## Low-dimensional behavior of the pattern formation Cahn-Hilliard equation.(English)Zbl 0581.35041

Trends in the theory and practice of non-linear analysis, Proc. 6th Int. Conf., Arlington/Tex. 1984, North-Holland Math. Stud. 110, 323-336 (1985).
[For the entire collection see Zbl 0564.00006.]
We investigate the fourth-Hilliard parabolic partial differential equation which describes pattern formation in phase transition. Neumann and periodic boundary conditions are considered for a domain in $$R^ n$$, $$1\leq n\leq 3$$. This equation is characterized by a negative (backward) second order diffusion and multiple steady states for the appropriate range of parameters. We establish compactness of the orbits in $$H^ 1(\Omega)$$ and convergence to some steady state. We demonstrate that the Cohn-Hilliard equation admits an intrinsic low dimensional behavior: in $$R^ 1$$, the number of determining modes (in a Galerkin expansion) is proportional to $$L^{3/2}$$; where $$L$$, the diameter of the domain, is also proportional to the number of unstable modes for the linearized equation. Similar results hold for $$n=2,3$$.

### MSC:

 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35K20 Initial-boundary value problems for second-order parabolic equations

Zbl 0564.00006