \varepsilon;\\
|s|^{l}+|g(s)|^{\frac{l}{l-1}}\leq G^{-1}(sg(s)),\quad \text{if }
|s|\leq \varepsilon;
\end{gather*}
with $p$ satisfying
\begin{gather*}
l-1\leq p\leq \frac{n+2}{n-2}, \quad \text{if }n>2; \\
l-1\leq p<\infty,\quad \text{if }n\leq 2\,.
\end{gather*}
\end{itemize}

Now we define the energy associated to the solution of the system \eqref{p1}
-\eqref{p3} by
\begin{equation}
E(t)=\frac{l-1}{l}\Vert u_t\Vert _{l}^{l}+\frac{1}{1+\gamma }\Vert \nabla
_xu\Vert _2^{2(\gamma +1)}  \label{E1}
\end{equation}

\begin{lemma} \label{lem2.1} 
Let $u$ be the solution of \eqref{p1}-\eqref{p3}.
Then
\begin{equation}
E'(t)=-\alpha (t) \int_{\Omega }u_tg(u_t)\,dx\leq 0.
\label{E2}
\end{equation}
\end{lemma}

\begin{proof}
 Multiplying \eqref{p1} by $u_t$ and integrating over $
\Omega $, using the boundary conditions, the assertion of the lemma follows.
\end{proof}

\section{Main Result}

To prove our main result, first prove the following lemma.

\begin{lemma} \label{lem3.1}
Assume that {\rm (H1), (H2)} hold and that $l\geq 2(\gamma +1)$. 
Then the functional 
\[
F(t)=ME(t)+ \int_{\Omega }u|u_t|^{l-2}u_t\,dx,
\]
defined along the solution of \eqref{p1}-\eqref{p3},
satisfies the following estimate,  for
some positive constants $M,c,m$:
\[
F'(t)\leq -mE(t)+c \int_{\Omega
}(|u_t|^{l}+|ug(u_t)|^{\frac{l}{l-1}})\,dx
\]
and $F(t)\sim E(t)$.
\end{lemma}

\begin{proof} Using  system \eqref{p1}-\eqref{p3}, \eqref{E1} and \eqref{E2}, we
obtain
\begin{align*}
F'(t) &= ME'(t)+ \int_{\Omega
}|u_t|^{l}\,dx+\int_{\Omega }u(|u_t|^{l}u_t)_t\,dx   \\
&\leq  \int_{\Omega }|u_t|^{l}\,dx+\Big(\int_{\Omega }|\nabla
u|^{2}\,dx\Big)^{\gamma }\int_{\Omega }u\Delta u\,dx-\alpha (t)\int_{\Omega
}ug(u_t)\,dx   \\
&\leq  \int_{\Omega }|u_t|^{l}\,dx-\int_{\Omega }|\nabla
u|^{2(\gamma +1)}\,dx-\alpha (t)\int_{\Omega }ug(u_t)\,dx \\
&\leq -mE(t)+c \int_{\Omega }[|u_t|^{l}+|ug(u_t)|]\,dx
\end{align*}

To prove that $F(t)\sim E(t)$, we show that 
for some positive constants $\lambda _1$ and $\lambda _2$,
\begin{equation}
\lambda _1E(t)\leq E(t)\leq \lambda _2E(t)\,.  \label{AM1}
\end{equation}
 We use \eqref{E1}, Poincar\'e's and Young's inequalities with exponents 
$\frac{l}{l-1}$ and $\frac{1}{l}$ and recall that
 $2\leq l\leq p+1\leq \frac{2n}{n+2}$, to obtain
\begin{align*}
\int_{\Omega }u|u_t|^{l-2}u_t\,dx 
&\leq C_{\varepsilon }\int_{\Omega
}|u|^{l}\,dx+\varepsilon \int_{\Omega }|u_t|^{l}\,dx \\
&\leq C_{\varepsilon }\Vert \nabla u\Vert _2^{l}+\varepsilon \Vert
u_t||_{l}^{l} \\
&\leq C_{\varepsilon }E^{\frac{l}{2(\gamma +1)}}(t)+c\varepsilon E(t) \\
&\leq C_{\varepsilon }E^{\frac{l-2(\gamma +1)}{2(\gamma +1)}
}(t)E(t)+c\varepsilon E(t).
\end{align*}
By noting that $l\geq 2(\gamma +1)$ and using \eqref{E2}, we have
\[
\int_{\Omega }u|u_t|^{l-2}u_t\,dx\leq C_{\varepsilon }E^{\frac{
l-2(\gamma +1)}{2(\gamma +1)}}(0)E(t)+c\varepsilon E(t),
\]
and
\begin{align*}
\int_{\Omega }u|u_t|^{l-2}u_t\,dx
&\geq -C_{\varepsilon }\int_{\Omega
}|u|^{l}\,dx-\varepsilon \int_{\Omega }|u_t|^{l}\,dx \\
&\geq -C_{\varepsilon }E^{\frac{l-2(\gamma +1)}{2(\gamma +1)}
}(t)E(t)-c\varepsilon E(t) \\
&\geq -C_{\varepsilon }E^{\frac{l-2(\gamma +1)}{2(\gamma +1)}
}(0)E(t)-c\varepsilon E(t)
\end{align*}
Then, for $M$ large enough, we obtain \eqref{AM1}. This completes the proof.
\end{proof}

Taking $0<\varepsilon _1<\varepsilon $ such that
\begin{equation} \label{e3.3}
sg(s)\leq \min \{\varepsilon ,G(\varepsilon )\},\quad \text{if }
|s|\leq \varepsilon _1,
\end{equation}
and
\begin{equation} \label{e3.4}
\begin{cases}
c_1'|s|^{l-1}\leq |g(s)|\leq c_2'|s|^{p}, &\text{if }
|s|\geq \varepsilon _1 \\
|s|^{l}+|g(s)|^{\frac{l}{l-1}}\leq G^{-1}(sg(s)), &\text{if }
|s|\leq \varepsilon _1.
\end{cases}
\end{equation}
Considering the following partition of $\Omega $,
\[
\Omega _1=\{x\in \Omega :|u_t|\leq \varepsilon _1\},\quad 
\Omega _2=\{x\in \Omega : |u_t|>\varepsilon _1\}
\]
and using the embedding $H_0^{1}(\Omega )\hookrightarrow L^{p+1}(\Omega )$
and H\"{o}lder's inequality, we obtain
\begin{align*}
\int_{\Omega _2}|ug(u_t)|\,dx 
&\leq \Big( \int_{\Omega_2}|u|^{p+1}\,dx\Big)^{\frac{1}{p+1}}
 \Big( \int_{\Omega _2}|g(u_t)|^{1+\frac{1}{p}}\,dx\Big)^{p/(p+1)} \\
&\leq c\Vert u\Vert _{H_0^{1}(\Omega )}\Big( \int_{\Omega
_2}|g(u_t)|^{1+\frac{1}{p}}\,dx\Big)^{p/(p+1)}
\end{align*}
Using  Poincar\'e's inequality and \eqref{e3.4} yields
\begin{align*}
&\int_{\Omega _2}[|u_t|^{l}+|ug(u_t)|]\,dx \\
&\leq c  \int_{\Omega _2}|u_t|^{l-1}|u_t|\,dx+c\Big(
\int_{\Omega }|\nabla u|^{2}\,dx\Big)^{1/2}\Big(
\int_{\Omega _2}|g(u_t)|^{1+\frac{1}{p}}\,dx\Big)^{p/(p+1)} \\
&\leq c \int_{\Omega _2}u_tg(u_t)\,dx+c\Big(
\int_{\Omega }|\nabla u|^{2}\,dx\Big)^{1/2}\Big(
\int_{\Omega _2}u_tg(u_t)\,dx\Big)^{p/(p+1)} \\
&\leq -cE'(t)+cE^{\frac{1}{2(\gamma +1)}}(-E'(t))^{p/(p+1)}.
\end{align*}
Then, we use Young's inequality and the fact that $p\geq l-1\geq 2\gamma +1$, 
 for any $\delta >0$, we have
\begin{equation} \label{e3.5}
\begin{aligned}
 \int_{\Omega _2}[|u_t|^{l}+|ug(u_t)|]\,dx
&\leq -cE'(t)+c\delta E^{\frac{p+1}{2(\gamma +1)}}(t)+C_{\delta
}(-E'(t))   \\
&\leq c\delta E^{\frac{p+1}{2(\gamma +1)}}(t)-C_{\delta }E'(t) \\
&\leq c\delta E^{\frac{p-(2\gamma +1)}{2(\gamma +1)}}(0)E(t)-C_{\delta
}E'(t)
\end{aligned}
\end{equation}
Similarly, using \eqref{E1} and Young's inequality, we have, for any
 $\delta >0$,
\begin{equation} \label{e3.6}
\begin{aligned}
 \int_{\Omega _1}[|u_t|^{l}+|ug(u_t)|]\,dx
&\leq  \int_{\Omega _1}|u_t|^{l}\,dx+\delta
\int_{\Omega _1}|u|^{l}\,dx+C_{\delta } \int_{\Omega
_1}|g(u_t)|^{\frac{l}{l-1}}\,dx   \\
&\leq  \int_{\Omega _1}|u_t|^{l}\,dx+c\delta E^{\frac{l}{
2(\gamma +1)}}(t)+C_{\delta } \int_{\Omega _1}|g(u_t)|^{
\frac{l}{l-1}}\,dx
\end{aligned}
\end{equation}
By Lemma 3.1, \eqref{e3.5} and \eqref{e3.6}, for $\delta $ small enough,
the function $ L=F+C_{\delta }E$ satisfies
\begin{equation} \label{e3.7}
\begin{aligned}
L'(t)
&\leq \Big(-m+c\delta E^{\frac{p-(2\gamma +1)}{2}
}(0)+c\delta E^{\frac{l-2(\gamma +1)}{2(\gamma +1)}}(0)\Big)E(t) \\
&\quad+ \int_{\Omega _1}|u_t|^{l}\,dx+C_{\delta }
\int_{\Omega _1}|g(u_t)|^{\frac{l}{l-1}}\,dx   \\
&\leq -dE(t)+c \int_{\Omega _1}\Big(|u_t|^{l}+|g(u_t)|^{
\frac{l}{l-1}}\Big)\,dx
\end{aligned}
\end{equation}
and
\begin{equation} \label{e3.8}
L(t)\sim E(t).
\end{equation}

\begin{theorem} \label{thm3.2}
 Assume that {\rm (H1),  (H2)} hold and $l\geq 2(\gamma +1)$. 
Then there exist positive constants $k_1,k_2,k_3$ and $\varepsilon _0$ 
such that the solution of \eqref{p1}-\ref{p3} satisfies
\begin{equation}
E(t)\leq k_3G_1^{-1}\Big(k_1 \int_0^{t}\alpha
(s)\,ds+k_2\Big)\quad \forall t\geq 0,  \label{3B}
\end{equation}
where
\begin{equation} \label{e3.10}
G_1(t)= \int_t^{1}\frac{1}{G_2(s)}\,ds,\quad 
G_2(t)=tG'(\varepsilon _0t).
\end{equation}
Here, $G_1$ is strictly decreasing and convex on $(0,1]$ with 
$\lim_{t\to 0}G_1(t)=+\infty $.
\end{theorem}

\begin{proof} 
Multiplying \eqref{e3.7} by $\alpha (t)$, we have
\begin{equation} \label{e3.11}
\alpha (t)L'(t)\leq -d\alpha (t)E(t)+c\alpha (t)\int_{\Omega _1}
\Big(|u_t|^{l}+|g(u_t)|^{\frac{l}{l-1}}\Big)\,dx
\end{equation}
\textbf{Case 1}. $G$ is linear on $[0,\varepsilon ]$, then we deduce that
\[
\alpha (t)L'(t)\leq -d\alpha (t)E(t)+c\alpha (t)\int_{\Omega
_1}u_tg(u_t)\,dx=-d\alpha (t)E(t)-cE'(t)
\]
Consequently, we arrive at
\[
(\alpha L+cE)'(t)\leq -d\alpha (t)E(t).
\]
Recalling that
\begin{equation}
\alpha L+cE\sim E,  \label{11}
\end{equation}
we obtain
\[
E(t)\leq c'e^{-c''\int_0^{t}\alpha (s)\,ds}
\]
Thus, we have
\[
E(t)\leq c'e^{-c''\int_0^{t}\alpha
(s)\,ds}=c'G_1^{-1}(c''\int_0^{t}\alpha (s)\,ds)
\]
by a simple computation.
\smallskip

\noindent\textbf{Case 2}. 
$G$ is nonlinear on $[0,\varepsilon ]$. In this case, we
define 
\[
I(t)=\frac{1}{|\Omega _1|}\int_{\Omega _1}u_tg(u_t)\,dx.
\]
and exploit Jensen's inequality and the concavity of $G^{-1}$ to obtain
\[ 
G^{-1}(I(t))\geq c\int_{\Omega _1}G^{-1}(u_tg(u_t))\,dx.
\]
By using this inequality and \eqref{e3.4}, we obtain
\begin{equation}
\alpha (t)\int_{\Omega _1}[|u_t|^{l}+|g(u_t)|^{\frac{l}{l-1}}]\,dx
\leq \alpha (t)\int_{\Omega _1}G^{-1}(u_tg(u_t))\,dx
\leq c\alpha (t)G^{-1}(I(t)).  \label{M}
\end{equation}
Let us set $H_0=\alpha L+E$ and exploit \eqref{E2}, \eqref{e3.11}, \eqref{M}, and
$\alpha $ being  nonincreasing, to obtain
\begin{equation}
\begin{aligned}
H_0'(t)
& \leq -d\alpha (t)E(t)+c\alpha (t)G^{-1}(I(t))+E'(t) \\
& \leq -d\alpha (t)E(t)+c\alpha (t)G^{-1}(I(t)),
\end{aligned}
\label{M1}
\end{equation}
and recall \eqref{e3.8}, to deduce that $H_0\sim E$.

For $\varepsilon _0<\varepsilon $ and $c_0>0$, we define $H_1$ by
\[
H_1(t)=G'(\varepsilon _0\frac{E(t)}{E(0)})H_0(t)+c_0E(t).
\]
Then, we see easily  that, for $a_1,a_2>0$,
\begin{equation}
a_1H_1(t)\leq E(t)\leq a_2H_1(t),  \label{e3.16}
\end{equation}
By recalling that $E'\leq 0$, $G'>0$, $G''>0$
on $(0,\varepsilon ]$ and making use of \eqref{E1} and \eqref{M1}, we obtain
\begin{equation} \label{e3.17}
\begin{aligned}
H_1'(t)
&= \varepsilon _0\frac{E'(t)}{E(0)}
G''(\varepsilon _0\frac{E(t)}{E(0)})H_0(t)+G'(\varepsilon
_0\frac{E(t)}{E(0)})H_0'(t)+c_0E'(t)   \\
&\leq -d\alpha (t)E(t)G'(\varepsilon _0\frac{E(t)}{E(0)}
)+c\alpha (t)G'(\varepsilon _0\frac{E(t)}{E(0)}
)G^{-1}(I(t))+c_0E'(t).
\end{aligned}
\end{equation}
Let $G^{*}$ be the convex conjugate of $G$ in the sense of Young (see
Arnold \cite[p. 61-64]{AR}), then
\begin{equation} \label{e3.19}
G^{*}(s)=s(G')^{-1}(s)-G[(G')^{-1}(s)],\quad \text{if } 
s\in (0,G'(\varepsilon )],  
\end{equation}
and $G^{*}$ satisfies the generalized Young's inequality
\begin{equation}
AB\leq G^{*}(A)+G(B),\quad\text{if }A\in (0,G'(\varepsilon)],\;
B\in (0,\varepsilon ].
\end{equation}
with $A=G'(\varepsilon _0 E(t)/E(0))$ and
$ B=G^{-1}(I(t))$, using \eqref{E2}, \eqref{e3.3} and
\eqref{e3.17}--\eqref{e3.19}, we obtain
\begin{align*}
H_1'(t)
&\leq -d\alpha (t)E(t)G'(\varepsilon _0\frac{
E(t)}{E(0)})+c\alpha (t)G^{*}\Big(G'(\varepsilon _0\frac{E(t)}{
E(0)})\Big) +c\alpha (t)I(t)+c_0E'(t) \\
&\leq -d\alpha (t)E(t)G'(\varepsilon _0\frac{E(t)}{E(0)}
)+c\varepsilon _0\alpha (t)\frac{E(t)}{E(0)}G'(\varepsilon _0
\frac{E(t)}{E(0)})-cE'(t)+c_0E'(t).
\end{align*}
Choosing $c_0>c$ and $\varepsilon _0$ small enough, we obtain
\begin{equation}
H_1'(t)\leq -k\alpha (t)\frac{E(t)}{E(0)}G'\Big(
\varepsilon _0\frac{E(t)}{E(0)}\Big)=-k\alpha (t)G_2\Big(\frac{E(t)}{E(0)
}\Big)  \label{E23}
\end{equation}
where
$G_2(t)=tG'(\varepsilon _0t)$.
Since
\[
G_2'(t)=G'(\varepsilon _0t)+\varepsilon _0tG''(\varepsilon _0t).
\]
and $G$ is convex on $(0,\varepsilon ]$, we find that $G_2'(t)>0$
and $G_2(t)>0$ on $(0,1]$. By setting
$  H(t)=\frac{a_1H_1(t)}{E(0)}$ ($a_1$ is given in \eqref{e3.16}),
 we easily see that, by \eqref{e3.16}, we have
\begin{equation}
H(t)\sim E(t)  \label{1B}
\end{equation}
Using \eqref{E23}, we arrive at
\[
H'(t)\leq -k_1\alpha (t)G_2(H(t))
\]
By recalling \eqref{e3.10}, we deduce $G_2(t)=-1/G_1'(t)$,
hence
\[
H'(t)\leq k_1\alpha (t)\frac{1}{G_1'(H(t))},
\]
which gives
\[
\lbrack G_1(H(t))]'=H'(t)G_1'(H(t))\leq
k_1\alpha (t)
\]
A simple integration leads to
\[
G_1(H(s))\leq k_1\int_0^{t}\alpha (s)\,ds+G_1(H(0))
\]
Consequently,
\begin{equation}
H(t)\leq G_1^{-1}(k_1\int_0^{t}\alpha (s)\,ds+k_2)  \label{2B}
\end{equation}
Using \eqref{1B} and \eqref{2B} we obtain \eqref{3B}.
 The proof is  complete.
\end{proof}

\subsection*{Acknowledgments} 
The second author wants to thank KFUPM for its financial support.

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\end{document}