The stability and exponential stabilization of the heat equation with memory

In this paper, we investigate the stability of heat equation with a constant memory kernel. We prove that the system does not decay for the positive kernel, and is polynomially stable but not exponentially for the negative kernel. In particular, a more detailed description of the decay property for the negative kernel is obtained by dividing initial values into two categories. Moreover, we show that the system can be exponentially stabilized by boundary control for any constant memory kernel.