A De Giorgi–Nash type theorem for time fractional diffusion equations

Mathematische Annalen volume 356, pages 99–146 (2013)Cite this article

Abstract

We study the regularity of weak solutions to linear time fractional diffusion equations in divergence form of arbitrary time order \(\alpha \in (0,1)\). The coefficients are merely assumed to be bounded and measurable, and they satisfy a uniform parabolicity condition. Our main result is a De Giorgi–Nash type theorem, which gives an interior Hölder estimate for bounded weak solutions in terms of the data and the \(L_\infty \)-bound of the solution. The proof relies on new a priori estimates for time fractional problems and uses De Giorgi’s technique and the method of non-local growth lemmas, which has been introduced recently in the context of nonlocal elliptic equations involving operators like the fractional Laplacian.

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Acknowledgments

The author would like to thank the anonymous referee for his valuable comments and suggestions, which certainly helped to improve the manuscript.

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  1. Institute of Mathematics, Martin-Luther University Halle-Wittenberg, Theodor-Lieser-Strasse 5, 06120,  Halle, Germany

    Rico Zacher

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  1. Rico Zacher
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Correspondence to Rico Zacher.

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This work was supported by the Deutsche Forschungsgemeinschaft (DFG), Bonn, Germany.

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Zacher, R. A De Giorgi–Nash type theorem for time fractional diffusion equations. Math. Ann. 356, 99–146 (2013). https://doi.org/10.1007/s00208-012-0834-9

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Mathematics Subject Classification (2010)

  • 35R09
  • 35K10
  • 45K05