Nonlinear thermoelastic plate equations – Global existence and decay rates for the Cauchy problem

Abstract

Abstract. We consider the Cauchy problem in R n 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.

Long-time behavior of quasilinear thermoelastic Kirchhoff–Love plates with second sound

“…Turning to the hyperbolic-hyperbolic case (τ > 0, γ > 0), linear well-posedness and exponential stability in bounded domains have been established and singular limits τ → 0, γ → 0 have been studied [36]. Similar investigations of the linear system in the full space were performed as well [46] and subsequently generalized to the nonlinear case [47].…”

“…Last but not least, a further contribution of this paper is a physical derivation of the thermoelastic plate model (1.1)- (1.5 [46,47,36], etc.) have implicitly 'conjectured' the physical model.…”

“…Therefore, one cannot obtain a superlinear bound for E 1 . Nevertheless, if seeking for a weaker result with a smallness condition on X(0) instead (for instance, as in [26] or [47]), the nonlinearity K(z) = z −z 2 would be admissible. This should not be surprising as our Assumption 2.5 is weakened, accordingly.…”

The Cauchy Problem for Thermoelastic Plates with Two Temperatures

“…The non-exponential stability for τ > 0 was proved in in [20]. We remark that related nonlinear problems have been discussed in [10,14,19,32]. Our main new contributions are • First discussion of the fourth-order thermoelastic plate system with two temperatures in all of R n .…”

Stability of abstract thermoelastic systems with inertial terms

“…We remark that nonlinear versions of these equations have been studied recently, for bounded domains see [21] with τ = 0 and µ = 0, and [16] for µ > 0. The Cauchy problem was investigated in [32].…”