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The complex Ginzburg-Landau equation on large and unbounded domains: Sharper bounds and attractors. (English) Zbl 0905.35043
Let \(\Omega\subset{\mathbb R}^d\) be a smooth (possibly unbounded) domain. The paper deals with the study of the complex Ginzburg-Landau equation \[ u_t=(1+i\alpha)\Delta u+Ru-(1+i\beta)| u| ^{2}u\qquad\text{in \(\Omega \times (0,\infty)\)}. \] Using an appropriate weighted \(L^p\)-space the author obtains new bounds on the long-time behaviour of solutions. These a priori estimates are essentially independent of the underlying domain, and they improve previous bounds related to a polynomial growth with respect to the instability parameter. The author’s analysis includes the standard case where \(u\) is periodic of period 1 with respect to each coordinate. There are also established sharp estimates on the maximal influence of the boundaries on the dynamics in the interior. This enables the author to prove the existence of global attractors and to compare two attractors associated to two semigroups on different domains in the case of a large joint domain. Moreover, every orbit in one of the attractors can be approximated by a pseudo-orbit inside the other attractor.
The paper is of significant relevance in the study of dynamics of parabolic equations on large or unbounded domains.

MSC:
35K55 Nonlinear parabolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B45 A priori estimates in context of PDEs
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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