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Some existence, uniqueness and nonuniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities

Published online by Cambridge University Press:  14 November 2011

Eduard Feireisl  and
John Norbury
Eduard Feireisl
Affiliation:
Institute of Mathematics ČSAV, Prague, Czechoslovakia
John Norbury
Affiliation:
Mathematical Institute, St Giles, Oxford, U.K.
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Synopsis

We consider the problem

where H stands for the maximal monotone graph associated with the Heaviside step function. It is shown that the problem possesses at least one (strong) solution belonging to an appropriate function space. Moreover, we prove:

(i) There is a smooth initial function u0, u0≧1, where the equality holds at exactly one point such that there are at least two different solutions corresponding to the initial data u0.

(ii) The comparison principle: The relation for any x ≠ 0, implies u1(t)>u2(t), t≧0 for any u1, u2 solving the problem with the initial data , respectively.

(iii) For a “reasonable” set of initial data the solution is uniquely determined. Moreover, the free boundary {(x, t)| u(x, t) = 1} is regular and on its complement the equation holds in a classical sense.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 119 , Issue 1-2 , 1991 , pp. 1 - 17
Copyright
Copyright © Royal Society of Edinburgh 1991

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