A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces

R. Laister, J. C. Robinson*, M. Sierżęga, A. Vidal-López

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

23 Citations (Scopus)

Abstract

We consider the scalar semilinear heat equation ut−Δu=f(u), where f:[0,∞)→[0,∞) is continuous and non-decreasing but need not be convex. We completely characterise those functions f for which the equation has a local solution bounded in Lq(Ω) for all non-negative initial data u0∈Lq(Ω), when Ω⊂Rd is a bounded domain with Dirichlet boundary conditions. For q∈(1,∞) this holds if and only if limsups→∞s−(1+2q/d)f(s)<∞; and for q=1 if and only if ∫1s−(1+2/d)F(s)ds<∞, where F(s)=sup1≤t≤s⁡f(t)/t. This shows for the first time that the model nonlinearity f(u)=u1+2q/d is truly the ‘boundary case’ when q∈(1,∞), but that this is not true for q=1. The same characterisations hold for the equation posed on the whole space Rd provided that limsups→0f(s)/s<∞.

Original languageEnglish
Pages (from-to)1519-1538
Number of pages20
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume33
Issue number6
DOIs
Publication statusPublished - 1 Nov 2016

Keywords

  • Dirichlet heat kernel
  • Dirichlet problem
  • Instantaneous blow-up
  • Local existence
  • Non-existence
  • Semilinear heat equation

Cite this