Existence of solutions for nonlinear parabolic problems

Nikolaos Halidias; Nikolaos S. Papageorgiou

Archivum Mathematicum (1999)

  • Volume: 035, Issue: 3, page 255-274
  • ISSN: 0044-8753

Abstract

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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 φ and of a lower solution ψ such that ψ ≤ φ , and using the theory of nonlinear operators of monotone type, we show that there exists a solution x ∈ [ ψ , φ ] and that the set of all such solutions is compact in W p q ( T ) . For the problem with a Caratheodory right hand side we show the existence of extremal solutions in [ ψ , φ ] .

How to cite

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Halidias, Nikolaos, and Papageorgiou, Nikolaos S.. "Existence of solutions for nonlinear parabolic problems." Archivum Mathematicum 035.3 (1999): 255-274. <http://eudml.org/doc/248364>.

@article{Halidias1999,
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 \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 $[\psi ,\phi ]$.},
author = {Halidias, Nikolaos, Papageorgiou, Nikolaos S.},
journal = {Archivum Mathematicum},
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; upper and lower solutions; evolution triple; operator of type ; operator of type ; -pseudomonotone operator; multivalued problem; extremal solutions; Zorn's lemma},
language = {eng},
number = {3},
pages = {255-274},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Existence of solutions for nonlinear parabolic problems},
url = {http://eudml.org/doc/248364},
volume = {035},
year = {1999},
}

TY - JOUR
AU - Halidias, Nikolaos
AU - Papageorgiou, Nikolaos S.
TI - Existence of solutions for nonlinear parabolic problems
JO - Archivum Mathematicum
PY - 1999
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 035
IS - 3
SP - 255
EP - 274
AB - 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 $[\psi ,\phi ]$.
LA - eng
KW - 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; upper and lower solutions; evolution triple; operator of type ; operator of type ; -pseudomonotone operator; multivalued problem; extremal solutions; Zorn's lemma
UR - http://eudml.org/doc/248364
ER -

References

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