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On the Harnack inequality for time-fractional and more general non-local in time subdiffusion equations

Katarzyna Ryszewska, Rico Zacher

TL;DR

The paper addresses whether a full Harnack inequality can hold for globally nonnegative local solutions to time-nonlocal subdiffusion equations in one space dimension. It combines Moser-type iterations, time-shifted regularisations via Yosida approximations, and a one-dimensional parabolic embedding to prove a full Harnack inequality for a broad class of kernels $k$ (class $\mathscr{PC}$), including time-fractional models, linking memory effects to pointwise bounds. The main contribution is establishing a 1D full Harnack inequality for nonlocal-in-time subdiffusions, filling a gap left by higher-dimensional counterexamples and clarifying the dimension-dependent nature of these estimates. The results provide rigorous insight into how memory terms influence qualitative behavior in subdiffusive processes and have potential implications for the study of anomalous diffusion models.

Abstract

In this paper we establish the Harnack inequality for globally positive local solutions to a general class of nonlocal in time subdiffusion equations in one space dimension, which includes time-fractional diffusion equations with time order less than one. It is already known that for these equations the classical Harnack inequality does not hold if the space dimension is greater than or equal to two. Here, we complete the analysis, by providing a positive result in one space dimension.

On the Harnack inequality for time-fractional and more general non-local in time subdiffusion equations

TL;DR

The paper addresses whether a full Harnack inequality can hold for globally nonnegative local solutions to time-nonlocal subdiffusion equations in one space dimension. It combines Moser-type iterations, time-shifted regularisations via Yosida approximations, and a one-dimensional parabolic embedding to prove a full Harnack inequality for a broad class of kernels (class ), including time-fractional models, linking memory effects to pointwise bounds. The main contribution is establishing a 1D full Harnack inequality for nonlocal-in-time subdiffusions, filling a gap left by higher-dimensional counterexamples and clarifying the dimension-dependent nature of these estimates. The results provide rigorous insight into how memory terms influence qualitative behavior in subdiffusive processes and have potential implications for the study of anomalous diffusion models.

Abstract

In this paper we establish the Harnack inequality for globally positive local solutions to a general class of nonlocal in time subdiffusion equations in one space dimension, which includes time-fractional diffusion equations with time order less than one. It is already known that for these equations the classical Harnack inequality does not hold if the space dimension is greater than or equal to two. Here, we complete the analysis, by providing a positive result in one space dimension.
Paper Structure (7 sections, 15 theorems, 196 equations)

This paper contains 7 sections, 15 theorems, 196 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Iterations and embedding
  4. Properties of the kernels
  5. Regularised weak formulation and time-shifts
  6. The improved weak Harnack inequality
  7. Estimates for subsolutions

Key Result

Lemma 1.1

There exists a unique strictly increasing function $\Phi \in C([0,r_0))\cap C^{1}((0,r_0))$, where $r_0$ is defined in (r0), such that $\Phi(0) = 0$, $\lim\limits_{r\rightarrow r_0}\Phi(r) = \infty$ and

Theorems & Definitions (22)

  • Remark 1.1
  • Lemma 1.1: krz
  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.1
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 12 more