On the MGT equation with memory of type II

Abstract

We consider the Moore-Gibson-Thompson equation with memory of type II $${\partial }_{ttt}u(t)+\alpha {\partial }_{tt}u(t)+\beta A{\partial }_{t}u(t)+\gamma Au(t)-{\int }_0^{t}g(t-s)A{\partial }_{t}u(s)\mathrm{d}s=0$$ 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 a well-posedness result without requiring any restriction on the total mass $\varrho$ of $g$. Then we show that 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. In particular, this provides the answer to a question raised in "F. Dell'Oro, I. Lasiecka, V. Pata, J. Differential Equations 261 (2016), 4188-4222".