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Non-local heat equations with moving boundary

Giacomo Ascione, Pierre Patie, Bruno Toaldo

TL;DR

This work develops a framework for non-local in time heat equations on time-dependent domains with moving boundaries, establishing existence, uniqueness, and regularity of solutions via a stochastic representation using the delayed Brownian motion killed on a moving boundary. A Dynkin-Hunt decomposition is derived for the killed process, with explicit density representations $q_Φ$ in terms of the uncensored density $p_Φ$ and boundary-crossing laws, including special constant-boundary formulas. The analysis leverages the semi-Markov structure of the time-changed Brownian motion and a weak maximum principle to obtain well-posedness and stability, while also uncovering anomalous diffusion through MSD asymptotics linked to the inverse-subordinator structure. The results generalize the classical fractional heat equation by substituting the power kernel with a general Bernstein function $Φ$, offering a broad stochastic-PDE connection and a path to Stefan-type problems with memory kernels.

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.

Non-local heat equations with moving boundary

TL;DR

This work develops a framework for non-local in time heat equations on time-dependent domains with moving boundaries, establishing existence, uniqueness, and regularity of solutions via a stochastic representation using the delayed Brownian motion killed on a moving boundary. A Dynkin-Hunt decomposition is derived for the killed process, with explicit density representations in terms of the uncensored density and boundary-crossing laws, including special constant-boundary formulas. The analysis leverages the semi-Markov structure of the time-changed Brownian motion and a weak maximum principle to obtain well-posedness and stability, while also uncovering anomalous diffusion through MSD asymptotics linked to the inverse-subordinator structure. The results generalize the classical fractional heat equation by substituting the power kernel with a general Bernstein function , offering a broad stochastic-PDE connection and a path to Stefan-type problems with memory kernels.

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.
Paper Structure (18 sections, 37 theorems, 414 equations)

This paper contains 18 sections, 37 theorems, 414 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on subordinators and non-local operators
  3. A weak maximum principle
  4. The Brownian motion delayed by an inverse subordinator
  5. Definition and regularity of the density
  6. The semi-Markov property
  7. The killed delayed Brownian motion and a Dynkin-Hunt formula
  8. The constant boundary case
  9. The non-decreasing and continuous boundary case
  10. Main result
  11. Anomalous diffusive behaviour
  12. Proofs of some technical results
  13. Proof of Proposition \ref{['derfL']}
  14. Proof of Theorem \ref{['thm:weakmax']}
  15. Proof of Proposition \ref{['prop:regx']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

Assume $\varphi$ is continuous and either constant or strictly increasing with $\varphi'(0+)>0$ and $f \in C_c(-\infty,\varphi(0))$. Then there exists a function $q_\alpha:(0,+\infty) \times \mathop{\mathrm{\mathbb{R}}}\nolimits \times \mathop{\mathrm{\mathbb{R}}}\nolimits$ such that for all Borel s In particular, the unique solution of eq:moving is given by

Theorems & Definitions (80)

  • Theorem 1.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • proof
  • ...and 70 more