Geodesic
Existence of solutions for nonlinear parabolic problems
Archivum mathematicum, Tome 35 (1999) no. 3, pp. 255-274 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 \le \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 $[\psi ,\phi ]$.
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 \le \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 $[\psi ,\phi ]$.
Classification : 35K55, 35K60
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 
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Halidias, Nikolaos; Papageorgiou, Nikolaos S. Existence of solutions for nonlinear parabolic problems. Archivum mathematicum, Tome 35 (1999) no. 3, pp. 255-274. http://geodesic.mathdoc.fr/item/ARM_1999_35_3_a5/

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