A note on the equations of a thermoelastic plate
We consider the equations \[ u_{tt} = - \Delta^2 u - \alpha \Delta \theta, \quad \theta_t = \beta \Delta \theta - \gamma \theta + \alpha \Delta u_t \] on a bounded domain \(\Omega \subset \mathbb{R}^2\) with a smooth boundary and subject to Dirichlet conditions \(u = {\partial u \over \partial n} = \theta = 0\), on \(\partial \Omega\). We prove that the equations are associated with an analytic semigroup. In particular, this implies exponential decay and further shows that the rate of decay is determined by the least stable eigenvalue. It also gives regularity properties of solutions which should be useful for studying nonlinear problems. We shall consider only the case of Dirichlet conditions, but the method should extend to other cases without major difficulty. One remarkable aspect of our result is that the analytic semigroup property is clearly introduced by the lower dimensional approximations which are involved in modeling plates and beams. The equations of thermoelasticity themselves do not generate an analytic semigroup.
scientific article; zbMATH DE number 775961