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On the parabolic Harnack inequality for non-local diffusion equations

  • Dominik Dier
  • Jukka Kemppainen
  • Juhana Siljander
  • Rico ZacherEmail author
Article
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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 \(d\ge \beta \), where \(\beta \in (0,2]\) 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 \(d\ge \beta \). 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 \(u_0 \in L^q_{loc}\) for q larger than the critical value \(\tfrac{d}{\beta }\) of the elliptic operator \((-\Delta )^{\beta /2}\), 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 \(\beta >1\), since we prove that the local Harnack inequality holds if \(d<\beta \).

Keywords

Non-local diffusion Harnack inequality Riemann–Liouville derivative Fractional Laplacian Fundamental solution H functions Asymptotics 

Mathematics Subject Classification

Primary 35R11 Secondary 45K05 45M05 35C15 26A33 35B40 33C60 

Notes

Acknowledgements

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 547/4-1.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Dominik Dier
    • 1
  • Jukka Kemppainen
    • 2
  • Juhana Siljander
    • 3
  • Rico Zacher
    • 1
    Email author
  1. 1.Institute of Applied AnalysisUniversity of UlmUlmGermany
  2. 2.Applied and Computational MathematicsUniversity of OuluOuluFinland
  3. 3.Department of Mathematics and StatisticsUniversity of JyväskyläJyvaskylaFinland

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