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

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• 作者：Jiří ; Benedikt^a ; ^{benedikt@kma.zcu.cz" class="auth_mail" title="E-mail the corresponding author} ; Petr Girg^a ; ^{pgirg@kma.zcu.cz" class="auth_mail" title="E-mail the corresponding author} ; Luká ; &scaron ; Kotrla^a ; ^{kotrla@kma.zcu.cz" class="auth_mail" title="E-mail the corresponding author} ; Peter Taká ; č^b ; ^{peter.takac@uni-rostock.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35B05 ; 35B30 ; 35K15 ; 35K55 ; 35K65
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 January 2016
• 年：2016
• 卷：260
• 期：2
• 页码：991-1009
• 全文大小：381 K

文摘

The validity of the weak and strong comparison principles for degenerate parabolic partial differential equations with the p  -Laplace operator Δ_p${\mathrm{\Delta }}_p$ is investigated for p>2$p>2$. This problem is reduced to the comparison of the trivial solution (≡0, by hypothesis) with a nontrivial nonnegative solution 05a20" title="Click to view the MathML source">u(x,t)$u(x,t)$. 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.