On the analyticity of the abstract MGT-Fourier system

Abstract

We consider the abstract MGT-Fourier system $${\left\{ \begin{array}{ll} u_{ttt}+\alpha u_{tt}+\beta A^\varrho u_t+\gamma A^\varrho u = \eta A \theta \\ \theta _t+\kappa A \theta = -\eta A u_{tt} - \eta \alpha A u_t \end{array}\right. }$$ u ttt + α u tt + β A ϱ u t + γ A ϱ u = η A θ θ t + κ A θ = - η A u tt - η α A u t in the subcritical regime $$\beta >\gamma /\alpha$$ β > γ / α , where the operator A is strictly positive selfadjoint. For any fixed $$\varrho \in [1,2]$$ ϱ ∈ [ 1 , 2 ] , we show that the associated solution semigroup S(t) is analytic and exponentially stable as well.

Uniform attractors for nonautonomous MGT‐Fourier system

“…The same equation also arises as a model for the temperature evolution in a type III heat conduction with a relaxation parameter (see Quintanilla [8]). The MGT equation is coupled with the classical Fourier heat equation by means of the coupling constant $$$ \eta $$$ describes the vibrations of viscoelastic heat conductor obeying the Fourier thermal law which forms the MGT‐Fourier system (1.1)–(1.2) by Dell'Oro and Pata [9]. In [10], the authors proved exponential stability whenever $$$ \left(\eta \right) $$$ in the supercritical case by using the semigroup method.…”