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Well-Posedness for Weak and Strong Solutions of Non-Homogeneous Initial Boundary Value Problems for Fractional Diffusion Equations

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Abstract

We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak solutions, we introduce a definition of solutions which allows to prove the existence of solution to the initial boundary value problems with non-zero initial and boundary values and non-homogeneous source terms lying in some negative-order Sobolev spaces. For strong solutions, we introduce an optimal compatibility condition and prove the existence of the solutions. We introduce also some sharp conditions guaranteeing the existence of solutions with more regularity in time and space.

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Authors and Affiliations

  1. Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

    Kian Yavar

  2. Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    Masahiro Yamamoto

  3. Honorary Member of Academy of Romanian Scientists, Splaiul Independentei Street, No 54, 050094, Bucharest, Romania

    Masahiro Yamamoto

  4. Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Str., Moscow, 117198, Russia

    Masahiro Yamamoto

Authors
  1. Kian Yavar
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  2. Masahiro Yamamoto
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Correspondence to Kian Yavar.

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Yavar, K., Yamamoto, M. Well-Posedness for Weak and Strong Solutions of Non-Homogeneous Initial Boundary Value Problems for Fractional Diffusion Equations. Fract Calc Appl Anal 24, 168–201 (2021). https://doi.org/10.1515/fca-2021-0008

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  • DOI: https://doi.org/10.1515/fca-2021-0008

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