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Attractors for modulation equations on unbounded domains – existence and comparison. (English) Zbl 0833.35016
The authors study the long time behaviour of solutions for a certain class of evolutionary equations on unbounded domains, in particular, the Swift-Hohenberg and Ginzburg-Landau equations on the whole \(\mathbb{R}^N\). It is proved that the equations possess a global attractor \(\mathcal A\) belonging to a space of “locally bounded” functions with respect to a suitable Sobolev norm. The set \(\mathcal A\) attracts bounded sets in a weighted Sobolev norm with a weight decreasing to zero at infinity. Moreover, some estimates of the diameter of the attractor are given in terms of the parameters appearing in the equations. Finally, the dependence of the attractors on a parameter is investigated, namely, the upper semicontinuity of a family of attractors is shown.
Reviewer: E.Feireisl (Praha)

35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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