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Local solvability and dilation-critical singularities of supercritical fractional heat equations

Yohei Fujishima, Kotaro Hisa, Kazuhiro Ishige, Robert Laister

Abstract

We consider the Cauchy problem for fractional semilinear heat equations with supercritical nonlinearities and establish both necessary conditions and sufficient conditions for local-in-time solvability. We introduce the notion of a dilation-critical singularity (DCS) of the initial data and show that such singularities always exist for a large class of supercritical nonlinearities. Moreover, we provide exact formulae for such singularities.

Local solvability and dilation-critical singularities of supercritical fractional heat equations

Abstract

We consider the Cauchy problem for fractional semilinear heat equations with supercritical nonlinearities and establish both necessary conditions and sufficient conditions for local-in-time solvability. We introduce the notion of a dilation-critical singularity (DCS) of the initial data and show that such singularities always exist for a large class of supercritical nonlinearities. Moreover, we provide exact formulae for such singularities.
Paper Structure (12 sections, 141 equations)

This paper contains 12 sections, 141 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Local solvability of the semilinear problem \ref{['eq:SHE']}
  5. Preliminaries
  6. A necessary condition for local solvability
  7. A sufficient condition for local solvability
  8. Proofs of the main results
  9. Preparatory lemmas
  10. Proof of Theorem \ref{['thm:main']}
  11. Proof of Theorem \ref{['thm:CS']}
  12. Application: a generalised exponential