Electron. J. Diff. Equ., Vol. 2015 (2015), No. 38, pp. 1-7.

Nonuniqueness of solutions of initial-value problems for parabolic p-Laplacian

Jiri Benedikt, Vladimir E. Bobkov, Petr Girg, Lukas Kotrla, Peter Takac

Abstract:
We construct a positive solution to a quasilinear parabolic problem in a bounded spatial domain with the p-Laplacian and a nonsmooth reaction function. We obtain nonuniqueness for zero initial data. Our method is based on sub- and supersolutions and the weak comparison principle. Using the method of sub- and supersolutions we construct a positive solution to a quasilinear parabolic problem with the p-Laplacian and a reaction function that is non-Lipschitz on a part of the spatial domain. Thereby we obtain nonuniqueness for zero initial data.

Submitted January 29, 2015. Published February 10, 2015.
Math Subject Classifications: 35B05, 35B30, 35K15, 35K55, 35K65.
Key Words: Quasilinear parabolic equations with p-Laplacian; nonuniqueness for initial-boundary value problem; sub- and supersolutions; comparison principle.

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Jiri Benedikt
Department of Mathematics and NTIS
Faculty of Applied Scences, University of West Bohemia
Univerzitni 22, CZ-306 14 Plzen, Czech Republic
email: benedikt@kma.zcu.cz
Vladimir E. Bobkov
Fachbereich Mathematik
Universitat Rostock, Germany
email: vladimir.bobkov@uni-rostock.de
Petr Girg
Department of Mathematics and NTIS
Faculty of Applied Scences, University of West Bohemia
Univerzitni 22, CZ-306 14 Plzen, Czech Republic
email: pgirg@kma.zcu.cz
Lukas Kotrla
Department of Mathematics and NTIS
Faculty of Applied Scences, University of West Bohemia
Univerzitni 22, CZ-306 14 Plzen, Czech Republic
email: kotrla@ntis.zcu.cz
Peter Takac
Fachbereich Mathematik
Universitat Rostock
Germany
email: peter.takac@uni-rostock.de

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