We reveal the hybrid of wave-like and heat-like phenomenon for the heat equation with time delay.  We prove that the solutions have both  the propagation of singularities and the smooth effects. Such propagation of singularities differs from that of the wave equation. The singularities here  propagats  along the time direction, periocically in some discrete points and with the gain/ loss of 2-order of space-derivative. That is the periodically jumping propagation of singularities for heat equation with time delay.

This is a joint work with K. Hunish, H. Yu and Y. Zhang