On the parabolic Harnack inequality for nonlocal diffusion equations 

Author:  Dier, Dominik^{1}; Kemppainen, Jukka^{2}; Siljander, Juhana^{3}; 
Organizations: 
^{1}Institute of Applied Analysis, University of Ulm, 89069 Ulm, Germany ^{2}Mathematics Division, P.O. Box 4500, 90014 University of Oulu, Finland ^{3}Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, 40014 Jyväskylä, Finland 
Format:  article 
Version:  accepted version 
Access:  embargoed 
Persistent link:  http://urn.fi/urn:nbn:fife2020051333133 
Language:  English 
Published: 
Springer Nature,
2020

Publish Date:  20201111 
Description: 
AbstractWe settle the open question concerning the Harnack inequality for globally positive solutions to nonlocal in time diffusion equations by constructing a counterexample for dimensions d ≥ β, where β ∈ (0,2] is the order of the equation with respect to the spatial variable. The equation can be nonlocal both in time and in space but for the counterexample 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 timefractional diffusion compared to the purely spacefractional 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 nonlocal 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. see all

Series: 
Mathematische Zeitschrift 
ISSN:  00255874 
ISSNE:  14321823 
ISSNL:  00255874 
Volume:  295 
Pages:  1751  1769 
DOI:  10.1007/s00209019024217 
OADOI:  https://oadoi.org/10.1007/s00209019024217 
Type of Publication: 
A1 Journal article – refereed 
Field of Science: 
111 Mathematics 
Subjects:  
Funding: 
J. S. was supported by the Academy of Finland Grant 259363 and a Väisälä foundation travel Grant. R. Z. was supported by a research grant of the German Research Foundation (DFG), GZ Za 547/41. 
Copyright information: 
© SpringerVerlag GmbH Germany, part of Springer Nature 2019. This is a postpeerreview, precopyedit version of an article published in Mathematische Zeitschrift. The final authenticated version is available online at: https://doi.org/10.1007/s00209019024217. 