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Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions. (English) Zbl 1068.35018
The paper contains the proof of the result announced in [Gakuto Int. Series; Math. Sci. Appl. 20, 382–390 (2004; Zbl 1068.35052)]. The initial boundary value problem is considered \begin{aligned} &\frac{\partial u}{\partial t}=\Delta\mu,\quad \mu=-\Delta u-u+u^3\quad t>0,\;x\in\Omega\\ &\frac{\partial \mu}{\partial \nu}=0,\quad \frac 1{\Gamma_s}\frac{\partial u}{\partial t}=\sigma_s\Delta_{\|}\,u-\frac{\partial u}{\partial \nu}+h_s-g_s u\quad t>0,\;x\in\partial\Omega\tag{1}\\ &u(x,0)=u_0(x), \quad x\in\Omega \end{aligned} where $$\Omega$$ is a bounded domain from $$\mathbb R^n$$, $$n\leq 3$$, $$\Gamma_s>0,\;\sigma_s>0,\;g_s>0$$, $$h_s$$ are given constants, $$\Delta_{\|}$$ is the tangential Laplacian operator, $$\nu$$ is the outward normal to $$\partial\Omega$$.
It is proved that $$\lim_{t\to\infty}\| u(\cdot,t)-\psi\|_{H^3(\Omega)}=0$$ where $$\psi$$ is an equilibrium to the problem (1) \begin{aligned} & -\Delta \psi-\psi+\psi^3=\text{const}\quad\;x\in\Omega\\ &\sigma_s\Delta_{\|}\,\psi-\frac{\partial \psi}{\partial \nu}+h_s-g_s \psi=0\quad t>0,\;x\in\partial\Omega\\ &\int_{\Omega}\psi(x)\,dx=\int_{\Omega}u_0(x)\,dx. \end{aligned}

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 37L30 Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 82C24 Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics 74N20 Dynamics of phase boundaries in solids 35K35 Initial-boundary value problems for higher-order parabolic equations
##### Keywords:
time asymptotic behavior; Simon-Lojasiewicz inequality
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##### References:
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