We characterize the waves in the heat equation with memory from the viewpoint of high frequency behaviors of the solutions. In particular, the solutions of the equation have the propagation of singularities along the time direction. This can be used to understand the memory effect inside.

Furthermore, under a sharp sufficient control condition, the difference between the reachable subspaces of the controlled heat equations, with and without memory, is a special Sobolev space. The appearance of such Sobolev space is mainly related to the hyperbolic nature of the heat equation with memory.