Periodic propagation of singularities for heat equations with time delay

TL;DR

This paper investigates how singularities propagate periodically and in a stepwise manner in heat equations with time delay, revealing unique behaviors distinct from classical heat and wave equations.

Contribution

It introduces the phenomenon of periodic and stepwise propagation of singularities in delayed heat equations, highlighting differences from standard heat and wave equations.

Findings

Singularities propagate periodically along the time axis.

Propagation occurs in a stepwise manner with derivative order changes.

Initial data and historical values influence singularity propagation.

Abstract

This paper presents two remarkable phenomena associated with the heat equation with a time delay: namely, the propagation of singularities and periodicity. These are manifested through a distinctive mode of propagation of singularities in the solutions. Precisely, the singularities of the solutions propagate periodically in a bidirectional fashion along the time axis. Furthermore, this propagation occurs in a stepwise manner. More specifically, when propagating in the positive time direction, the order of the joint derivatives of the solution increases by 2 for each period; conversely, when propagating in the reverse time direction, the order of the joint derivatives decreases by 2 per period. Additionally, we elucidate the way in which the initial data and historical values impact such a propagation of singularities. The phenomena we have discerned not only corroborate the pronounced differences between heat equations with and without time delay but also vividly illustrate the substantial divergence between the heat equation with a time delay and the wave equation, especially when viewed from the point of view of singularity propagation.