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Exponential attractors for a phase-field model with memory and quadratic nonlinearities. (English) Zbl 1070.37056

The authors deal with long-time behaviour of solutions for a phase-field model with memory and quadratic nonlinearities. They show that this problem can be interpreted as infinite-dimensional dynamical systems. The main difficulty encountered in this paper concerns the construction of a “nice” compact invariant set for the semigroup representing the dynamical system, which absorbs sufficiently large sets of initial data. To the best of the authors knowledge this paper is the first example of existence of an exponential attractor for an infinite-dimensional dynamical system with memory. Note that when memory effects are present, the additional variable does not enjoy any regularising effect, so that the existence of a universal (exponential) attractor has to be proved by using a suitable decomposition.

MSC:

37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35B41 Attractors
74N99 Phase transformations in solids
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
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