On the existence of solutions for nonlinear parabolic problems with nonmonotone discontinuities

Abstract

In this paper we consider a nonlinear parabolic problem with a discontinuous, nonmonotone nonlinearity. We assume the existence of an upper solution φ and a lower solution ψ such that ψ≤φ. Using results from the theory of pseudomonotone operators and from the theory of multivalued analysis together with truncation and penalization techniques, we show that there exists at least one solutionx(t,z) such that ψ(t,z)≤x(t,z)≤φ(t,z) a.e. onT×Z. © 1997 Academic Press.