Cauchy problem for thermoelastic plate equations with different damping mechanisms

In this paper we study the Cauchy problem for thermoelastic plate equations with friction or structural damping in $\mathbb{R}^n , n \geq 1$, where the heat conduction is modeled by Fourier’s law. We explain some qualitative properties of solutions influenced by different damping mechanisms. We show which damping in the model has a dominant influence on smoothing effect, energy estimates, $L^p - L^q$ estimates not necessary on the conjugate line, and on diffusion phenomena. Moreover, we derive asymptotic profiles of solutions in a framework of weighted $L^1$ data. In particular, sharp decay estimates for lower bounds and upper bounds of solutions in the $\dot{H}^s$ norm $(s \geq 0)$ are shown.

Keywords

thermoelastic plate equations, Fourier’s law, friction, structural damping, diffusion phenomena, asymptotic profiles

2010 Mathematics Subject Classification

35B40, 35Q99, 74F05

Full Text (PDF format)

Received 17 January 2019

Accepted 19 October 2019

Published 20 June 2022