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

Laister, Robert; Robinson, James C.; Sierzega, Mikolaj; Vidal-Lopez, Alejandro

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)

Citation

Laister, R., Robinson, J. C., Sierzega, M., & Vidal-Lopez, A. (2016). A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces. Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 33(6), 1519-1538. https://doi.org/10.1016/j.anihpc.2015.06.005

• Journal Article Type: Article
• Acceptance Date: Jun 25, 2015
• Online Publication Date: Jul 26, 2015
• Publication Date: Nov 1, 2016
• Deposit Date: Jan 15, 2015
• Publicly Available Date: Jul 26, 2016
• Journal: Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
• Print ISSN: 0294-1449
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 33
• Issue: 6
• Pages: 1519-1538
• DOI: https://doi.org/10.1016/j.anihpc.2015.06.005
• Keywords: Semilinear heat equation; Dirichlet problem; Local existence; Non-existence; Instantaneous blow-up; Dirichlet heat kernel
• Public URL: https://uwe-repository.worktribe.com/output/909531
• Publisher URL: http://dx.doi.org/10.1016/j.anihpc.2015.06.005