The finite dimensional attractor for a 4th order system of Cahn-Hilliard type with a supercritical nonlinearity
- Efendiev, Messoud A.
- Gajewski, Herbert
- Zelik, Sergei
2010 Mathematics Subject Classification
- 35B40 35B45
- Cahn-Hilliard system, global attractors, fractal dimension
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.
- Adv. Differential Equations, 7 (2002) pp. 1073--1100.