UNSPECIFIED (1996) Bounded solutions for non-autonomous parabolic equations. DYNAMICS AND STABILITY OF SYSTEMS, 11 (2). pp. 109-120. ISSN 0268-1110

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Abstract

The existence of bounded solutions (including in particular homoclinic and heteroclinic solutions) is studied for non-autonomous perturbed parabolic partial differential equations, without the restriction that the linear variational equation has a unique non-trivial bounded solution. Specifically, an idea applied to ordinary differential equations by Hale (1984) and by Battelli and Laari (1990) is realised in an infinite-dimensional setting. Like other work on related problems, the main technique is Lyapunov-Schmidt reduction; we use that technique here in the context of bounded solutions, rather than the more usual setting of periodic or homoclinic solutions. Moreover, several technical obstacles are circumvented in the infinite-dimensional setting-in particular in the proof of the existence of a solution to the reduced bifurcation equation. Non-uniqueness is shown to occur for the Kuramoto-Sivashinsky equation, demonstrating the need to remove the uniqueness restriction.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics T Technology > TJ Mechanical engineering and machinery
Journal or Publication Title:	DYNAMICS AND STABILITY OF SYSTEMS
Publisher:	CARFAX PUBL CO
ISSN:	0268-1110
Date:	June 1996
Volume:	11
Number:	2
Number of Pages:	12
Page Range:	pp. 109-120
Publication Status:	Published
URI:	http://wrap.warwick.ac.uk/id/eprint/18565

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