Regularity analysis for an abstract thermoelastic system with inertial term

In this paper, we provide a complete regularity analysis for the following abstract thermoelastic system with inertial term ρutt+lAγu tt+σAu-mAαθ =0, cθt+mAαu t+kAβθ =0, u(0)=u0,ut(0)=v0, θ(0)=θ0, where A is a self-adjoint, positive definite operator on a complex Hilbert space H and (α,β,γ) E=[0,β+12]×[0,1]×[0,1]. It is regarded as the second part of Fernández Sare et al. [J. Diff. Eqs. 267 (2019) 7085-7134]. where the asymptotic stability of this model was investigated. We are able to decompose the region E into three parts where the associated semigroups are analytic, of Gevrey classes of specific order, and non-smoothing, respectively. Moreover, by a detailed spectral analysis, we will show that the orders of Gevrey class are sharp, under proper conditions. We also show that the orders of polynomial stability obtained in Fernández Sare et al. [J. Diff. Eqs. 267 (2019) 7085-7134] are optimal.