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

Laister, R.; Robinson, J. C.; Sierżęga, M.; Vidal-López, A.; Laister, Robert; Robinson, James C.; Sierzega, Mikolaj; Vidal-Lopez, Alejandro

Authors

R. Laister

J. C. Robinson

M. Sierżęga

A. Vidal-López

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
Publication Date Nov 1, 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
APA6 Citation Vidal-López, A., Sierżęga, M., Robinson, J. C., Laister, R., Laister, R., Robinson, J. C., …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
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
Publisher URL http://dx.doi.org/10.1016/j.anihpc.2015.06.005

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