Well-posedness and global dynamics for the critical Hardy–Sobolev parabolic equation

Noboru Chikami, Masahiro Ikeda, Koichi Taniguchi

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)

Abstract

We study the Cauchy problem for the semilinear heat equation with the singular potential, called the Hardy–Sobolev parabolic equation, in the energy space. The aim of this paper is to determine a necessary and sufficient condition on initial data below or at the ground state, under which the behavior of solutions is completely dichotomized. More precisely, the solution exists globally in time and its energy decays to zero in time, or it blows up in finite or infinite time. The result on the dichotomy for the corresponding Dirichlet problem is also shown as a by-product via comparison principle.

Original languageEnglish
Pages (from-to)8094-8142
Number of pages49
JournalNonlinearity
Volume34
Issue number11
DOIs
Publication statusPublished - 2021 Nov

Keywords

  • Blow-up
  • Dissipation
  • Global existence
  • Hardy–Sobolev parabolic equation
  • Well-posedness

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