Global Non-Existence of a Coupled Parabolic–Hyperbolic System of Thermoelastic Type with History

Abstract

We consider two abstract systems of parabolic–hyperbolic type that model thermoelastic problems. We study the influence of the physical constants and the initial data on the nonexistence of global solutions that, in our framework, are produced by the blow-up in finite time of the norm of the solution in the phase space. We employ a differential inequality to find sufficient conditions that produce the blow-up. To that end, we construct a set that is positive-invariant for any positive value of the initial energy. As a result, we found that the coupling with the parabolic equation stabilizes the system, as well as the damping term in the hyperbolic equation. Moreover, for any pair of positive values (ξ,ϵ), there exist initial data, such that the corresponding solution with initial energy ξ blows up at a finite time less than ϵ. Our purpose is to improve results previously published in the literature.