Attractors for modulation equations on unbounded domains-existence and comparison

Abstract

We are interested in the long-time behaviour of nonlinear parabolic PDEs defined on unbounded cylindrical domains. For dissipative systems defined on bounded domains, the longtime behaviour can often be described by the dynamics in their finite-dimensional attractors. For systems defined on the infinite line, very little is known at present, since the lack of compactness prevents application of the standard existence theory for attractors. We develop an abstract theorem based on the interaction of a uniform and a localizing (weighted) norm which allows us to define global attractors for some dissipative problems on unbounded domains such as the Swift-Hohenberg and the Ginzburg-Landau equation. The second aim of this paper is the comparison of attractors. The so-called Ginzburg-Landau formalism allows us to approximate solutions of weakly unstable systems which exhibit modulated periodic patterns. Here we show that the attractor of the Swift-Hohenberg equation is upper semicontinuous in a particular limit to the attractor of the associated Ginzburg-Landau equation.