New decay rates for Cauchy problem of Timoshenko thermoelastic systems
with past history: Cattaneo and Fourier law
Abstract
In this paper, we investigate the decay properties of the thermoelastic
Timoshenko system with past history in the whole space where the thermal
effects are given by Cattaneo and Fourier laws. We obtain that both
systems, Timoshenko-Fourier and Timoshenko-Cattaneo, have the same rate
of decay (1+t)^{-(1/4)} and the regularity-loss type property is
not present in some cases. Moreover, new stability number χ is
introduced, such new number controls the decay rate of the solution with
respect to the regularity of the initial data. To prove our results, we
use the energy method in Fourier space to build an appropriate Lyapunov
functionals that give the desired results.