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\font\eightrm=cmr8 \font\eighti=cmti8 \font\eightbf=cmbx8
\headline={\ifnum\pageno=1 \hfill\else%
{\tenrm\ifodd\pageno\rightheadline \else
\leftheadline\fi}\fi}
\def\rightheadline{EJDE--1998/18\hfil Second order evolution 
equation\hfil\folio}
\def\leftheadline{\folio\hfil Mohammed Aassila
\hfil EJDE--1998/18}
\voffset=2\baselineskip
\vbox {\eightrm\noindent\baselineskip 9pt %
 Electronic Journal of Differential Equations,
Vol.\ {\eightbf 1998}(1998) No.~18, pp. 1--6.\hfill\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfil\break ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
\footnote{}{\vbox{\hsize=10cm\eightrm\noindent\baselineskip 9pt %
1991 {\eighti Subject Classification:} 35B37, 35L70, 35B40.
\hfil\break
{\eighti Key words and phrases:} asymptotic behavior, strong stabilization.
\hfil\break
\copyright 1998 Southwest Texas State University  and
University of North Texas.\hfil\break
Submitted April 15, 1998. Published July 2, 1998.
} }

\bigskip\bigskip

\centerline{SOME REMARKS ON A SECOND ORDER EVOLUTION EQUATION} 
\medskip
\centerline{Mohammed Aassila}
\bigskip\bigskip

{\eightrm\baselineskip=10pt \narrower
\centerline{\eightbf Abstract}
We prove the strong asymptotic stability of solutions to a second order 
evolution equation when the LaSalle's invariance principle
cannot be applied due to the lack of monotonicity and compactness.
\bigskip}


\def\diver{\mathop{\rm div}}

\bigbreak
\centerline{{\bf \S 1. Introduction and statement of the main result}}
\medskip\nobreak
In recent papers [1, 2] we studied the asymptotic stability for some
dissipative wave systems. Earlier work in the same direction is due to Nakao
[7] who treated particularly the case of abstract evolution equations. In
this work we give a new asymptotic stability theorem which extends the analysis
 in [5, 8] by taking into account the new approach introduced in [1, 2].

We focus on abstract equations of the form
$$\displaylines{
u''-\diver\bigl((1+|\nabla u|^a)^b |\nabla u|^{c-2}\nabla u\bigr)+g(u')=0
\quad\hbox{in}\quad \Omega\times {\Bbb R}_+,\cr
\hfill u(x,0)=u_0(x), u'(x,0)=u_1(x)\quad\hbox{in}\quad\Omega,\hfill (P)\cr
u(x,t)=0\quad\hbox{on}\quad\partial\Omega\times {\Bbb R}_+,\cr}
$$
where $\Omega$ is a domain in ${\Bbb R}^n$ of {\it finite measure} with smooth 
boundary
$\partial\Omega$ and $a\geq 1$, $b, c>1$ are real numbers such that 
$ab+c\geq 1$. Concrete examples of (P) include the dissipative wave 
equation
$$\displaylines{
u''-\Delta u+g(u')=0
\quad\hbox{in}\quad \Omega\times {\Bbb R}_+,\cr
\hfill u(x,0)=u_0(x), u'(x,0)=u_1(x)\quad\hbox{in}\quad\Omega,\hfill (P1)\cr
u(x,t)=0\quad\hbox{on}\quad\partial\Omega\times {\Bbb R}_+,\cr}
$$
when $a=b=0,\,c=2$. The degenerate Laplace operator\medskip
$$\displaylines{
u''-\diver\bigl( |\nabla u|^{c-2}\nabla u\bigr)+g(u')=0
\quad\hbox{in}\quad \Omega\times {\Bbb R}_+,\cr
\hfill u(x,0)=u_0(x), u'(x,0)=u_1(x)\quad\hbox{in}\quad\Omega,\hfill (P2)\cr
u(x,t)=0\quad\hbox{on}\quad\partial\Omega\times {\Bbb R}_+,\cr}
$$
when $a=b=0$, $c>1$. And the quasilinear wave equation
$$\displaylines{
u''-\diver\Bigl({{\nabla u}\over {\sqrt{1+|\nabla u|^2}}}\Bigr)+g(u')=0
\quad\hbox{in}\quad \Omega\times {\Bbb R}_+,\cr
\hfill u(x,0)=u_0(x), u'(x,0)=u_1(x)\quad\hbox{in}\quad\Omega,\hfill (P3)\cr
u(x,t)=0\quad\hbox{on}\quad\partial\Omega\times {\Bbb R}_+,\cr}
$$
when $a=2$, $b=-1/2$ and $c=2$.
Problem (P3), with $-\Delta u'$ instead of $g(u')$, describes the motion of
fixed membrane with strong viscosity. This problem with $n=1$ was proposed by
Greenberg [3] and Greenberg-MacCamy-Mizel [4] as a model of
quasilinear wave equation which admits a global solution for large data. Quite
recently, Kobayashi-Pecher-Shibata [6] have treated such nonlinearity
and proved the global existence of smooth solutions. Subsequently, Nakao [8]
has derived a decay estimate of the solutions under the assumption that
the mean curvature of $\partial\Omega$ is non-positive. The object of this paper
is to study the asymptotic behavior of the solution $u$ of (P) which is assumed
to exist in the class
$$
u\in C({\Bbb R}_+, W_0^{1, ab+c}(\Omega))\cap C^1({\Bbb R}_+, L^2(\Omega))
\eqno (1.1)
$$
without any boundedness or geometrical conditions on $\Omega$.

We make the following assumptions on the nonlinear function $g$:
\item{(H1)} $ g: {\Bbb R}\to {\Bbb R}$ is locally Lipschitz continuous

\item{(H2)} $xg(x)>0$ for all $x\neq 0$ 

\item{(H3)} There exists a number $q\geq 1$ satisfying 
$$\displaylines{
(n-2)q\leq n+2\quad\hbox{for}\quad (P1)\cr
(n-c)q\leq n(c-1)+c\quad\hbox{for}\quad (P2)\cr
(n-1)q\leq 1\quad\hbox{for}\quad (P3)\,,\cr }
$$
 and there exist positive constants $c_1$, $c_2$ such that
$$ c_{1}|x|\leq |g(x)|\leq c_{2}|x|^{q}\quad \hbox{for all}\quad |x|\geq 1\,.
$$

We define the energy associated to the solution given by (1.1) by the
following formula
$$
E(u(t)):={1\over 2}\Vert u'(t)\Vert_2^2+\Vert {\cal A}(\nabla u)\Vert_1\,,
\eqno (1.2)
$$
where ${{\partial{\cal A}(v)}\over {\partial v}}:=(1+|v|^a)^b|v|^{c-2}v$.
\medskip
\noindent Our main result is the following

\proclaim{Main Theorem}. It holds that
$$
E(u(t))\to 0\,,\quad\hbox{as}\quad t\to +\infty\,,
$$
for every solution $u$ satisfying (1.1).

\bigbreak
\centerline{{\bf \S 2. Proof of the main theorem}} \medskip\nobreak

For the proof we need the two following lemmas. 
\proclaim Lemma 2.1.  It holds that
$$\int_{0}^{t}\int_{\Omega}|ug(u')|\,dx\,ds=o(t)\,,\quad t\to +\infty\,.$$

\proclaim Lemma 2.2. It holds that
 $$\int_{0}^{t}\int_{\Omega}|u'|^{2}\,dx\,ds=o(t)\,,\quad t\to +\infty\,.$$
\bigskip
\noindent{\bf Proof of lemma 2.1.} As $g$ is locally Lipschitz continuous 
we have
$$\eqalign{
\int_{|u'|\leq 1}|ug(u')|\,dx\leq & c\int_{\Omega}(|u'|\,|g(u')|)^{1/2}
|u|\,dx\cr
\leq & c\,\Bigl(\int_{\Omega}u'g(u')\,dx\Bigr)^{1/2}\,\Vert u
\Vert_{L^{2}(\Omega)}\,. \cr}
$$
Similarly, by (H3) we have
$$\int_{|u'|>1}|ug(u')|\,dx\leq 
c\,\Bigl(\int_{\Omega}u'g(u')\,dx\Bigr)^{{1}\over {(q+1)'}}\,
\Vert u\Vert_{L^{q+1}(\Omega)}
$$
where $(q+1)'={{q}\over {q+1}}$ is the H\"older conjugate of $q+1$.
\medskip
Then from the H\"older's inequality we obtain
$$\eqalign{
\int_{0}^{t}\int_{\Omega}|ug(u')|&\,dx\,ds\cr
\leq& c\,\Bigl(\int_{0}^{t}\int_{\Omega}
u'g(u')\,dx\,ds\Bigr)^{1/2}\sqrt {t}\sup_{[0,t]}\Vert u(s)\Vert_{L^{2}
(\Omega)} \cr 
&+c t^{{1}\over 
{q+1}}\Bigl(\int_{0}^{t}\int_{\Omega}u'g(u')\,dx,ds\Bigr)^{{1}\over 
{(q+1)'}}\,\sup_{[0,t]}\Vert u(s)\Vert_{L^{q+1}(\Omega)}\,. \cr }
$$
Using the H\"older, Sobolev, and Poincar\' e inequalities we have
$$
\Vert u(s)\Vert_{L^{2}(\Omega)}\leq c\,\Vert u(s)\Vert_{L^{q+1}(\Omega)}\leq
c\,E(s)^{1/2}\leq c\,E(0)^{1/2}\quad \hbox{for all}\quad s
\geq 0.
$$
From these estimates it follows that
$$
\int_{0}^{t}\int_{\Omega}|ug(u')|\,dx,ds\leq c\sqrt {t}+c t^{{1}\over {q+1}}
=o(t),\quad t\to +\infty\,.
$$
\bigskip
\noindent{\bf Proof of lemma 2.2.}\quad Let $\varepsilon>0$ be an arbitrarily 
small real and set
$$
M(\varepsilon)=\sup\,\, \{{{x}\over {g(x)}};\quad |x|\geq \sqrt{{{\varepsilon}
\over {|\Omega|}}}\,\,\}
$$
by hypotheses (H1)-(H3), we have $M(\varepsilon)<+\infty$.

Clearly, $$ \int_{|u'|<\sqrt{{\varepsilon}\over {|\Omega|}}}|u'|^{2}
\,dx\leq \varepsilon.$$

On the other hand
$$
\int_{|u'|\geq \sqrt{{{\varepsilon}\over {|\Omega|}}}}|u'|^{2}\,dx =
\int_{|u'|\geq \sqrt{{{\varepsilon}\over {|\Omega|}}}}{{u'}\over {g(u')}}\,
u' g(u')\,dx\leq M(\varepsilon)\int_{\Omega}u'g(u')\,dx\,.
$$
As
$$
\int_{|u'|\geq \sqrt{{{\varepsilon}\over {|\Omega|}}}}|u'|^{2}\,dx\leq\sqrt{
2E(0)}\,\,\Bigl(\int_{|u'|\geq \sqrt{{{\varepsilon}\over 
{|\Omega|}}}}|u'|^{2}\,dx\Bigr)^
{1/2},
$$
we deduce that
$$
\int_{\Omega}|u'|^{2}\,dx\leq \varepsilon 
+\sqrt{2E(0)M(\varepsilon)}\,\,\Bigl(\int_
{\Omega}u'g(u')\,dx\Bigr)^{1/2},
$$
and then by the H\"older inequality
$$\eqalign{
\int_{0}^{t}\int_{\Omega}|u'|^{2}\,dx\,ds\leq& \varepsilon t+\sqrt{2E(0)M(
\varepsilon)}\,\sqrt{t}\,\,\Bigl(\int_{0}^{t}\,\,\int_{\Omega}u'g(u')\,dx,ds\Bigr)^{1/2} \cr
\leq & \varepsilon t+E(0)\sqrt{2M(\varepsilon)}\,\sqrt{t}=o(t),\quad t\to +
\infty\,. \cr }
$$

\bigskip\noindent{\bf Proof of the main theorem}\hfill\break
Assume on the contrary that $l:=\displaystyle\lim_{t\to +\infty}E(t) >0$. 
Then we have
$$
\int_0^t\int_\Omega uu'\,dx\,ds=\int_0^t\int_\Omega |u'|^2-A(\nabla u)\nabla u-
g(u')u\,dx\,ds
$$
where $A(\nabla u):=(1+|\nabla u|^a)^b|\nabla u|^{c-2}\nabla u.$
Following the approach introduced in [1, 2], we shall prove 
that
$$
\Vert u'\Vert_2^2+\int_\Omega A(\nabla u)\nabla u\,dx\geq c_3>0\,.\eqno (2.1)
$$
We have
$$
\Vert u'(t)\Vert_2^2+2\Vert {\cal A}(\nabla u)\Vert_1\geq l\,;
$$
hence, if $\Vert u'(t)\Vert_2^2\geq {l\over 2}$ we get (2.1) with $c_3=
{l\over 2}$. And, if we have $\Vert {\cal A}(\nabla u)\Vert_1\geq {l\over 4}$,
then
$$
c_4\Bigl(\Vert \nabla u\Vert_1+\Vert \nabla u\Vert_{ab+c}^{ab+c}\Bigr)\geq
{l\over 4}$$
that is
$$
\Vert \nabla u\Vert_{ab+c}\geq c_5>0\,.
$$
Since $A$ is coercive (that is $(A(v),v)_{L^2}\geq c_6 |v|^{ab+c}$ with $|v|
\geq |v_0|$), we get (2.1) with a positive constant $c_7>0$.

Thanks to lemmas 1,2, and the relation (2.1), we arrive by the same 
arguments in [1, 2] to
$$\phi (t)\to -\infty\quad\hbox{as}\quad t\to +\infty\,,$$
where $\phi (t)=\int_\Omega uu'\,dx$. This is a contradiction to the fact that
$|\phi (t)|\leq c_8E(0)$. Thus
$$
\lim_{t\to +\infty}E(t)=0\,.
$$
\medskip\noindent{\bf Remark.}
If $g$ is linear or superlinear near the origin, then it is sufficient to
consider a domain $\Omega\subset {\Bbb R}^n$ in which the Poincar\'e's 
inequality
holds.

\bigbreak
\centerline{\bf References}\medskip\nobreak

\item{[1]} M. Aassila, Nouvelle approche \`a la stabilisation forte des
syst\`emes distribu\'es, C. R. Acad. Sci. Paris {\bf 324} (1997), 43--48.

\item{[2]} M. Aassila, A new approach of strong stabilization of distributed
 systems, Differential and Integral Equations {\bf 11}(1998), 369--376.

\item{[3]} J. Greenberg, On the existence, uniqueness and stability of the
equation $\rho_0 X_{tt}=E(X_x)X_{xx}+X_{xxt}$, J. Math. Anal. Appl. {\bf 25}
(1969), 575--591.

\item{[4]} J. Greenberg, R. MacCamy and V. Mizel, On the existence, uniqueness
and stability of the equation $\sigma'(u_x)u_{xx}+\lambda u_{xtx}=\rho_0 
u_{tt}$, J. Math. Mech. {\bf 17} (1968), 707--728.

\item{[5]} R. Ikehata, T. Matsuyama and M. Nakao, Global solutions to the
initial boundary value problem for the quasilinear viscoelastic wave equation
with a perturbation, Funkcialaj Ekva. {\bf 40} (1997), 293--312.

\item{[6]} T. Kobayashi, H. Pecher and Y. Shibata, On a global in time existence
theorem of smooth solutions to nonlinear wave equation with viscosity, Math. 
Ann. {\bf 296} (1993), 215--234.

\item{[7]} M. Nakao, Asymptotic stability for some nonlinear evolution equations
of second order with unbounded dissipative terms, J. Diff. Eqns. {\bf 30} 
(1978), 54--63.

\item{[8]} M. Nakao, Energy decay for the quasilinear wave equation with 
viscosity, Math. Z. {\bf 219} (1995), 289--299.


\bigskip\noindent
Mohammed Aassila\hfil\break
Institut de Recherche Math\'ematique Avanc\'ee\hfil\break
Universit\'e Louis Pasteur et C.N.R.S.\hfil\break
7, rue Ren\'e Descartes\hfil\break
67084 Strasbourg C\'edex, France.\hfil\break
E-mail: aassila@math.u-strasbg.fr

\bye 

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