WIAS Preprint No. 634, (2001)

The finite dimensional attractor for a 4th order system of Cahn-Hilliard type with a supercritical nonlinearity



Authors

  • Efendiev, Messoud A.
  • Gajewski, Herbert
  • Zelik, Sergei

2010 Mathematics Subject Classification

  • 35B40 35B45

Keywords

  • Cahn-Hilliard system, global attractors, fractal dimension

Abstract

The paper is devoted to study the long-time behaviour of solutions of the following 4th order parabolic system in a bounded smooth domain Ω ⊂ ⊂ ℝn:

(1) b∂tu = - Δxu(aΔxu - α∂tu - ƒ(u) + g̃),

where u = (u1,...uk) is an unknown vector-valued function, a and b are given constant matrices such that a + a* > 0, b = b* > 0, α > 0 is a positive number, and ƒ and g are given functions. Note that the nonlinearity ƒ is not assumed to be subordinated to the Laplacian. The existence of a finite dimensional global attractor for the system (1) is proved under some natural assumptions on the nonlinear term ƒ.

Appeared in

  • Adv. Differential Equations, 7 (2002) pp. 1073--1100.

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