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

We consider the MGT equation with memory $$\partial_{ttt} u + \alpha \partial_{tt} - \beta \Delta \partial_{t} \gamma\Delta \int_{0}^{t}g(s) u(t-s) ds = 0.$$ prove an existence and uniqueness result removing convexity assumption on convolution kernel $g$, usually adopted in literature. In subcritical case $\alpha\beta>\gamma$, we establish exponential decay of energy, without leaning classical differential inequality involving $g$ its derivative $g'$, namely, $$g'+\delta g\leq 0,\quad\delta>0,$$ but only asking that vanishes exponentially fast.