On the MGT equation with memory of type II

TL;DR

This paper investigates the Moore-Gibson-Thompson equation with type II memory, establishing well-posedness without restrictions on the memory kernel's total mass and demonstrating conditions for exponential energy growth, addressing a prior open question.

Contribution

It proves well-posedness of the MGT equation with type II memory without mass restrictions and constructs kernels leading to exponential energy growth, answering an open problem.

Findings

Well-posedness holds without restrictions on the total mass of g.

Existence of kernels with mass less than β causing exponential energy growth.

Addresses an open question from previous literature.

Abstract

We consider the Moore-Gibson-Thompson equation with memory of type II $$ \partial_{ttt} u(t) + \alpha \partial_{tt} u(t) + \beta A \partial_t u(t) + \gamma Au(t)-\int_0^t g(t-s) A \partial_t u(s){\rm d} s=0 $$ where $A$ is a strictly positive selfadjoint linear operator (bounded or unbounded) and $\alpha,\beta,\gamma>0$ satisfy the relation $\gamma\leq\alpha\beta$. 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".