Non-local heat equations with moving boundary

TL;DR

This paper studies non-local in time heat equations on expanding domains with moving boundaries, establishing well-posedness, maximum principles, and stochastic representations involving delayed Brownian motion, revealing anomalous diffusion behavior.

Contribution

It introduces a solution framework for non-local heat equations on moving boundaries and links solutions to stochastic delayed Brownian motion, providing new insights into their properties.

Findings

Established well-posedness and uniqueness of solutions.

Derived a stochastic representation using delayed Brownian motion.

Identified anomalous diffusive behavior in the process.

Abstract

In this paper we consider non-local (in time) heat equations on time-increasing parabolic sets whose boundary is determined by a suitable curve. We provide a notion of solution for these equations and we study well-posedness under Dirichlet conditions outside the domain. A maximum principle is proved and used to derive uniqueness and continuity with respect to the initial datum of the solutions of the Dirichlet problem. Existence is proved by showing a stochastic representation based on the delayed Brownian motion killed on the boundary. Several related distributional properties of the delayed Brownian motion and its crossing probabilities are also obtained. The asymptotic behaviour of the mean square displacement of the process is determined, showing that the diffusive behaviour is anomalous.