On the Cauchy problem for the evolution \(p\)-Laplacian equations with gradient term and source
type: Item
instance of: scholarly article
title: On the Cauchy problem for the evolution \(p\)-Laplacian equations with gradient term and source
zbMATH Open document ID: 1128.35059
author: Hongjun Yuan
author: Chunling Cao
author: Xiaojing Xu
author: Songzhe Lian
author: Wenjie Gao
published in: Journal of Differential Equations
publication date: 2007-04-26T00:00:00Z
review text: The authors study local and global existence of solutions of the Cauchy problem for the equation \[ u_t=\text{div}\,(| Du| ^{p-2}Du)+| u| ^{q-1}u-\lambda| Du| ^l,\qquad x\in\mathbb R^N,\;t>0, \] where \(p>2\), \(q\geq1\), \(l\geq1\) and \(\lambda\in\mathbb R\). They prove local existence and uniqueness of solutions for suitable class of initial data provided \(l\leq p-1\) and \(l\geq p/2\), respectively. If \(p/2\leq l\leq p-1<q<p-1+p/N\) then positive solutions with large initial data blow up in finite time. On the other hand, if \(l\leq p-1\) and \(q>p-1+p/N\) then positive solutions with small initial data exist globally.
reviewed by: Pavol Quittner
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 5146783
zbMATH Keywords: blow-up
zbMATH Keywords: global existence
zbMATH Keywords: large initial data
zbMATH Keywords: small initial data
MaRDI profile type: Publication
full work available at URL: j.jde.2006.11.014
OpenAlex ID: W1997306252
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DOI: 10.1016/J.JDE.2006.11.014
description: scientific article; zbMATH DE number 5146783