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Fractional semilinear heat equations with singular and nondecaying initial data

Revista Matemática Complutense volume 35, pages 415–445 (2022)Cite this article

Abstract

We study integrability conditions for existence and nonexistence of a local-in-time integral solution of fractional semilinear heat equations with rather general growing nonlinearities in uniformly local \(L^p\) spaces. Our main results about this matter consist of Theorems 1.4, 1.6, 5.1 and 5.3. We introduce a supersolution of an integral equation which can be applied to a nonlocal parabolic equation. When the nonlinear term is \(u^p\) or \(e^u\), a local-in-time solution can be constructed in the critical case, and integrability conditions for the existence and nonexistence are completely classified. Our analysis is based on the comparison principle, Jensen’s inequality and \(L^p\)-\(L^q\) type estimates.

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Acknowledgements

The second author was supported by JSPS KAKENHI Grant Number 19H01797.

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Authors and Affiliations

  1. École Nationale des Ponts et Chaussées, 6-8 avenue Blaise-Pascal Champs-sur-Marne, 77455, Marne-la-Vallée, France

    Théo Giraudon

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

    Yasuhito Miyamoto

Authors
  1. Théo Giraudon
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  2. Yasuhito Miyamoto
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Correspondence to Yasuhito Miyamoto.

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Giraudon, T., Miyamoto, Y. Fractional semilinear heat equations with singular and nondecaying initial data. Rev Mat Complut 35, 415–445 (2022). https://doi.org/10.1007/s13163-021-00389-9

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  • DOI: https://doi.org/10.1007/s13163-021-00389-9

Keywords

  • Local-in-time solution
  • Optimal singularity
  • Supersolutions
  • Fractional Laplacian

Mathematics Subject Classification

  • Primary 35K55
  • 35R11
  • Secondary 35A01
  • 46E30