Existence and nonexistence of global solutions to fast diffusions with source
type: Item
instance of: scholarly article
title: Existence and nonexistence of global solutions to fast diffusions with source
zbMATH Open document ID: 0832.35083
author: K. Mochizuki
author: Kentaro Mukai
published in: Methods and Applications of Analysis
publication date: 1995-09-20T00:00:00Z
review text: The authors consider positive solutions of the Cauchy problem \[ \partial_t u= \Delta u^m+ u^p,\quad (x, t)\in \mathbb{R}^N\times (0, T),\tag{1} \] \[ u(x, 0)= u_0(x),\quad x\in \mathbb{R}^N,\tag{2} \] where \(\max\{0,1- 2/N\}< m< 1< p\) and \(u_0(x)\geq 0\). The main result of the authors reflects the fact that if \(p\leq m+ 2/N\) then all positive solutions of (1), (2) blow up in a finite time, and if \(p> m+ 2/N\) for sufficiently small \(u_0(x)\), the Cauchy problem (1), (2) has a global solution. The proofs are based on the comparison principle for parabolic equations, construction of supersolutions and some results of A. Friedman and S. Kamin.
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 798946
zbMATH Keywords: fast diffusions
zbMATH Keywords: blow up
zbMATH Keywords: global solution
reviewed by: Emil Minchev
MaRDI profile type: Publication
DOI: 10.4310/MAA.1995.V2.N1.A6
description: scientific article; zbMATH DE number 798946