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EXISTENCE OF SOLUTIONS FOR NONLINEAR PARABOLIC PROBLEMS

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*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