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

We consider the MGT equation with memory $$partial_{ttt} u + alpha partial_{tt} u - beta Delta partial_{t} u - gammaDelta u + int_{0}^{t}g(s) Delta u(t-s) ds = 0.$$ 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 $alphabeta>gamma$, we establish the exponential decay of the energy, without leaning on the classical differential inequality involving $g$ and its derivative $g$, namely, $$g+delta gleq 0,quaddelta>0,$$ but only asking that $g$ vanishes exponentially fast.