Spectral analysis and stability of the Moore-Gibson-Thompson-Fourier model

We consider the linear evolution system $$ \begin{cases} u_{ttt}+\alpha u_{tt} + \beta \Delta^2 u_t + \gamma \Delta^2 u =- \eta \Delta \theta \\ \noalign{\vskip1mm} \theta_t - \kappa \Delta \theta = \eta \Delta u_{tt} + \alpha\eta \Delta u_t \end{cases} $$ describing the dynamics of a thermoviscoelastic plate of MGT type with Fourier heat conduction. The focus is the analysis of the energy transfer between the two equations, particularly when the first one stands in the supercritical regime, and exhibits an antidissipative character. The principal actor becomes then the coupling constant $\eta$, ruling the competition between the Fourier damping and the MGT antidamping. Indeed, we will show that a sufficiently large $\eta$ is always able to stabilize the system exponentially fast. One of the features of this model is the presence of the bilaplacian in the first equation. With respect to the analogous model with the Laplacian, this introduces some differences in the mathematical approach. From the one side, the energy estimate method does not seem to apply in a direct way, from the other side, there is a gain of regularity allowing to rely on analytic semigroup techniques.