Moore-Gibson-Thompson equation with memory, part I: exponential decay of energy

We are interested in the Moore-Gibson-Thompson(MGT) equation with memory \begin{equation}\nonumber \tau u_{ttt}+ \alpha u_{tt}+c^2\A u+b\A u_t -\int_0^tg(t-s)\A w(s)ds=0. \end{equation} We first classify the memory into three types. Then we study how a memory term creates damping mechanism and how the memory causes energy decay.