Nikolaos Halidias and Nikolaos S. Papageorgiou

Address. National Technical University, Department of Mathematics, Zografou Campus, Athens 157 80, GREECE

E-mail: npapg@math.ntua.gr

Abstract. We consider nonlinear parabolic boundary value problems. First we assume that the right hand side term is discontinuous and nonmonotone and in order to have an existence theory we pass to a multivalued version by filling in the gaps at the discontinuity points. Assuming the existence of an upper solution $\phi$ and of a lower solution $\psi$ such that $\psi \leq \phi$, and using the theory of nonlinear operators of monotone type, we show that there exists a solution $x \in [\psi,\phi]$ and that the set of all such solutions is compact in $W_{pq}(T)$. For the problem with a Caratheodory right hand side we show the existence of extremal solutions in $[\psi,\phi]$.

AMSclassification. 35K55

Keywords. Upper and lower solutions, weak solution, evolution triple, compact embedding, distributional derivative, operator of type $(S)_{+}$, operator of type $L-(S)_{+}$, $L-$ pseudomonotone operator, multivalued problem, extremal solutions, Zorn's lemma