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\markboth{\hfil Decay of solutions\hfil EJDE--1998/21}%
{EJDE--1998/21\hfil Julio G. Dix \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol.\ {\bf 1998}(1998), No.~21, pp. 1--10. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
\vspace{\bigskipamount} \\
Decay of solutions of a degenerate hyperbolic equation
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35L05, 35B40.
\hfil\break\indent
{\em Key words and phrases:} Degenerate hyperbolic equation, asymptotic behavior.
\hfil\break\indent
\copyright 1998 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Submitted January 29, 1998. August, 28, 1998.} }
\date{}
\author{Julio G. Dix}
\maketitle
\begin{abstract}
This article studies the asymptotic behavior of solutions to
the damped, non-linear wave equation
$$
\ddot u +\gamma \dot u -m(\|\nabla u\|^2)\Delta u = f(x,t)\,,
$$
which is known as degenerate if the greatest lower bound for $m$
is zero, and non-degenerate if the greatest lower bound is positive.
For the non-degenerate case, it is already known that solutions decay
exponentially, but for the degenerate case exponential decay has remained
an open question. In an attempt to answer this question,
we show that in general solutions can not decay with exponential order,
but that $\|\dot u\|$ is square integrable on $[0, \infty)$.
We extend our results to systems and to related equations.
\end{abstract}
\newtheorem{theorem}{Theorem}
\section{Introduction}
This article presents a study of the asymptotic behavior of solutions to
the initial value problem
\begin{eqnarray}
&\ddot u +\gamma \dot u -m(\|\nabla u\|^2)\Delta u = f(x,t)
\,, \quad\mbox{for }x\in\Omega,\ t\geq 0& \label{eqi} \\
&u(x,0)=g(x)\,, \quad \dot u(x,0)=h(x)\,, \quad\mbox{for }x\in\Omega& \nonumber\\
&u(x,t)=0\,, \quad\mbox{for } x\in\partial\Omega\,\ t\geq 0\,;& \nonumber
\end{eqnarray}
where $\Omega$ is a bounded domain in ${\mathbb R}^n$, with smooth boundary
$\partial\Omega$; $\gamma$ is a positive constant;
$m$ is a non-negative, bounded, and continuous function;
$\dot u$ denotes the derivative of $u$ with respect to time; and as usual
$$
\Delta u=\sum_{i=1}^n \frac{\partial^2 u}{\partial x_i^2}\,,\quad
\|\nabla u\|^2=\sum_{i=1}^n \int_\Omega |\frac{\partial u}{\partial x_i}|^2\,dx\,.
$$
This equation appears in mathematical physics as the Carrier or Kirchoff
equation, when modeling planar vibrations. For a background and physical
properties of this model, we refer the reader to \cite{Ca}, \cite{Dy},
\cite{Na}, \cite{Po}, and their references.
When the greatest lower bound for $m$ is positive, (\ref{eqi}) is
known as non-degenerate, and it has been the subject of many publications
(see \cite{Li}, \cite{Ni}, \cite{Ni2}, \cite{Po}, and \cite{Ya}).
Global solutions have been obtained under various assumptions on the data,
of which we are interested in the $(-\Delta)$-analyticity introduced by Pohozaev
(\cite{Po}).
Exponential decay in the non-degenerate case
for (\ref{eqi}) and for related
problems has been obtained in \cite{Ni} and \cite{Bi}, respectively.
When the greatest lower bound for $m$ is zero, (\ref{eqi}) is known as
degenerate, and has been considered in just a few publications.
In the special case $m(r)=r^\alpha, \alpha\geq 1$, existence of solutions
and polynomial decay has been shown in \cite{Ma} and \cite{Ni2}.
For general $m$, assumed only to be continuous and bounded below by zero,
the existence of global solutions was shown by Arosio and Spagnolo
(\cite{Ar}). Their article assumes that the initial data are
$(-\Delta)$-analytic, and that $f\equiv 0, \gamma=0$.
Using the same analyticity assumption, and replacing the damping term
$\gamma \dot u$ by a memory term, existence of global solutions has been
proven in \cite{Di}.
In spite of these developments, decay for degenerate problems
remains an open question.
The outline of this article is as follows:
Section 1 sketches the proof of existence of global solutions.
Section 2 proves exponential decay for the non-degenerate case, and explains
why these estimates can not be used in the degenerate case.
Section 3 shows that if $m\equiv 0$, the decay is exponential.
In general degenerate-problem solutions do not decay exponentially, as
we indicate with an example, but $\|\dot u\|$ is square integrable on
$[0,\infty)$. Section 4 extends our results to related systems.
\subsection*{Notation}
For the remainder of this article, $H$ denotes the standard Hilbert space
$L^2(\Omega)$, with norm $\|.\|$ and inner product $\langle.,.\rangle$.
We define the self-adjoint operator $A$ as the negative Laplacian, with
domain
$${\cal D}(A)\subset H^2(\Omega)\cap H_0^1(\Omega)\,,
$$
where $H^2$, $H^1$ are the usual Hilbert Sobolev spaces. The negative
Laplacian, with zero boundary conditions, has eigenvectors
denoted as follows
$$
A\phi_i=\lambda_i^2\phi_i\,,\quad \mbox{with }0<\lambda_1\leq\lambda_2 \dots\,,
\quad \lim_{i\to \infty}\lambda_i=+\infty\,. $$
Furthermore, these eigenvectors can be chosen to form an orthogonal basis for
$L^2(\Omega)$, in which functions have Fourier expansions of the form
$$
u(x)=\sum_{i=1}^\infty u_i\phi_i(x)\,, \quad\mbox{with }
u_i=\langle u, \phi_i\rangle\,.$$
Using this spectral decomposition, powers of $A$ are defined by
$$
A^ku(x)= \sum_{i=1}^\infty \lambda_i^{2k}u_i \phi_i(x)\,,\quad
\mbox{provided that } \sum_{i=1}^\infty \lambda_i^{4k}|u_i|^2 < +\infty\,.
$$
Notice that
\begin{equation} \label{l1}
\lambda_1^2\|u\|\leq \|Au\|\,, \quad
\lambda_1\|u\|\leq \|\nabla u\|=\|A^{1/2}u\|\,,
\end{equation}
and that (\ref{eqi}) can be rewritten as
\begin{eqnarray} \label{eq1}
&\ddot u +\gamma \dot u +m(\|A^{1/2} u\|^2)A u = f(x,t)& \\
&u(x,0)=g(x)\,, \quad \dot u(x,0)=h(x)\,.& \nonumber
\end{eqnarray}
A function $u$ is said to be A-analytic,
or $(-\Delta)$-analytic, if there
exists a positive constant $\eta$ such that
$\sum_{i=1}^\infty e^{2\eta\lambda_i}|u_i|^2 <\infty$\,.
Notice that every A-analytic function is in the domain of all powers of $A$.
Properties of A-analytic functions and an equivalent definition can be found
in \cite{Ar} and \cite{Di}.
By a solution on $[0,T)$ we mean a function that satisfies
(\ref{eq1}) and belongs to $C^2(0,T;H^{-1})\cap C(0,T;H^1)$.
\section{Existence of global solutions}
\begin{theorem} \label{thmex}
Assume that $m$ is bounded or that
$\int_0^\infty m=+\infty$; and that $f(.,t)$, $g$, and $h$ are
A-analytic for all $t\geq 0$. Then there exists a global solution to
(\ref{eqi}). \end{theorem}
\paragraph{Proof} Solutions are obtained by the use of energy estimates and
the Galerkin method which is a standard technique described in the book
by Temam (\cite{Te}).
To accommodate the fact that $m$ may attain zero values,
we follow the procedure used in \cite{Ar}.
However the presence of $\gamma\dot u$ and $f$ requires some algebraic manipulations
of the type shown in the proof of Theorem~\ref{thmp}. Since the proof
is basically the same as the one in \cite{Ar}, we shall indicate the main steps
in this proof, and refer the reader to the original source.
Step 1. Replace $m(\|\nabla u\|^2)$ by a non-negative bounded continuous
function $a(t)$. Then show that if $u$ is a solution to the new equation
on the interval $[0,T)$, then $u$ admits a limit as $t\to T^-$, and
$u$ and $\dot u$ are A-analytic on $[0,T]$.
Step 2. Use the Galerkin method and a compactness imbedding argument to
obtain a local solution. Then show that $m(\|\nabla u\|^2)$ is
bounded in its domain of definition.
Step 3. Use Zorn's Lemma and the result in step 1 to show that the maximal domain
of definition is $[0,+\infty)$.
\paragraph{Remark} Uniqueness of solutions has been shown
under the additional assumption that $m$ is Lipschitz; see \cite{Ar}.
\section{Decay in the non-degenerate case}
The following statement is already known for $f=0$ (see \cite{Ni}).
We present a proof for general $f$ and show how the rate of decay depends
on the lower bound for $m$.
\begin{theorem} \label{thme}
Assume that $00\,.\label{csi} \end{equation}
Also we will use the following two equations that arise from
taking the inner product of each term in (\ref{eq1}) with $2\dot u$,
and with $u$, respectively.
\begin{eqnarray}
&\frac d{dt}\|\dot u\|^2+2\gamma\|\dot u\|^2+ \frac d{dt}\int_0^{\|A^{1/2}u\|^2} m(r)\,dr
=2\langle f, \dot u\rangle\,, & \label{mut}\\
&\langle \ddot u,u\rangle + \gamma\langle \dot u,u\rangle +
m(\|A^{1/2}u\|^2)\|A^{1/2}u\|^2=\langle f,u\rangle\,. & \label{mu}
\end{eqnarray}
Now, we differentiate $F$ and build a first-order linear inequality that yields
the desired estimates.
\begin{eqnarray*}
F'(t)&=&\frac d{dt}\|\dot u\|^2+ \frac d{dt}\int_0^{\|A^{1/2}u\|^2} m(r)\,dr
+\delta \langle\ddot u,u\rangle+\delta \|\dot u\|^2 \\
&=&-(2\gamma-\delta)\|\dot u\|^2+2\langle f, \dot u\rangle
-\gamma\delta\langle \dot u, u\rangle -\delta m(.)\|A^{1/2}u\|^2
+\delta \langle f, u\rangle\,,
\end{eqnarray*}
where we have used (\ref{mut}) and (\ref{mu}). Now from
(\ref{l1}) and (\ref{csi}) we obtain
$2\langle f, \dot u\rangle\leq \frac2\gamma\|f\|^2+\frac\gamma2\|\dot u\|^2$
and $$
\delta \langle f, u\rangle\leq \frac{\delta}{\lambda_1}\|f\|\,\|A^{1/2}u\|
\leq \frac{\delta}{2m_0\lambda_1^2}\|f\|^2+\frac{\delta m_0}2\|A^{1/2}u\|^2\,.
$$
Using the two inequalities above, and the fact that $m_0\leq m(.)\leq M_0$,
we obtain
\begin{eqnarray*}
F'(t)&\leq&-\gamma\|\dot u\|^2-\frac{\delta m_0}{2M_0}\int_0^{\|A^{1/2}u\|^2}
m(r)\,dr- \delta \gamma\langle\dot u,u\rangle +
(\frac2\gamma +\frac\delta{2m_0\lambda_1^2})\|f\|^2 \\
&\leq& -c_1F(t)+(\frac2\gamma +\frac\delta{2m_0\lambda_1^2})\|f\|^2 \,,
\end{eqnarray*}
where $c_1=\min\{\gamma, \frac{\delta m_0}{2M_0}\}$. From this first-order differential
inequality, it follows that
$$
F(t)\leq e^{-c_1t}\left(F(0)+(\frac2\gamma +\frac\delta{2m_0\lambda_1^2})\int_0^t
e^{c_1s}\|f(.,s)\|^2\,ds\right)\,.
$$
From the assumption that $\|f(.,s)\|$ decays exponentially
follows the existence of positive constants
$c_2, c_3$, such that $F(t)\leq c_2e^{-c_3t}$, with $c_3\beta$,
and $p$ is a bounded and
continuous function. Existence of solutions is proven as in
Theorem~\ref{thmex}, with the assumption that $0\leq p(.)\leq P_0$.
Theorem~\ref{thmp} is proven under the assumption $0\leq p(.)\leq P_0$, in
which case, the energy functional (\ref{F}) is redefined to be
$$
E(t)=\|\dot u\|^2 + \int_0^{\|A^{\alpha/2}u\|^2} m(r)\,dr
+ \int_0^{\|A^{\beta/2}u\|^2} p(r)\,dr\,.
$$
Estimates for this functional require some algebraic manipulations,
but otherwise the proof is the same as before.
Exponential decay is proven under the conditions of Theorem~\ref{thme} and the
assumption that $-m_0\delta\lambda_1^{\alpha-\beta} \leq p(.)\leq P_0$.
Notice that $p$ is allowed to assume negative values.
When proving this statement, the energy functional (\ref{E}) remains
the same, but extra algebraic manipulations are required.
\subsection*{Systems of equations}
Let ${\bf u,f,g,h}$ be functions with values in ${\mathbb R}^k$, and
$m,\gamma$ be $k\times k$ diagonal matrices. Then rewrite (\ref{eq1}) as
\begin{eqnarray}
&\ddot {\bf u} +\gamma \dot {\bf u} +m(\|A^{1/2}{\bf u}\|^2)A {\bf u} =
{\bf f}(x,t)\,, & \label{eqs} \\
&{\bf u}(x,0)={\bf g}(x)\,, \quad \dot {\bf u}(x,0)={\bf h}(x)\,.& \nonumber
\end{eqnarray}
Now, components of vectors are denoted with
sub-indices; derivatives are computed component-wise,
$$
\dot {\bf u}= (\dot u_1,\dots ,\dot u_k)\,,\quad
A{\bf u}=(Au_1,\dots ,Au_1)\,;
$$
inner products and norms are redefined as
$$
\langle {\bf u}, {\bf v}\rangle= \langle u_1,v_1\rangle+\dots +
\langle u_k,v_k\rangle\,,\quad \|{\bf u}\|^2=\|u_1\|^2+\dots +\|u_k\|^2\,;
$$
functions are A-analytic if their components are A-analytic; and
eigenvectors of $A$ have the form $e_i\phi_j$, $i=1,\dots, k$, $j=1,2,\dots$,
where $e_i$'s are the standard basis for ${\mathbb R}^k$, and $\phi_j$'s are
the eigenvectors defined in \S 1.
Because most of the previous estimates hold with very little modification,
Theorems~\ref{thmex},~\ref{thme},~\ref{thmp}
are proven along similar lines to the previous proofs.
For example the energy functional (\ref{F}) is rewritten as
$$
F(t)=\|\dot{\bf u}\|^2+\sum_{i=1}^k \int_0^{\|A^{1/2}u_i\|^2}m_{ii}(r)\,dr
+\delta \langle\dot{\bf u},{\bf u}\rangle\,.
$$
Computation of estimates for this functional depends on the constants
$m_0=\min_i\inf_r m_{ii}(r)$, $\gamma_0=\min\gamma_{ii}$, and the inequality
$$
2\langle{\bf v},{\bf w}\rangle\leq \alpha \|{\bf v}\|^2+
\frac 1\alpha \|{\bf w}\|^2\quad \forall \alpha >0\,.
$$
Notice that small modifications of the system (\ref{eqs}) lead to a variety
of control problems for which decay of solutions is of great interest.
For example pre-multiply $\ddot{\bf u}$
by a diagonal matrix that has some entries equal to zero, and
substitute $m$ and $\gamma$ by positive-definite matrices
(instead of diagonal matrices).
\subsection*{Modified Carrier model}
For equations in which the powers of $A$ in the coefficient and in the argument
of $m$ are not in the ratio two to one, we introduce
\begin{equation} \label{eqm}
\ddot u +\gamma \dot u +m(\|A^{\alpha/2} u\|^2)A^\beta u = f(x,t)\,,
\end{equation}
where $\alpha$ and $\beta$ are non-negative integers.
Global solutions are obtained by the same method as the one used in
Theorem~\ref{thmex}.
Exponential decay is proven under the assumption that
$\|A^{\alpha/2-\beta}f\|$ decays exponentially as $t\to\infty$.
In proving this statement, we follow the proof of Theorem~\ref{thme}, with
the energy functional
$$
F(t)=\|A^{(\alpha-\beta)/2}\dot u\|^2+ \int_0^{\|A^{\alpha/2}u\|^2} m(r)\,dr
+\delta \langle A^{(\alpha-\beta)/2}\dot u, A^{(\alpha-\beta)/2} u\rangle\,.
$$
To estimate $F'(t)$, we use the following two equations that come from taking
the inner product of (\ref{eqm}) with $2A^{\alpha-\beta}\dot u$ and
$A^{\alpha-\beta} u$, respectively.
\begin{eqnarray*}
&\frac d{dt}\|A^{(\alpha-\beta)/2}\dot u\|^2
+2\gamma\|A^{(\alpha-\beta)/2}\dot u\|^2+ m(\|A^{\alpha/2}u\|)
\frac d{dt}\|A^{\alpha/2}u\|^2& \\
&=2\langle A^{(\alpha-\beta)/2}\dot u, A^{(\alpha-\beta)/2}f\rangle\,, & \\
&\langle A^{(\alpha-\beta)/2}\ddot u,A^{(\alpha-\beta)/2}u\rangle
+ \gamma\langle A^{(\alpha-\beta)/2}\dot u, A^{(\alpha-\beta)/2}u\rangle
+m(\|A^{\alpha/2}u\|^2)\|A^{\alpha/2}u\|^2 & \\
&=\langle A^{\alpha/2}u,A^{\alpha/2-\beta}f\rangle\,. &
\end{eqnarray*}
As in Theorem~\ref{thme}, we obtain positive constants $c_2,c_3$ such that
$$
\|A^{(\alpha-\beta)/2}\dot u\|^2 \leq c_2e^{-c_3t}\,, \quad
\|A^{\alpha/2}u\|\leq \frac{c_2}{m_o}e^{-c_3t}\,.
$$
As in Theorem~\ref{thmp}, under the assumption
$\int_0^\infty \|A^{(\alpha-\beta)/2}f\|^2<\infty$,
and using the energy functional
$$
E(t)=\|A^{(\alpha-\beta)/2}\dot u\|^2+ \int_0^{\|A^{\alpha/2}u\|^2} m(r)\,dr\,,
$$
we obtain $\int_0^\infty \|A^{(\alpha-\beta)/2}\dot u\|^2 < \infty$.
Notice that the larger the difference $\alpha-\beta$, the higher the
order of the decaying derivative. Also notice that the earlier example can
be used to show that in systems the decay is not necessarily
exponential. In fact, the same $u$ and $m$ satisfy (\ref{eqm}) with
$\alpha=0$ and $\beta=1$.
\paragraph{Acknowledgments} The author wants to express its gratitude to
the anonymous referee, to M.~L.~Oliveira, and to S.~Welsh for their suggestions;
also to Prof. H. `~Warchall who accepted this article for publication.
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\bigskip
{\sc Julio G. Dix}\\
Department of Mathematics\\
Southwest Texas State University\\
San Marcos, TX 78666 USA\\
E-mail address: jd01@swt.edu
\end{document}