Maximal regularity and asymptotic behavior of solutions for the Cahn–Hilliard equation with dynamic boundary conditions

AbstractThis paper deals with the Cahn–Hilliard equation $$\psi_t=\Delta \mu, \quad \mu = -\Delta \psi-\psi+\psi^3,\quad (t, x)\in J\times \Omega,$$ subject to the boundary conditions$$\frac{1}{\Gamma_s}\psi_t = \sigma_s \Delta_{||} \psi-\partial_\nu \psi - g_s \psi +h,\quad \partial_\nu \mu=0,$$ and the initial condition ψ(0,x) = ψ0(x) where J = (0,∞), and Ω ⊂ ℝn is a bounded domain with smooth boundary Γ = ∂ G, n≤ 3, and Γs,σs,gs > 0, h are constants.This problem has already been considered in the recent paper of R. Racke and S. Zheng (The Cahn–Hilliard equation with dynamic boundary conditions. Adv. Diff. Eq. 8, 83–110, 2003), where global existence and uniqueness were obtained. In this paper we first obtain results on the maximal Lp-regularity of the solution. We then study the asymptotic behavior of the solution of this problem and prove the existence of a global attractor.