Attractors for parabolic problems with nonlinear boundary conditions
type: Item
instance of: scholarly article
title: Attractors for parabolic problems with nonlinear boundary conditions
zbMATH Open document ID: 0876.35059
author: S. M. Oliva
author: Anibal Rodriguez-Bernal
author: A. N. Carvalho
author: Antônio L. Pereira
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-11-25T00:00:00Z
review text: The authors investigate the long time behaviour of solutions of reaction diffusion systems of the following structure: \[ u_t-\text{div}(a(x)\nabla u)+ \sum B_j(x)\partial_ju+ \lambda u+ f(u)=0\text{ in }\Omega,\quad {\partial u\over\partial n_a}+ g(u)=0\text{ on }\partial\Omega.\tag{1} \] Here \(u=(u_1,\dots,u_m)\), \(m\geq 1\), \(a(x)=\text{diag}(a_1(x),\dots,a_m(x))\), \(a_i\in C^1(\overline\Omega)\), \(a_i(x)\geq m_0>0\). Moreover, \(\partial u/\partial n_a=\langle a\nabla u,n\rangle\) with \(n\) the outward unit normal to \(\partial\Omega\) and \(B_j=\text{diag}(b^1_j,\dots,b^m_j)\) continuous on \(\overline\Omega\) \((j=1,\dots,N)\). Moreover, \(\Omega\subset\mathbb{R}^N\) is a smooth, bounded domain, \(\lambda\in\mathbb{R}\), and \(f=(f_1,\dots,f_m)\), \(g=(g_1,\dots,g_m)\) are nonlinearities subject to suitable growth conditions, stated in the course of the arguments. In case \(N=3\), \(f\), \(g\) have to satisfy among others \[ |f(u)- f(v)|\leq L(1+|u|^2+ |v|^2)|u-v|,\;|g(u)- g(v)|\leq L(1+|u|^{1+p}+ |v|^{1+p})|u-v|,\;u,v\in\mathbb{R}^m, \] supplied by a similar condition for the derivative \(Dg(u)\); here \(p\in [0,1]\). In addition, certain sign conditions are imposed on \(f,g\) and their introduction justified. In order to handle equation (1), i.e. to take care of the nonlinear boundary conditions, the authors treat (1) not as an equation on \(L^2(\Omega)\) but, after suitable reinterpretation, on \(H^{-1}(\Omega)\), what allows them to subsume the nonlinear boundary conditions into an abstract parabolic evolution equation of type: \[ u_t+ Au+ h(u)=0, \qquad u(0)= u_0\in X^\alpha.\tag{2} \] Here \(X\) is a Hilbert space, \(A\) a sectorial operator and \(h(\cdot)\) a mapping from the fractional power space \(X^\alpha\) to \(X^\beta\) which is assumed to be locally Lipschitz; thereby \(0\leq\alpha-\beta< 1\). Section 3 contains an extended discussion of the conditions which have to be satisfied so that system (1) can be cast into the form (2). The advantag…
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 1005688
zbMATH Keywords: abstract parabolic evolution equation
zbMATH Keywords: sectorial operator
zbMATH Keywords: local existence and uniqueness
zbMATH Keywords: global attractor
Wikidata QID: Q60221683
reviewed by: Bruno Scarpellini
MaRDI profile type: Publication
OpenAlex ID: W1995861146
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DOI: 10.1006/JMAA.1997.5282
description: scientific article; zbMATH DE number 1005688