Measure-valued solutions and the phenomenon of blow-down in logarithmic diffusion
type: Item
instance of: scholarly article
title: Measure-valued solutions and the phenomenon of blow-down in logarithmic diffusion
zbMATH Open document ID: 1197.35153
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-04-14T00:00:00Z
review text: The author studies the existence, uniqueness and behavior of generalized solutions of an initial value problem for the equation \(u_t=\Delta\log(u+c)\), where \(x\in\mathbb R^2\), \(t>0\) and \(c\geq0\) is a constant. The initial value is a general nonnegative Radon measure of the form \(\mu(x)=\sum_{i=1}^\infty M_i\delta(x-x_i)+\mu_0(x)\), where \(\mu_0\) is non-atomic. In particular, the author shows that the solution becomes classical after a finite time and the singular part of the solution has the form \(u_{\text{sing}}(x,t)=\sum_i(M_i-4\pi t)_+\delta(x-x_i)\). The total mass of the solution is preserved if \(c>0\), while \(\int u(x,t)\,dx=(\mu(\mathbb R^2)-4\pi t)_+\) if \(c=0\).
reviewed by: Pavol Quittner
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 35K59
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 5543869
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: nonnegative Radon measure
author: Juan Luis Vázquez
MaRDI profile type: Publication
Wikidata QID: Q124817681
full work available at URL: j.jmaa.2008.06.032
OpenAlex ID: W2071838492
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Measure-valued solutions and the phenomenon of blow-down in logarithmic diffusion (DOI)
DOI: 10.1016/J.JMAA.2008.06.032