On the regularized Moore-Gibson-Thompson equation

Abstract

We study the regularized MGT equation \begin{document}$ u_{ttt} + \alpha u_{tt} +\beta Au_t +\gamma A u +\delta A u_{tt} = 0 $\end{document} where $ A $ is a strictly positive unbounded operator and $ \alpha, \beta, \gamma, \delta>0 $. The effect of the regularizing term $ \delta A u_{tt} $ translates into having an analytic semigroup $ S(t) = e^{t{{\mathbb A}}} $ of solutions. Moreover, the asymptotic properties of the semigroup are ruled by the stability number \begin{document}$ \varkappa = \beta - \frac{\gamma}{\alpha +\delta \lambda_0} $\end{document} which, contrary to the case of the standard MGT equation, depends also on the minimum $ \lambda_0>0 $ of the spectrum of $ A $.