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A necessary and sufficient condition for uniqueness of the trivial solution in semilinear parabolic equations
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Laister, Robert, Robinson, James C. and Sierżęga, Mikołaj (2017) A necessary and sufficient condition for uniqueness of the trivial solution in semilinear parabolic equations. Journal of Differential Equations, 262 (10). pp. 4979-4987. doi:10.1016/j.jde.2017.01.014 ISSN 0022-0396.
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Official URL: http://dx.doi.org/10.1016/j.jde.2017.01.014
Abstract
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.
Item Type: | Journal Article | ||||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||
Library of Congress Subject Headings (LCSH): | Differential equations, Parabolic, Differential equations, Partial, Differential equations | ||||||||||
Journal or Publication Title: | Journal of Differential Equations | ||||||||||
Publisher: | Academic Press | ||||||||||
ISSN: | 0022-0396 | ||||||||||
Official Date: | 15 May 2017 | ||||||||||
Dates: |
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Volume: | 262 | ||||||||||
Number: | 10 | ||||||||||
Page Range: | pp. 4979-4987 | ||||||||||
DOI: | 10.1016/j.jde.2017.01.014 | ||||||||||
Status: | Peer Reviewed | ||||||||||
Publication Status: | Published | ||||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||||
Date of first compliant deposit: | 2 March 2017 | ||||||||||
Date of first compliant Open Access: | 30 January 2018 | ||||||||||
Adapted As: |
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