Well-posedness and exponential decay for the Moore-Gibson-Thompson equation with time-dependent memory kernel

We are interested in the well-posedness and exponential decay of Moore-Gibson-Thompson equation with memory tau & part;(ttt)u + alpha & part;(tt)u + h(t)(0)Au + bA & part;(t)u -integral(infinity)(0) h ' t(s)Au(t - s)ds = 0.Here, the main feature is that the memory kernel h(t)(middot) depends on time. By using the linear semigroup theory, we prove that the above system is global well-posedness. In addition, the exponential decay of the related energy is shown to occur, provided that the memory kernel is controlled by a negative exponential.