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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 $ϵ{u}_{tt}$, where $ϵ\ge 0$ 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 ${ℳ}_{ϵ}$, whose basins of attraction are the whole phase space, and whose fractal dimension is bounded, uniformly in $ϵ$.
##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 35B25 Singular perturbations (PDE) 35B41 Attractors (PDE) 35G30 Boundary value problems for nonlinear higher-order PDE 74N99 Phase transformations in solids 82C24 Interface problems (dynamic and non-equilibrium); diffusion-limited aggregation