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Nonuniqueness and multi-bump solutions in parabolic problems with the p-Laplacian

Citace:
BENEDIKT, J., GIRG, P., KOTRLA, L., TAKÁČ, P. Nonuniqueness and multi-bump solutions in parabolic problems with the p-Laplacian. JOURNAL OF DIFFERENTIAL EQUATIONS, 2016, roč. 260, č. 2, s. 991?1009. ISSN: 0022-0396
Druh: ČLÁNEK
Jazyk publikace: eng
Anglický název: Nonuniqueness and multi-bump solutions in parabolic problems with the p-Laplacian
Rok vydání: 2016
Autoři: RNDr. Jiří Benedikt Ph.D. , Doc. Ing. Petr Girg Ph.D. , Ing. Lukáš Kotrla , Prof. Dr. Peter Takáč Ph.D.
Abstrakt CZ: V článku je studována platnost srovnávacího principu pro degenerované parabolické rovnice s p-laplaciánem. Jsou nalezeny protipříklady k tomuto principu. Je použita metoda horních a dolních řešení a speciální konstrukce horních řešení pomocí funkcí Barenblattova typu.
Abstrakt EN: The validity of the weak and strong comparison principles for degenerate parabolic partial differential equations with the p-Laplace operator is investigated for p>2. This problem is reduced to the comparison of the trivial solution (?0, by hypothesis) with a nontrivial nonnegative solution u(x,t). The problem is closely related also to the question of uniqueness of a nonnegative solution via the weak comparison principle. In this article, realistic counterexamples to the uniqueness of a nonnegative solution, the weak comparison principle, and the strong maximum principle are constructed with a nonsmooth reaction function that satisfies neither a Lipschitz nor an Osgood standard ?uniqueness? condition. Nonnegative multi-bump solutions with spatially disconnected compact supports and zero initial data are constructed between sub- and supersolutions that have supports of the same type.
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