Optimal rates of decay in 2-d thermoelasticity with second sound (Q2865499)

The authors study a thermoelastic plate model written as: \(\rho u_{tt}-\mu \Delta u_{tt}+\gamma \Delta ^{2}u+\alpha \Delta \theta =0\), \(c\theta _{t}+\kappa \text{div}q-\alpha \Delta u_{t}=0\), \(\tau q_{t}+\kappa _{0}q+\kappa \nabla \theta =0\), posed in a bounded domain \(\Omega \) of \( \mathbb{R}^{2}\). Here \(q\) is the heat flux which satisfies a Cattaneo's law (the last above one). The different constants are positive except \(\mu \) which may be equal to 0. The boundary conditions \(u=\Delta u=\theta =0\) are imposed on \( \partial \Omega \) and initial conditions are imposed on \(u\), \(u_{t}\), \( \theta \) and \(q\). The authors first prove a global in time solution of this evolution linear and coupled system. The proof of this existence result is essentially based on the construction of a semigroup framework for this problem. The main result of the paper proves that for \(\mu >0\) the associated semigroup of contractions is exponentially stable, while for \(\mu =0\) the energy associated to this system is of order \(\frac{1}{t^{2}}\). This is proved studying the properties of the semigroups of contractions associated to the above system.