A necessary and sufficient condition for uniqueness of the trivial solution in semilinear parabolic equations

Sier??ga, M.; Laister, Robert; Robinson, J. C.

Abstract

© 2017 Elsevier Inc. In their (1968) paper Fujita and Watanabe considered the issue of uniqueness of the trivial solution of semilinear parabolic equations with respect to the class of bounded, non-negative solutions. In particular they showed that if the underlying ODE has non-unique solutions (as characterised via an Osgood-type condition) and the nonlinearity f satisfies a concavity condition, then the parabolic PDE also inherits the non-uniqueness property. This concavity assumption has remained in place either implicitly or explicitly in all subsequent work in the literature relating to this and other, similar, non-uniqueness phenomena in parabolic equations. In this paper we provide an elementary proof of non-uniqueness for the PDE without any such concavity assumption on f. An important consequence of our result is that uniqueness of the trivial solution of the PDE is equivalent to uniqueness of the trivial solution of the corresponding ODE, which in turn is known to be equivalent to an Osgood-type integral condition on f.

• Journal Article Type: Article
• Acceptance Date: Jan 19, 2017
• Online Publication Date: Jan 30, 2017
• Publication Date: May 15, 2017
• Deposit Date: Jan 23, 2017
• Publicly Available Date: Jan 30, 2018
• Journal: Journal of Differential Equations
• Print ISSN: 0022-0396
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 262
• Issue: 10
• Pages: 4979-4987
• DOI: https://doi.org/10.1016/j.jde.2017.01.014
• Keywords: semilinear, parabolic, Osgood, non-uniqueness, uniqueness, lower solution
• Public URL: https://uwe-repository.worktribe.com/output/887759
• Publisher URL: http://dx.doi.org/10.1016/j.jde.2017.01.014
• Contract Date: Jan 23, 2017