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On the hyperbolic relaxation of the Cahn-Hilliard equation in 3D: approximation and long time behaviour. (English) Zbl 1131.35082
In Ω×(0,+), Ω is a bounded regular domain in 3 it is considered the mixed problem for the nonlinear partial differential equation εU tt +U tt -Δ(-ΔU+φ(U))=0; U(x,0)=U 0 (x), U t (x,0)=U 1 (x); U(x,t)=ΔU(x,t)=0, xΩ×(0,) describing the evolution of the relative concentration U of one component of a binary alloy system. The term of singular perturbation accounts the relaxation of the diffusion flux. The aim of this article is to prove a global in time existence result and the asymptotic stability from the point of view of global attractors. The arising here difficulties connected with the low regularity of weak solutions is overcoming by using of a density argument based on the Galerkin approximation scheme and Ball’s theory of generalized semiflows [J. M. Ball, J. Nonlinear Sci. 7, No. 5, 475–502 (1997); erratum ibid. 8, 233 (1998; Zbl 0903.58020)].

MSC:
35Q72Other PDE from mechanics (MSC2000)
35B41Attractors (PDE)
35B45A priori estimates for solutions of PDE
35L70Nonlinear second-order hyperbolic equations
37L30Attractors and their dimensions, Lyapunov exponents