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Modulus of continuity for solutions of non-local heat equations

Ben Andrews, Sophie Chen

TL;DR

The paper develops a modulus-of-continuity framework for non-local heat equations, proving preservation of an initial modulus on $\mathbb{R}^n$ and, in the one-dimensional regional setting, highlighting a dimension-dependent limitation. The authors employ a maximum-principle argument with an auxiliary function $Z_{\varepsilon}$ and a coupling-by-reflection technique, together with a one-dimensional reduction $\tilde{\varphi}$ that satisfies a related non-local equation. They show that, under a globally bounded regulator $\Psi$, the modulus persists and derive explicit control via a constant $C$ relative to the kernel-bound $K$. A key finding is that the 1D modulus result does not generalize to higher dimensions for the regional problem, supported by a counterexample and a discussion of a non-local Payne-Weinberger inequality, while the nonlinear non-local extension is also established under suitable Lipschitz conditions on $q$. The work lays groundwork for sharp, dimension-sensitive oscillation control for non-local diffusions and informs potential eigenvalue bounds and non-local analogues of classical inequalities.

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.

Modulus of continuity for solutions of non-local heat equations

TL;DR

The paper develops a modulus-of-continuity framework for non-local heat equations, proving preservation of an initial modulus on and, in the one-dimensional regional setting, highlighting a dimension-dependent limitation. The authors employ a maximum-principle argument with an auxiliary function and a coupling-by-reflection technique, together with a one-dimensional reduction that satisfies a related non-local equation. They show that, under a globally bounded regulator , the modulus persists and derive explicit control via a constant relative to the kernel-bound . A key finding is that the 1D modulus result does not generalize to higher dimensions for the regional problem, supported by a counterexample and a discussion of a non-local Payne-Weinberger inequality, while the nonlinear non-local extension is also established under suitable Lipschitz conditions on . The work lays groundwork for sharp, dimension-sensitive oscillation control for non-local diffusions and informs potential eigenvalue bounds and non-local analogues of classical inequalities.

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.

Paper Structure

This paper contains 8 sections, 3 theorems, 70 equations, 5 figures.

Table of Contents

  1. Introduction and main results
  2. Non-local heat equation on $\mathbb R^n$
  3. Regional heat equation
  4. One-dimensional case
  5. General case
  6. Counterexample
  7. Non-local Payne-Weinberger inequality
  8. Non-linear non-local heat equation

Key Result

Theorem 2.3

Suppose $u:\mathbb R^n\times[0,\infty)\rightarrow\mathbb R$ satisfies the non-local heat equation eq:nonlocal_heat and $\|u\|_{L^{\infty}(\mathbb R^n)}\le 1$. Let $\varphi:[0,\infty)\times[0,\infty)\rightarrow[0,\infty)$ be a function with the following properties: Furthermore, assume we can define a suitable regularistion function: Then $\varphi(\cdot,t)$ is a modulus of continuity for $u(\cdot

Figures (5)

  • Figure 1: Change of variables using a 'coupling-by-reflection' technique.
  • Figure 2: $|x-y|\geq 2\delta$
  • Figure 3: $|x-y|<2\delta$
  • Figure 4: Original asymmetric configuration
  • Figure 5: Configuration after translation by $m$ units to the right. Notice that $m<0$.

Theorems & Definitions (8)

  • Theorem 2.3: Non-local modulus of continuity on $\mathbb R^n$
  • Remark 2.7
  • proof
  • Theorem 3.3: Non-local modulus of continuity on a compact interval
  • proof
  • Conjecture 3.20: Non-local modulus of continuity on a bounded convex domain
  • Theorem 4.2: Non-linear non-local modulus of continuity on $\mathbb R^n$
  • proof