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Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems. (English) Zbl 1088.37005
Summary: We suggest a new explicit algorithm allowing us to construct exponential attractors which are uniformly Hölder continuous with respect to the variation of the dynamical system in some natural large class. Moreover, we extend this construction to nonautonomous dynamical systems (dynamical processes) treating in that case the exponential attractor as a uniformly exponentially attracting, finite-dimensional and time-dependent set in the phase space. In particular, this result shows that, for a wide class of nonautonomous equations of mathematical physics, the limit dynamics remains finite-dimensional no matter how complicated the dependence of the external forces on time is. We illustrate the main results on the model example of a nonautonomous reaction-diffusion system in a bounded domain.
MSC:
37B55Nonautonomous dynamical systems
37L30Attractors and their dimensions, Lyapunov exponents
37C70Attractors and repellers, topological structure
35K57Reaction-diffusion equations
35B41Attractors (PDE)