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Bi-spaces global attractors in abstract parabolic equations. (English) Zbl 1024.35058
Picard, Rainer (ed.) et al., Evolution equations. Propagation phenomena, global existence, influence on non-linearities. Based on the workshop, Warsaw, Poland, July 1-July 7, 2001. Warsaw: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 60, 13-26 (2003).
Summary: An abstract semilinear parabolic equation in a Banach space \(X\) is considered. Under general assumptions on nonlinearity this problem is shown to generate a bounded dissipative semigroup on \(X^\alpha\). This semigroup possesses an \((X^\alpha-Z)\)-global attractor \(\mathcal A\) that is closed, bounded, invariant in \(X^\alpha\), and attracts bounded subsets of \(X^\alpha\) in a ‘weaker’ topology of an auxiliary Banach space \(Z\). The abstract approach is finally applied to the scalar parabolic equation in \(\mathbb{R}^n\) and to the partly dissipative system.
For the entire collection see [Zbl 1012.00020].

35K90 Abstract parabolic equations
35B41 Attractors
35K15 Initial value problems for second-order parabolic equations
35K45 Initial value problems for second-order parabolic systems
35B40 Asymptotic behavior of solutions to PDEs
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