On the hyperbolic relaxation of the one-dimensional Cahn–Hilliard equation

Abstract

We consider the one-dimensional Cahn-Hilliard equation with an inertial term εu tt , for ε 0. This equation, endowed with proper boundary conditions, generates a strongly continuous semigroup S ε (t) which acts on a suitable phase-space and possesses a global attractor. Our main result is the construction of a robust family of exponential attractors {M ε }, whose common basins of attraction are the whole phase-space.  2005 Elsevier Inc. All rights reserved.

Singularly perturbed 1D Cahn–Hilliard equation revisited

“…We now report a well-posedness result for P which is proven in [7] (see also [22] and [46] Thanks to Theorem 2.2, we can define the semigroup…”

“…where ρ(t) is the solution to P at time t. Moreover, arguing as in [22], for every ∈ (0, 0 ], we can prove the existence of the global attractor A ,δ for S (t) on K δ that is bounded in K 4 δ . In order to compare the dynamics of problems (2.1) and (2.2) (i.e., P ), we introduce the canonical lifting of the exponential attractor in the unperturbed case (see [30]; cf.…”

“…Then, arguing as in [22], we can prove that B 5,δ is an exponentially attracting set in K δ . Thus we can apply Theorem 5.1 with B = B 5,δ and we can extend the basin of attraction to the whole phase-space by using the transitivity of the exponential attraction.…”

“…This set is exponentially attracting in H 0 as well (that is, any bounded set B ⊂ H 0 is attracted exponentially to B 5 with respect to metric induced by the H 0 −norm), due to the transitivity of exponential attraction property (see [22,Lemma 4.4]), first devised in [14, Theorem 5.1]. Then, it was constructed a robust family of exponential attractors M on B 5 with respect to the metric induced by the H 0 −norm, whose common basin of attraction is the whole phase-space H 0 , with a uniform upper bound on the fractal dimension and also a uniform rate of attraction of trajectories.…”

“…We recall that inertial manifolds and their dependence on were also analyzed in [2,3] in the viscous case. The existence a family of robust exponential attractors with no restriction on was proven in [22] by using the new construction mentioned above which is no longer based on the squeezing property (see also [23] for a version with memory). However, in [22], the convergence of the exponential attractors, as goes to 0, was obtained with respect to a norm that depends on .…”