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Existence, uniqueness and non-uniqueness of weak solutions of parabolic initial-value problems with discontinuous nonlinearities

Published online by Cambridge University Press:  12 July 2007

Hideo Deguchi
Hideo Deguchi
Affiliation:
Institute of Mathematics, University of Tsukuba, Tsukuba 305-8571, Japan (hdegu@math.tsukuba.ac.jp)
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Abstract

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We deal with the initial-value problem for parabolic equations with discontinuous nonlinearities and establish the existence of its weak solution. Next, we show that for a suitable class of initial data, the weak solution is locally or globally unique in time. Lastly, we prove that there exist at least two different weak solutions in general if initial data do not belong to this class.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 135 , Issue 6 , December 2005 , pp. 1139 - 1167
Copyright
Copyright © Royal Society of Edinburgh 2005