On the regularized Moore-Gibson-Thompson equation

We study the regularized MGT equation $$u_ttt + \alpha u_tt + \beta A u_t + \gamma Au + \delta Au_tt = 0$$ where $A$ is a strictly positive unbounded operator and $\alpha, \beta, \gamma, \delta > 0$. The effect of the regularizing term $\delta Au_tt$ translates into having an analytic semigroup $S(t) = e^t\A$ of solutions. Moreover, the asymptotic properties of the semigroup are ruled by the stability number $$\kappa = \beta - \gamma \alpha + \delta \lambda 0_$$ which, contrary to the case of the standard MGT equation, depends also on the minimum $\lambda_0 > 0$ of the spectrum of $A$.