具有快速增长非线性项的Cahn-Hilliard方程的近似惯性流形

Approximate Inertial Manifold to the Cahn-Hilliard Equation with Fast Growing Nonlinearity

讨论了具有快速增长非线性项的Cahn Hilliard方程ut +ΓΔ2 u-ΔG(u) =0 ,G(u) = uΦ(u) , xunx∈ Ω = x(Δu)nx∈ Ω =0 ,u(0 ,x) =u0 (x)解的长时间行为 .构造了一个新系统 ,利用压缩映象原理 ,得到了该系统解的存在唯一性和一个m维光滑流形 ,即近似惯性流形 ,证明了Cahn

In this paper, the long-time behaviour of solutions of the Cahn-Hilliard equations with fast growing nonlinearityu t+ΓΔ 2u-ΔG(u)=0, G(u)= uΦ(u),  xun x∈Ω= x(Δu)n x∈Ω=0, u(0,x)=u 0(x),is considered. A new system is constructed. Existence and uniqueness of the solution of the new system and a smooth m-dimensional manifold (i.e. approximate inertial manifold) are given based on the contraction principle. It is proved that arbitrary trajectory of Cahn-Hilliard equation goes into a small neighbourhood of such manifold after large time.