Nonlinear thermoelastic plate equations : global existence and decay rates for the Cauchy problem

Abstract

We consider the Cauchy problem in Rn for some fully nonlinear thermoelastic Kirchhoff type plate equations where heat conduction is modeled by either the Cattaneo law or by the Fourier law. Additionally, we take into account possible inertial effects. Considering nonlinearities which are of fourth-order in the space variable, we deal with a fully nonlinear system which triggers difficulties typical for nonlinear Schrödinger equations. The different models considered are systems of mixed type comparable to Schrödinger-parabolic or Schrödinger-hyperbolic systems. The main task consists in proving sophisticated a priori estimates with the achievement of obtaining the global existence of solutions for small data, neither known nor expected for the Cauchy problem in pure plate theory nor available before for the coupled system under investigation, where only special cases (bounded domains with analytic semigroup setting, or the Cauchy problem with semilinear nonlinearities) had been treated before.