Dissipative structures for thermoelastic plate equations in $\mathbb R^n$

Abstract

We consider the Cauchy problem in $\mathbb R^n$ for linear thermoelastic plate equations where heat conduction is modeled by either the Cattaneo law or by the Fourier law -- described by the relaxation parameter $\tau$, where $\tau>0$ corresponds to Cattaneo's law and $ \tau=0 $ corresponds to Fourier's law. Additionally, we take into account possible inertial effects characterized by a parameter $\mu\geq 0$, where $\mu=0$ corresponds to the situation without inertial terms. For the Catteneo system without inertial term, being a coupling of a Schrödinger type equation (the elastic plate equation) with a hyperbolic system for the temperature and the heat flux, we shall show that a regularity-loss phenomenon appears in the asymptotic behavior as time tends to infinity, while this is not given in the standard model where the Cattaneo law is replaced by the standard Fourier law. This kind of effect of changing the qualitative behavior when moving from Fourier to Cattaneo reflects the effect known for bounded domains, where the system with Fourier law is exponentially stable while this property is lost when going to the Cattaneo law. In particular, we shall describe in detail the singular limit as $\tau\to 0$. For the system with inertial term we demonstrate that it is of standard type, not of regularity loss type. The corresponding limit of a vanishing inertial term is also described. All constants appearing in the main results are given explicitly, allowing for quantitative estimates. The optimality of the estimates is also proved.