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General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems

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Abstract

In this paper, we deal with the initial-boundary-value problems for a general time-fractional diffusion equation which generalizes the single- and the multi-term time-fractional diffusion equations as well as the time-fractional diffusion equation of the distributed order. First, important estimates for the general time-fractional derivatives of the Riemann-Liouville and the Caputo type of a function at its maximum point are derived. These estimates are applied to prove a weak maximum principle for the general time-fractional diffusion equation. As an application of the maximum principle, the uniqueness of both the strong and the weak solutions to the initial-boundary-value problem for this equation with the Dirichlet boundary conditions is established. Finally, the existence of a suitably defined generalized solution to the the initial-boundary-value problem with the homogeneous boundary conditions is proved.

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Authors and Affiliations

  1. Dept. of Mathematics, Physics, and Chemistry, Beuth Technical University of Applied Sciences, Luxemburger Str. 10, Berlin -, 13353, Germany

    Yuri Luchko

  2. Dept. of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo -, 153, Japan

    Masahiro Yamamoto

Authors
  1. Yuri Luchko
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  2. Masahiro Yamamoto
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Correspondence to Yuri Luchko.

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Luchko, Y., Yamamoto, M. General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems. FCAA 19, 676–695 (2016). https://doi.org/10.1515/fca-2016-0036

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  • DOI: https://doi.org/10.1515/fca-2016-0036

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