Variational approach to complicated similarity solutions pf higher-order nonlinear PDEs. I

2009/02/09 by Victor A. Galaktionov, V. A. Galaktionov, Galaktionov, V. A. +6
Mathematics · #35K40 #35K55 #35K65 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Numerical methods for differential equations #math.AP #msc:35K40 #msc:35K55 #msc:35K65

paper · pdf · doi:10.48550/arxiv.0902.1425

41 pages, 27 figures

arxiv created 2009/02/09 · openalex publication_date 2009/02/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Three classes of higher-order nonlinear parabolic hyperbolic, and nonlinear dispersion equations are shown to admit exact blow-up or compacton solutions, which are induced by elliptic equations with non-Lipschitz nonlinearities. Variational techniques give countable families of various compactly supported solutions with oscllatory behaviour cloase to the interfaces. Overall, the whole set of such compact patterns reveals certain chaotic properties and contains solutions of arbitrary complexity.

Citations

• Three tupes of self-similar blow-up for the fourth-order p-Laplacian equaiton with source: variational and branching approaches
• Third-order nonlinear dispersion PDEs: shocks, rarefaction, and blow-up waves

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• Multi-scale Analysis of Nonlinear Equations
• Self-similar blow-up in parabolic equations of Monge--Ampère type