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

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

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Authors

James C. Robinson

Mikolaj Sierzega

Alejandro Vidal-Lopez



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)

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
Contract Date May 22, 2016

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