Decay properties for the Cauchy problem of the linear JMGT-viscoelastic plate with heat conduction

Abstract

We consider the Cauchy problem related to the JMGT-viscoelastic plate coupled with a heat equation with two kinds of thermal laws, which are thermoelasticity of type III and the Gurtin-Pipkin thermal law, respectively. We prove optimal results on decay rates for both the thermoelasticity type III system and the Gurtin-Pipkin thermal law system. More precisely, for the type III system, we show that the decay property is not of regularity-loss type in both the subcritical and critical cases. The result matches with the system in a bounded domain, where the system is known to be exponentially stable in the subcritical case. For the Gurtin-Pipkin thermal law system, there is a regularity-loss phenomenon in the critical case. We also study the asymptotic expansion of the eigenvalues to prove the optimality of the obtained decay rates for both models.