Maximum and minimum solutions for nonlinear parabolic problems with discontinuities |
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Authors: | Dimitrios A. Kandilakis Nikolaos S. Papageorgiou |
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Affiliation: | (1) Department of Mathematics, University of the Aegean, 83200 Karlovassi Samos, Greece;(2) Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece |
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Abstract: | In this paper we examine nonlinear parabolic problems with a discontinuous right hand side. Assuming the existence of an upper solution φ and a lower solution ψ such that ψ ≤ φ, we establish the existence of a maximum and a minimum solution in the order interval [ψ, φ]. Our approach does not consider the multivalued interpretation of the problem, but a weak one side “Lipschitz” condition on the discontinuous term. By employing a fixed point theorem for nondecreasing maps, we prove the existence of extremal solutions in [ψ, φ for the original single valued version of the problem. |
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Keywords: | Upper solution lower solution evolution triple compact embedding integration by parts Sobolev space regular cone |
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