## 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].

### MSC:

 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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