Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity
type: Item
instance of: scholarly article
title: Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity
zbMATH Open document ID: 1358.35063
author: Shanming Ji
author: Jingxue Yin
author: Yang Cao
published in: Journal of Differential Equations
publication date: 2016-10-17T00:00:00Z
reviewed by: Georg Hetzer
Mathematics Subject Classification ID: 35K70
Mathematics Subject Classification ID: 35K58
Mathematics Subject Classification ID: 35B10
zbMATH DE Number: 6639797
zbMATH Keywords: pseudo-parabolic equation
zbMATH Keywords: logarithmic nonlinearity
zbMATH Keywords: periodic solutions
zbMATH Keywords: instability
MaRDI profile type: Publication
full work available at URL: j.jde.2016.08.017
OpenAlex ID: W2507597187
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DOI: 10.1016/J.JDE.2016.08.017
review text: The authors investigate the existence and stability of \(T\)-periodic solutions of NEWLINE\[NEWLINEu_t-k\partial_t \Delta u=m(t)u\log(\left|u\right|),\quad x\in\Omega,\;t\in\mathbb{R}; \qquad u(x,t)=0, \quad x\in\partial\Omega,\;t\in\mathbb{R}NEWLINE\]NEWLINE under the following assumptions: \(T>0\), \(k\in\{0,1\}\) (\(k=1\) in pseudo-parabolic case), \(\Omega\subset \mathbb{R}^n\) bounded domain with smooth boundary, \(m\) is a smooth, positive and T -periodic function. They find that existence results are similar to those for right hand sides \(m(t)u^p\), \(p>1\), whereas there are differences between the instability results for the linear case, the `logarithmic case', and the `power case'. The existence proofs rely on regularity results, a priori estimates, and topological degree arguments; the stability results use the comparison method.
description: scientific article; zbMATH DE number 6639797