On the Moore-Gibson-Thompson Equation with Memory with Nonconvex Kernels

We consider the MGT equation with memory partial derivative(ttt)u + alpha partial derivative(tt)u - beta Delta partial derivative(t)u - gamma Delta u + integral(t)(0) 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', namely, g' + delta g <= 0, delta > 0, but we ask only that g vanish exponentially fast.