Existence and non-existence of a fast diffusion equation in \(\mathbb{R}^ n\)
type: Item
instance of: scholarly article
title: Existence and non-existence of a fast diffusion equation in \(\mathbb{R}^ n\)
zbMATH Open document ID: 0876.35062
published in: Journal of Differential Equations
publication date: 1997-11-02T00:00:00Z
review text: For the Cauchy problem to a fast diffusion equation of the form \(u_t = \Delta \log u, x\in \mathbb{R}^n, t>0\) the author studies the global existence, finite time extinction, and asymptotic behaviour of solutions. The main results are summarized as follows: Let \(n\geq 3\) then (a) there exists a non-trivial global solution \(u\) of the Cauchy problem which has the property that \(u(\cdot,t) \in L^p(\mathbb{R}^n),\) for any \(p>n/2\) and for all \(t>0\) but it does not belong to any \(L^p(\mathbb{R}^n)\) if \(p\leq n/2;\) (b) for any \(T>0,\) there exists a solution \(u(\cdot,t)\) which has a finite maximal existence time \(T\) and such that for any \(p>n/2, u(\cdot,t)\in L^p(\mathbb{R}^n), 0<t<T\); moreover, \(u\) becomes extinct at \(t=T,\) i.e., \(u(\cdot,T)\equiv 0;\) (c) any non-trivial solution \(u\) satisfies \(\int_{R^n}u^p(x,t)dx = \infty, 0<t<t_{\max},\) where \(1<p\leq n/2,\) and \(t_{\max}>0\) is the maximal existence time.
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 1025925
zbMATH Keywords: global existence
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: finite extinction
author: Yuanwei Qi
reviewed by: Vladimir N. Grebenev
MaRDI profile type: Publication
full work available at URL: jdeq.1996.3246
OpenAlex ID: W2048347051
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DOI: 10.1006/JDEQ.1996.3246
description: scientific article; zbMATH DE number 1025925