A note on the Moore–Gibson–Thompson equation with memory of type II

Abstract

We consider the Moore–Gibson–Thompson equation with memory of type II

$$\begin{aligned} \partial _{ttt} u(t) + \alpha \partial _{tt} u(t) + \beta A \partial _t u(t) + \gamma Au(t)-\int _0^t g(t-s) A \partial _t u(s)\mathrm{d}s=0 \end{aligned}$$

where A is a strictly positive selfadjoint linear operator (bounded or unbounded) and $\alpha ,\beta ,\gamma >0$ satisfy the relation $\gamma \le \alpha \beta$. First, we prove well-posedness of finite energy solutions, without requiring any restriction on the total mass $\varrho$ of g. This extends previous results in the literature, where such a restriction was imposed. Second, we address an open question within the context of longtime behavior of solutions. We show that an “overdamping” in the memory term can destabilize the originally stable dynamics. In fact, it is always possible to find memory kernels g, complying with the usual mass restriction $\varrho <\beta$, such that the equation admits solutions with energy growing exponentially fast, even in the regime $\gamma < \alpha \beta$ where the corresponding model without memory is exponentially stable. In particular, this provides an answer to a question recently raised in the literature.