| Contents |
| We study the long time behavior of solutions to the linear thermoelastic plate in abstract setting:
\[
\left \{
\begin{array}{cccl}
u''(t) + A^2u(t) -A \theta(t) & = & 0, & \quad t \in (0, \infty),\\
\theta'(t) - A \theta(t) + A u'(t) & = & 0, & \quad t \in (0, \infty),
\end{array} \right.
\]
with the initial condition
\[
u(0) = u_0, \quad u'(0) = v_0, \quad \theta(0) =\theta_0.
\]
Here $A : \mathcal{D}(A) \subset H \rightarrow H$ is a non-negative self-adjoint operator in
$(H, \|\cdot\|)$ with a dense domain $\mathcal{D}(A)$. The total energy of $(u(t),\theta(t))$
is given by
\[E(t) = \|u'(t)\|^2 + \|Au(t)\|^2 + \|\theta(t)\|^2,\quad t\in(0,\infty). \]
We show that the behavior of $E(t)$ is determined by the abstract diffusion semigroup $e^{-tA}$ as $t\rightarrow\infty.$
This is yet another example of the diffusion phenomenon in dissipative wave equations. |
|