On the Moore-Gibson-Thompson equation with memory with nonconvex kernels

Abstract

We consider the MGT equation with memory $${\partial }_{ttt}u+\alpha {\partial }_{tt}u-\beta \Delta {\partial }_{t}u-\gamma \Delta u+{\int }_0^{t}g(s)\Delta u(t-s)ds=0.$$ We prove an existence and uniqueness result removing the convexity assumption on the convolution kernel $g$, usually adopted in the literature. In the subcritical case $\alpha \beta >\gamma$, we establish the exponential decay of the energy, without leaning on the classical differential inequality involving $g$ and its derivative $g^{\prime }$, namely, $$g^{\prime }+\delta g\le 0,\delta >0,$$ but only asking that $g$ vanishes exponentially fast.