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On the hyperbolic relaxation of the one-dimensional Cahn–Hilliard equation. (English) Zbl 1160.35518
The authors consider the one-dimensional Cahn-Hilliard equation with an inertial term ϵu tt , where ϵ0 is small. It is known that this equation, endowed with proper boundary conditions, generates a strongly continuous semigroup which is dissipative. The first contribution is the proof of the existence of a connected global attractor. Then they show, under non restrictive condition, that the functions belonging to the attractor enjoy good smoothness properties. The main result is the construction of a robust family of exponential attractors ϵ , whose basins of attraction are the whole phase space, and whose fractal dimension is bounded, uniformly in ϵ.
MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35B25Singular perturbations (PDE)
35B41Attractors (PDE)
35G30Boundary value problems for nonlinear higher-order PDE
74N99Phase transformations in solids
82C24Interface problems (dynamic and non-equilibrium); diffusion-limited aggregation