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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 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
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

Citations:

Zbl 1068.35052
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References:

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