Attractors and their stability on Boussinesq type equations with gentle dissipation
type: Item
instance of: scholarly article
title: Attractors and their stability on Boussinesq type equations with gentle dissipation
zbMATH Open document ID: 1409.35035
author: Zhijian Yang
author: Pengyan Ding
author: Xiao-Bin Liu
published in: Communications on Pure and Applied Analysis
publication date: 2018-12-20T00:00:00Z
review text: The authors consider the Boussinesq type equations \[ \begin{cases} u_{tt}+\Delta^2u+(-\Delta)^\alpha u_t-\Delta f(u)=g(x) \quad \text{in}\;\Omega\times\mathbb{R}^+, \\ u_{\mid\partial\Omega}=\Delta u_{\mid\partial\Omega}=0, \quad u(x,0)=u_0(x),\;u_t(x,0)=u_1(x). \end{cases} \] Notably, \(\alpha\in(0,1)\) indicates the so-called gentile dissipation regime. Careful effort is given to clarifying the critical exponent of the nonlinearity in terms of \(\alpha\) and the spatial dimension \(N\), for \(\Omega\subset\mathbb{R}^N\). Indeed, the critical exponent is \(p_\alpha:=\frac{N+2(2\alpha-1)}{N-2}\), and the admissible polynomial growth exponents range \(1\leq p<p_\alpha\). Also under consideration is the asymptotic behavior in terms of global attractors and the continuity properties of these attractors with respect to the parameter \(\alpha.\) The interested reader can read the case for \(\alpha\in[1,2)\) in [the first author, J. Math. Anal. Appl. 399, No. 1, 180--190 (2013; Zbl 1256.35094)] and [the first and second author, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 161, 108--130 (2017; Zbl 1469.35193)].
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 37L15
Mathematics Subject Classification ID: 35B33
zbMATH DE Number: 6996073
zbMATH Keywords: well-posedness
zbMATH Keywords: global attractor
zbMATH Keywords: exponential attractor
zbMATH Keywords: upper semicontinuity of attractors
reviewed by: Joseph Shomberg
MaRDI profile type: Publication
full work available at URL: cpaa.2019044
OpenAlex ID: W2896622492
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Wikidata QID: Q129037149
DOI: 10.3934/CPAA.2019044
description: scientific article; zbMATH DE number 6996073