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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 $0<m_0\leq m(.) \leq M_0$, and that $\|f(.,t)\|$ decays 
exponentially to zero as $t\to \infty$.
Then for every solution $u$ of (\ref{eqi}), $\|\dot u\|$ and $\|\nabla u\|$
decay exponentially to zero.
\end{theorem}

\paragraph{Proof} 
We shall find bounds for $\dot u$ and $A^{1/2} u$ 
by estimating  the  energy functional 
\begin{equation} \label{F}
F(t)=\|\dot u\|^2 + \int_0^{\|A^{1/2}u\|^2} m(r)\,dr +\delta \langle\dot u,u\rangle\,,
\end{equation}
where $\delta=\min\{2\lambda_1\sqrt{m_0}, \gamma/2\}$.
This choice of $\delta$ ensures that $F$ is non-negative. In fact,
$$
F(t)\geq \|\dot u\|^2 + m_0\|A^{1/2}u\|^2-\|\dot u\|^2 -\frac{\delta^2}4\|u\|^2 
   \geq (m_0-\frac{\delta^2}{4\lambda_1^2})\|A^{1/2}u\|^2\geq 0\,.
$$
Here and in several expressions to follow, we use inequalities of the form
\begin{equation}
2|\langle v,w\rangle|\leq 2\|v\|\,\|w\|\leq \alpha \|v\|^2+\frac1\alpha\|w\|^2
\quad \forall \alpha>0\,.\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 $c_3<c_1$. 
Therefore,
$$
\|\dot u\|^2\leq  c_2e^{-c_3t}\,,\quad 
\|A^{1/2} u\|^2\leq  \frac{c_2}{m_0}e^{-c_3t}\,, \quad\forall t\geq 0\,,
$$
which concludes this proof \hfill$\diamondsuit$\medskip

\paragraph{Remark} The order of exponential decay approaches zero as the 
lower bound $m_0$ approaches zero. This is so because the constant $c_3$ in 
Theorem~\ref{thme}  satisfies 
$$c_3<c_1\leq \frac{\lambda_1}{2M_0}(m_0)^{3/2}$$
the right side of which approaches zero as $m_0$ approaches $0$.


\section{Decay in the degenerate case}

We start with a positive statement about exponential decay.
\begin{theorem} \label{thmd}
Assume that $m\equiv 0$ and that $\|f(.,t)\|$ decays exponentially to zero 
as $t\to \infty$. Then for any solution $u$ of (\ref{eqi}), 
$\|\dot u\|$ decays exponentially to zero.
\end{theorem}

\paragraph{Proof} Since $m\equiv 0$, Equation (\ref{eqi}) reduces to
$\ddot u+\gamma\dot u=f$. By computing the inner product of $2\dot u$ with
each term in this equation, we have
\begin{eqnarray*}
&\frac d{dt}\|\dot u\|^2+2\gamma\|\dot u\|=2\langle f, \dot u\rangle
\leq \frac 1\gamma\|f\|^2+\gamma\|\dot u\|^2\,, & \\
&\frac d{dt}\|\dot u\|^2+\gamma\|\dot u\| \leq \frac 1\gamma\|f\|^2\,.&
\end{eqnarray*}
This first-order differential inequality and the initial conditions yield the
inequality
$$
\|\dot u\|^2\leq e^{-\gamma t}\bigl(\|h\|^2+\frac1\gamma \int_0^t
e^{\gamma s}\|f(.,s)\|^2\,ds\bigr)\,.
$$
From the assumption that $\|f(.,s)\|$ decays exponentially 
there exist positive constants $c_2, c_3$, such that
$\|\dot u\|^2\leq c_2e^{-c_3t}$, with $c_3<\gamma$.
Which concludes this proof. \hfill$\diamondsuit$\medskip

The following example shows that decay of solutions is not necessarily
exponential. 

\paragraph{Example} Consider the initial value problem
 \begin{eqnarray*}
&\ddot u +\dot u -m(\|u_x\|^2)\ u_{xx} = 0
\,, \quad\mbox{ for }0\leq x\leq 2\pi,\ t\geq 1+\sqrt 2& \\
&u(x,1+\sqrt 2)=\frac1{\sqrt\pi} e^{1/(1+\sqrt 2)}\sin x\,, \quad 
\dot u(x,1+\sqrt 2)=\frac1{9\sqrt\pi} e^{1/(1+\sqrt 2)}\sin x&\\ 
&u(0,t)=0\,, \quad u(2\pi,t)=0\,,\quad \mbox{for } t\geq 1+\sqrt 2\,, &
\end{eqnarray*} 
where $m$ is the non-negative and continuous function defined as
$$
m(r)=\left\{\begin{array}{ll} 
 \frac1{16}(\ln r)^2(4-4\ln r - (\ln r)^2)& 
 \mbox{if }1\leq r\leq e^{2/(1+\sqrt 2)}\,, \\[5pt]
 0 & \mbox{Otherwise}\,. \end{array} \right.
$$
Then $ u(x,t)=\frac1{\sqrt\pi} e^{1/t}\sin x$ is a solution. Since  
\begin{eqnarray*}
&\dot u = -\frac 1{t^2} u\,, \quad \ddot u =(\frac 1{t^4} + \frac 2{t^3})u\,,&
 u_x=\frac1{\sqrt\pi}e^{1/t}\cos x\,,\quad u_{xx}=-u\,,
\end{eqnarray*}
$\|u_x\|^2=e^{2/t}$, and $m(e^{2/t})=\frac 1{t^2}-2\frac 1{t^3}-\frac 1{t^4}$ 
for $ t\geq 1+\sqrt 2$, it follows that
$u$ satisfies the initial-value problem.
Notice that $\|\dot u\|$ decays polynomially rather than exponentially as
$t\to \infty$.
In fact, 
$$
\|\dot u\|^2=\frac1{t^4}e^{2/t}=O(t^{-4})\,.
$$ 
\smallskip

For non-constant $m$, the convergence of $\|\dot u\|$ to zero remains illusive:
we are unable to prove it, and unable to give a counter-example. So far, our 
best result is: 

\begin{theorem} \label{thmp}
If $\|f(.,t)\|$ is square integrable on $[0,\infty)$ and $u$ is a solution
to (\ref{eqi}), then  $\|\dot u\|$ is square integrable on $[0,\infty)$. 
\end{theorem}
\paragraph{Proof} The desired integral is obtained by estimating the growth 
of the energy functional
\begin{equation} \label{E}
E(t)=\|\dot u\|^2 + \int_0^{\|A^{1/2}u\|^2} m(r)\,dr \,.
\end{equation}
Using (\ref{csi}) and (\ref{mut}), it follows that the derivative of $E$ satisfies
$$
E'(t)=-2\gamma\|\dot u\|^2+2\langle f,\dot u\rangle \leq 
-\gamma\|\dot u\|^2+ \frac 1\gamma\|f\|^2\,.
$$
From this inequality and the Fundamental Theorem of Calculus, we obtain
$$
E(t)+\gamma\int_0^t\|\dot u\|^2\,ds \leq E(0)+ \frac 1\gamma 
\int_0^t\|f(.,s)\|^2\,ds\,.
$$
Since by hypothesis $\int_0^\infty \|f\|^2<\infty$, it follows that
$\int_0^\infty \|\dot u\|^2<\infty$, and 
the proof is complete.  \hfill$\diamondsuit$


\paragraph{Remark}
From the physics point of view, Theorems~\ref{thme} and \ref{thmd} state that
the energy $\|\dot u\|^2 + \|\nabla u\|^2$ decays as time goes by.
In terms of non-linear dynamics (see \cite{Te}), these two theorems indicate
that in the  space of A-analytic functions, every ball of center zero and finite radius 
is an absorbent set (under the norm $\|\dot u\|^2 + \|\nabla u\|^2$).
This means that given a ball of center zero,
the orbit of every bounded set enters and stays in this ball after a certain 
time.

\section{Extension of results}

\subsection*{Higher order derivatives} 
To extend the previous analysis to equations that involve powers of
$A$, for example $\Delta^2$ which appears in modeling  non-planar vibrations,
we introduce the equation
\begin{equation} \label{eqg} 
\ddot u +\gamma \dot u +m(\|A^{\alpha/2} u\|^2)A^\alpha u
+ p(\|A^{\beta/2} u\|^2)A^\beta u = f(x,t)\,,
\end{equation}
where $\alpha$ and $\beta$ are non-negative integers, $\alpha>\beta$,
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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\end{thebibliography}
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{\sc Julio G. Dix}\\
Department of Mathematics\\
Southwest Texas State University\\
San Marcos, TX 78666 USA\\
E-mail address: jd01@swt.edu

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