Classification of razor blades to the filtration equation -- the sublinear case.
type: Item
instance of: scholarly article
title: Classification of razor blades to the filtration equation -- the sublinear case.
zbMATH Open document ID: 1043.35087
DOI: 10.1016/S0022-0396(02)00019-0
author: E. Chasseigne
published in: Journal of Differential Equations
publication date: 2003-03-18T00:00:00Z
review text: The author studies non-negative solutions in the sense of \(D'(\mathbb R^n \setminus \{0\} \times (0, + \infty)) \) of the equation \(u_t=\Delta \phi (u).\) Here \(n>2\) and \(\phi\) is bounded between two powers (that are smaller than 1) near the origin. The initial data are identically zero on \(\mathbb R^n \setminus\{0\}.\) A \textit{razor blade} is a solution which is not zero nor a fundamental solution. Depending on the behavior of \(\phi\) near \(u=0\), different situations are found: (A) non-existence of razor blades, (B) existence of razor blades that are for all times not locally integrable near the origin, (C) solutions that satisfy in \(D'(\mathbb R^n \times (0, + \infty))\) the equation \(u_t=\Delta \phi(u) +\mu\) with \(\mu\) a Radon measure supported on \(\{0\} \times (0, \tau)\) (this measure is attained as a trace, and called the axial trace) while having a standing singularity for times greater than \(\tau\). The existence of razor blades with a prescribed axial trace is shown, and that the razor blade is uniquely determined by its axial trace. The author then classifies the behavior in time of all possible (self-similar) razor blade solutions in the case \(\phi(u)=u^m.\)
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35A20
zbMATH DE Number: 1884216
zbMATH Keywords: fast diffusion equation
zbMATH Keywords: sublinear filtration
zbMATH Keywords: Borel measures
zbMATH Keywords: singular solutions
reviewed by: Marianne Korten
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1884216