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Maximal regularity and asymptotic behavior of solutions for the Cahn-Hilliard equation with dynamic boundary conditions. (English) Zbl 1232.35081
Summary: This 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 \(\psi (0,x) = \psi _{0}(x)\) where \(J = (0,\infty )\), and \(\Omega \subset \mathbb R^{ n }\) is a bounded domain with smooth boundary \(\Gamma = \partial G\), \(n\leq 3\), and \(\Gamma _{ s },\sigma _{ s },g _{ s } > 0\), \(h\) are constants. This problem has already been considered in the recent paper by the two last authors [Adv. Diff. Eq. 8, 83–110 (2003; Zbl 1035.35050)], where global existence and uniqueness were obtained. In this paper we first obtain results on the maximal \(L _{ p }\)-regularity of the solution. We then study the asymptotic behavior of the solution of this problem and prove the existence of a global attractor.

MSC:
35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
35B41 Attractors
35B65 Smoothness and regularity of solutions to PDEs
35K57 Reaction-diffusion equations
37L30 Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
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