Dier, D., Kemppainen, J., Siljander, J. et al. On the parabolic Harnack inequality for non-local diffusion equations. Math. Z. 295, 1751–1769 (2020). https://doi.org/10.1007/s00209-019-02421-7

### On the parabolic Harnack inequality for non-local diffusion equations

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Author: Dier, Dominik1; Kemppainen, Jukka2; Siljander, Juhana3;
Organizations: 1Institute of Applied Analysis, University of Ulm, 89069 Ulm, Germany
2Mathematics Division, P.O. Box 4500, 90014 University of Oulu, Finland
3Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, 40014 Jyväskylä, Finland
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.2 MB)
Language: English
Published: Springer Nature, 2020
Publish Date: 2020-11-11
Description:

# Abstract

We settle the open question concerning the Harnack inequality for globally positive solutions to non-local in time diffusion equations by constructing a counter-example for dimensions d ≥ β, where β ∈ (0,2] is the order of the equation with respect to the spatial variable. The equation can be non-local both in time and in space but for the counter-example it is important that the equation has a fractional time derivative. In this case, the fundamental solution is singular at the origin for all times t > 0 in dimensions d ≥ β. This underlines the markedly different behavior of time-fractional diffusion compared to the purely space-fractional case, where a local Harnack inequality is known. The key observation is that the memory strongly affects the estimates. In particular, if the initial data $$\mathit{u}_0 \in \mathit{L}_{loc}^{q}$$ for q larger than the critical value $$\frac{d}{\beta}$$ of the elliptic operator (−Δ)β/2, a non-local version of the Harnack inequality is still valid as we show. We also observe the critical dimension phenomenon already known from other contexts: the diffusion behavior is substantially different in higher dimensions than d = 1 provided β > 1, since we prove that the local Harnack inequality holds if d < β.

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Series: Mathematische Zeitschrift
ISSN: 0025-5874
ISSN-E: 1432-1823
ISSN-L: 0025-5874
Volume: 295
Pages: 1751 - 1769
DOI: 10.1007/s00209-019-02421-7