Modulus of continuity for solutions of non-local heat equations

TL;DR

This paper extends the modulus of continuity method to non-local heat equations, demonstrating preservation of initial regularity and revealing complexities in non-local inequalities.

Contribution

It generalizes the modulus of continuity approach to non-local heat equations and explores boundary conditions and domain effects.

Findings

Modulus of continuity is preserved for solutions with suitable initial conditions.

Counterexample indicates non-local inequalities depend on more than domain diameter.

Method applies to equations on R^n and in one dimension with non-local boundary conditions.

Abstract

We extend the method of modulus of continuity for solutions of parabolic equations--as used, for instance, to prove the Fundamental Gap Conjecture--to solutions of non-local heat equations on R^n and in dimension one with a non-local Neumann boundary condition. Specifically, we show that if a solution of a non-local heat equation has an initial modulus of continuity satisfying simple criteria, then this modulus of continuity is preserved at all subsequent times. In the process of trying to generalise our result in one dimension, we found a counterexample suggesting that a non-local analogue of the Payne-Weinberger inequality would depend on more than the diameter of a bounded (convex) domain.