\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 132, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/132\hfil Loss of exponential stability]
{Loss of exponential stability for a thermoelastic system with memory}

\author[B. F. Alves,  W. D. Bastos, C. A. Raposo \hfil EJDE-2010/132\hfilneg]
{Bruno Ferreira Alves, Waldemar Donizete Bastos,
Carlos Alberto Raposo}  % in alphabetical order

\address{Bruno Ferreira Alves \newline
Departament of Mathematics, Federal University  of S. J. del-Rey\\
Pra\c{c}a Frei Orlando, 170,
36300-000, S\~ao Jo\~ao del Rei, MG, Brazil}
\email{bruno.fa@ibest.com.br}

\address{Waldemar Donizete Bastos \newline
Departament of Mathematics, S\~ao Paulo State University, UNESP \newline
Rua Crist\'ov\~ao Colombo, 2265,
15054-000, S\~ao Jos\'e do Rio Preto, SP, Brazil}
\email{waldemar@ibilce.unesp.br}

\address{Carlos Alberto Raposo \newline
 Departament of Mathematics,
Federal University  of S. J. del-Rey\\
Pra\c{c}a Frei Orlando, 170, 36300-000,
S\~ao Jo\~ao del Rei, MG, Brazil}
\email{raposo@ufsj.edu.br}

\thanks{Submitted February 23, 2010. Published September 14, 2010.}
\thanks{C. A. Raposo was supported by grants
573523/2008-8 - INCTMat from CNPq, and
\hfill\break\indent  620025/2006-9 from CNPq}
\subjclass[2000]{35B40, 35Q80, 35L05, 47D06}
\keywords{Heat conduction with memory;
$C_0$-semigroup; decay of solutions; \hfill\break\indent
thermoelastic system}

\begin{abstract}
 In this article we study a thermoelastic system considering the
 linearized model proposed by Gurtin and Pipkin \cite{GP}
 instead of the Fourier's law for the heat flux. We use theory of
 semigroups \cite{Pazy, ZL} combining Pruss' Theorem \cite{JamP} and
 the idea developed in \cite{GR} to show that the system is not
 exponentially stable.
\end{abstract}

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

\section{Introduction}

We  study a partial differential equation
that models an elastic string:
\begin{gather}
u_{tt} - u_{xx} + \theta_{xx} = 0 \quad  \text{in }
(0,L)\times(0,\infty), \label{1.1}\\
\theta_{t} - g*\theta_{xx} + c\,g*\theta - u_{xxt}= 0\quad
\text{in } (0,L)\times(0,\infty), \label{1.2}
\end{gather}
with initial data
$$
u(x,0)=u_0(x),\quad u_{t}(x,0)=u_1(x),\quad \theta(x,0)=\theta_0(x)\,.
$$
The function $u=u(x,t)$ is the small transversal vibration of the
elastic string of reference configuration of length $L$, and
$\theta=\theta(x,t)$ is the temperature difference from the material
and natural ambient. To fix ideas we assume that the string
is held fixed at both ends, $x = 0$ and $x = L$.
 We impose the boundary conditions
\begin{gather*}
u(0, t) = u(L, t) = 0, \\
\theta(0, t) = \theta(L, t) = 0.
\end{gather*}
In this model, $c$ is a positive constant, and
 $g: \mathbb{R}^{+}\to \mathbb{R}^{+}$ is the relaxation function.
We assume that $g$ is differentiable and
satisfies $g(0)>0$, $g'(t) <0$ and
$$
1 - \int_0^{\infty}g(s) ds = \ell > 0.
$$
We introduce the  convolution product
$$
(g*u)(t) :=\int_0^{t}g(t-\tau)u(\cdot,\tau) d\tau\, .
$$

Now we observe that when $c=0$ the thermoelastic system has
exponential decay, as can be seen in \cite{Luci}, when we replace
$g*u$ by $\theta$ in \eqref{1.2} we also have exponential decay, see
\cite{FBR}. The similar situation is valid for thermoelastic plate,
see \cite{GR} and \cite{VP}.

The article is organized as follows, in the Section $2$ we introduce
the notation and the functional spaces, in the Section $3$ we obtain
the semigroup of solutions and finally, in the Section  $4$ we prove
the loss of exponential stability for the thermoelastic system with
memory.

\section{Functional setting and notation}

 We use the standard Lebesgue spaces  and Sobolev spaces  with their
usual proprieties as in \cite{Adams}.
Consider the positive operators A and B on $L^2(0,L)$ defined by
$A=-(\cdot)_{xx}$  and $B = cI-(\,\cdot\,)_{xx}$, with domains
$D(A) = D(B) = (H^2\cap H^1_0 )(0,L)$. Now, for $r \in \mathbb{R}$,
we introduce the scale of Hilbert spaces $H_{r}=D(A^{r/2})$ with the
usual inner products $ \langle v_1, v_2\rangle_{H_r} = \langle
A^{r/2} v_1,A^{r/2}v_2\rangle$ and we have $H_{r_1}\hookrightarrow
H_{r_2}$ are compact whenever $r_1 > r_2$. Concerning the memory
kernel $g$, we make the substitution $\mu(s)=-g(s)$ and we require
\begin{equation} \label{2.1}
\mu \in  C^{1}(\mathbb{R}^{+}) \cap L^{1}(\mathbb{R}^{+}),\quad
\mu(s) > 0,\quad  \mu' (s) \leq 0,\quad
 g(0) =\int_0^{\infty} \mu(s) ds > 0.
\end{equation}
Calling  $\sigma_{\infty}
=\sup  \{s :\mu (s)  > 0  \}  $, we infer that, dual to \eqref{2.1},
 for each $\sigma >0$, there exists a set
$\mathcal{O}_{\sigma} \subset ( \sigma,\sigma_{\infty}) $
of positive Lebesgue measure such that $ \mu' (s)< 0$,
in $\mathcal{O}_{\sigma}$. Now for $r \in \mathbb{R}$
consider the weighted Hilbert spaces:
$$
\mathcal{M }_{r} = L^{2}_{\mu}(\mathbb{R}^{+};H_{r})
$$
with the inner product
 \begin{equation} \label{IP}
\langle \nu , \eta\rangle_{\mathcal{M}_r}
= \int_0^{\infty}\mu(s)\langle
B^{r/2}\nu (s),B^{r/2}\eta(s)\rangle\,ds
\end{equation}
and we introduce as in \cite{GVP} the linear operator $T$ on
$ \mathcal{M}_1$ defined by $ T\eta = -\eta_{s}$ with domain
$$
D( T) = \{
\eta \in \mathcal{M }_1 : \eta _{s}\in \mathcal{M }_1, \,
\eta(0)=0\},
$$
where $\eta_s$ is the distributional derivative of
$\eta$ with respect to the internal variable $s$, and then the
operator $T$ is the infinitesimal generator of a $C_0$-semigroup of
contractions. In particular, there holds
\begin{equation} \label{2.2}
\langle T\eta, \eta\rangle_{\mathcal{M }_1} =
\int_0^{\infty}\mu'(s)\| B^{1/2} \eta (s)\|\,ds \leq 0,
\quad\text{for  all }  \eta \in D(T).
\end{equation}
Finally, we define with the usual inner products, the following
Hilbert spaces:
$$
\mathcal{H}_r = H_{r+2} \times H_r \times H_r \times M_{r+1},\quad
r \in \mathbb{R}.
$$


\section{The semigroup of solutions}

To describe properly the solutions of the system
\eqref{1.1}-\eqref{1.2} by means of a $C_0$-semigroup of linear
operators acting on the phase-space $\mathcal{H}_0$, we will follow
the ideas of \cite{Adams}. In this direction  we introduce  an additional
variable, namely, the summed past history of $\theta$  defined as
$$
\eta^{t}(s) = \int_0^{s} \theta(t-y)dy, \quad \text{with }
t,s \geq 0.
$$
Observe that we have formally
$ (\frac{d}{dt} + \frac{d}{ds})(\eta^t(s)) =
\theta$ in $ (0,L)$ subject to the boundary and initial conditions
$\eta^{t}(0) = 0$ in $(0,L)$, $t\geq 0 $,
$$
\eta^{0}(s) = \int_0^{s}\theta(-y)dy,\quad  s \geq 0.
$$
For the rest of this article, we consider the
vectors $U(t) = ( u(t), v(t), \theta(t), \eta^{t} )^{T}$ and
$U(0) = ( u_0, v_0, \theta_0, \eta_0 )^{T} \in \mathcal{H}_0$.
We obtain the linear evolution equation, in $ \mathcal{H}_0$,
\begin{gather}
U_{t} - L\,U = 0   \label{3.1}\\
U(0) = U_0 \label{3.2}
\end{gather}
where the linear operator $L$ is defined as
\[
L\,U =  \begin{pmatrix}
v\\
u_{xx} - \theta_{xx}\\
u_{xx} - \int_0^{\infty}g(s)[ c\theta(t-s) -
\theta_{xx}(t-s)]ds\\
\eta
\end{pmatrix}.
\]
with domain
$D(L)= \{(u, v, \theta, \eta )^{T}  \in \mathcal{H}_0 \}$
such that
$v   \in H_{2}$, $u_{xx} - \theta_{xx} \in H_0$,
$$
u_{xx} - \int_0^{\infty}g(s)[ c\theta(t-s) -
\theta_{xx}(t-s)]ds \in H_0, \quad  \eta \in D(T).
$$

\begin{theorem} \label{thm3.1}
System \eqref{3.1} defines a $C_0$-semigroup of contractions $S(t) =
e^{tL }$ on the phase-space $\mathcal{H}_0$.
\end{theorem}

The  proof is done by using the Lumer - Phillips
theorem \cite[Theorem 4.3]{Pazy}.

\section{Loss of exponential stability}

To prove the loss of exponential  stability we
 use the following result.

\begin{theorem} \label{thm4.1}
Let $S(t)= e^{tL}$ be a $C_0$-semigroup of contractions in a
Hilbert space. Then $S(t)$ is  exponentially stable if and only if,
\begin{equation}
i\mathbb{R}= \{ i\beta : \beta \in \mathbb{R}\}
\subset \rho(L) \label{g1}
\end{equation}
and
\begin{equation}
\|(\lambda I - L )^{-1}\| \leq C,  \quad \text{for  every }
\lambda \in i \mathbb{R}. \label{g2}
\end{equation}
 \end{theorem}

The proof of the above theorem can be found in \cite{JamP}
and  in \cite{ZL}.

We note that \eqref{3.1}-\eqref{3.2} is
 dissipative, because \eqref{2.2} implies
\begin{equation}
\langle LU, U\rangle_{\mathcal{H }_0} = \langle T\eta,
\eta\rangle_{\mathcal{M }_1}  \leq 0, \quad \text{for  all }
U \in D(L),
\end{equation}
and it is standard matter to show that $(I - L)$ maps $D(L)$ onto
$\mathcal{H}_0$, see \cite{FBR}, where a similar case is treated.

Then, using  $\langle Tu,u\rangle  <0 $ for all nonzero $u$ in
$D(T)$, one can show that
the solution of thermoelastic system \eqref{1.1}-\eqref{1.2} decays  to
zero as time approaches $\infty$.

Now we are in position of to show our main result.

\begin{theorem} \label{thm4.2}
The semigroup $S(t)= e^{tL}$ on $\mathcal{H}_0 $ defined by
\eqref{3.1}-\eqref{3.2} is not exponentially stable.
\end{theorem}

\begin{proof}
For $i\lambda \in \rho(L)$ and $ V = (0,0,0, \eta)^{T} \in
\mathcal{H}_0$, consider the complex equation
\begin{equation} \label{w}
 (i\lambda\,I - L)U = V
\end{equation}
 that when written explicitly reads
\begin{gather}
i\lambda u - v = 0 \label{4.4}  \\
i\lambda v - u_{xx} + \theta_{xx} = 0 \label{4.5}
\end{gather}
Consider an orthonormal basis  $\{w_{j}\}_{n  \in \mathbb{N}} $
of eigenvectors of the operator $A$ and the respective
eigenvalues $\{ \alpha_{n}\}_{n \in \mathbb{N}}$.
We recall that $\alpha_n \to \infty$ as $n \to \infty$.  We set
$$
\eta_{n}(s) = \frac{w_{n}}{\sqrt{ c + \alpha_{n}}}
$$
and
$$
V_{n} = (0,0,0,\eta_{n})^{T}.
$$
Notice that, using \eqref{2.1} and \eqref{IP} we have
\[
\| V_{n}\|_{\mathcal{H}_0}
= \|\eta_{n}\|_{\mathcal{M}_1}
= \frac{1}{(c + \alpha_{n})}
\]
\begin{align*}
\int_0^{\infty} \mu(s) \|B^{1/2} w_n(s) \|^{2} ds
&= \frac{1}{(c + \alpha_{n})} \int_0^{\infty} \mu(s) ( c +
\alpha_n)\| w_n(s) \|^{2} ds
\\
&= \int_0^{\infty} \mu(s)ds = g(0).
\end{align*}
Now  we  build a sequence of $\lambda_{n}$ such that
the corresponding solution $ U_{n}$   of
\begin{equation} \label{w2}
(i\lambda_nI - L)U_n = V_n
\end{equation}
satisfies
  $\| U_{n}  \|_{\mathcal{H}_0} \to \infty $
as $ n \to \infty$. In this direction we look for a solution
$U_{n}=(w_{n},w_{n},s_nw_{n},w_{n} )$ where $ s_n \in \mathbb{C}$.
Then, from \eqref{4.4} and \eqref{4.5}  we have
\begin{equation}
-\lambda_{n}^{2} -\alpha_{n} + s_{n}\alpha_{n} = 0 \label{4.8}
\end{equation}
that implies
$$
s_{n} = 1 + \frac{\lambda_{n}^{2}}{\alpha_n}.
$$
Choosing $\lambda_n = |\alpha_n|$ we finally have
$$
\|U_n\|_{\mathcal{H}_0} \geq \|s_n\,w_n\|_{H_0}= |s_n| \geq
\frac{\lambda_n^{2}}{|\alpha_n|} = |\alpha_n| \to
\infty\quad \text{as } n \to \infty.
$$
which yields the conclusion.
\end{proof}

\begin{thebibliography}{00}

\bibitem{Adams} R. A. Adams;
\emph{Sobolev Spaces}, Academic Press,
New York, (1975).

\bibitem{CMD} C. M. Dafermos;
\emph{Asymptotic stability in viscoelasticity}, Arch.
Ration. Mech. Anal.  37 (1970), 297–308.

\bibitem{FBR}  M. Fabrizio, B. Lazzari, J. E. Mu\~noz Rivera;
\emph{Asymptotic Behavior in Linear Thermoelasticity},
  Journal of Mathematical Analysis and Applications.
  232 (1999) 138-165.

\bibitem{Luci} L. H. Fatore, J. E. Mu\~noz Rivera;
\emph{Energy decay for hyperbolic thermoelastic systems of memory type}.
Quartely Of Applied Mathematics. 59(3) (2001) 441-458.

\bibitem{GR} M. Grasselli, J. E. Mu\~noz Rivera, V. Pata;
\emph{ On the energy decay of
the linear thermoelastic plate with memory}.  J. Math. Anal.
Appl. 309 (2005) 1–14.

\bibitem{GVP} M. Grasselli, V. Pata;
\emph{Uniform attractors of nonautonomous systems
with memory}, in: A. Lorenzi, B. Ruf (Eds.), Evolution Equations,
Semigroups and Functional Analysis, in: Progr. Nonlinear
Differential Equations Appl., vol. 50, Birkhäuser, Boston, (2002).

\bibitem{VP} C. Giorgi, V. Pata;
 \emph{ Stability of linear thermoelastic systems with
memory}.  Math. Models Methods Appl. Sci. 11 (2001) 627–644.

\bibitem {GP} M. E. Gurtin, A. C. Pipkin;
\emph{A general theory of heat conduction with finite wave speeds}.
Arch. Rational Mech. Anal. 31 (1968) 113-126.

\bibitem{Pazy} A. Pazy;
\emph{Semigroups of Linear Operators and Applications to Partial
Differential Equations}, Springer- Verlag, New York, (1983).

\bibitem{JamP}  J. Prüss;
\emph{On the spectrum of C0-semigroups}
 Trans. Amer. Math. Soc.  284 (1984) 847–857.

\bibitem{ZL} Z. Liu, S. Zheng;
\emph{Semigroups Associated with Dissipative Systems}.
Chapman \& Hall/CRC Press, Boca Raton, FL, (1999).

\end{thebibliography}

\end{document}