On the Existence of Solutions for Nonlinear Parabolic Problems with Nonmonotone Discontinuities

In this paper we consider a nonlinear parabolic problem with a discontinuous, nonmonotone nonlinearity. We assume the existence of an upper solution w and a lower solution c such that c Fw. 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 solution x t, z.such that c t, z.Fx t, z.Fw t, z.a.e. on T=Z.