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

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

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


Authors

M. Sier??ga

J. C. Robinson



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

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