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Maximum and minimum solutions for nonlinear parabolic problems with discontinuities

Kandylakis Dimitrios, Papageorgiou Nikolaos S.

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Year 1998
Type of Item Peer-Reviewed Journal Publication
Bibliographic Citation D.A. Kandilakis, N.S. Papageorgiou, "Maximun and minimum solutions for nonlinear parabolic problems with discontinuities," Proceedings of the Indian Academy of Sciences - Mathematical Sciences, vol. 108, no. 2, pp. 179-187, Jun. 1998. doi: 10.1007/BF02841551
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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.