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Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions. (English) Zbl 1068.35020
The initial boundary value problem is considered
\[ \begin{aligned} &\frac{\partial u}{\partial t}=\Delta\mu,\quad \mu=-\Delta u+\varepsilon\frac{\partial u}{\partial t}+f(u),\quad t\in(0,T),\;x\in\Omega,\\ &\frac{\partial \mu}{\partial \nu}=0,\quad \frac{\partial u}{\partial t}=\Delta_{\|}\,u-\frac{\partial u}{\partial \nu}-\lambda u -g( u),\quad t\in(0,T),\;x\in\partial\Omega,\\ &u(x,0)=u_0(x), \quad x\in\Omega, \end{aligned} \] where \(\Omega\) is a bounded domain from \(\mathbb R^3\) with smooth boundary, \(\varepsilon,\;\lambda\) are given positive constants, \(\Delta_{\|}\) is the Laplace-Beltrami operator on the \(\partial\Omega\), \(\nu\) is the outward normal to \(\partial\Omega\), \(f\) and \(g\) are given nonlinear functions.
The existence and uniqueness of the solution to the problem is obtained by the Leray-Schauder principle. A robust family of exponential attractors are constructed.

MSC:
35B41 Attractors
35K55 Nonlinear parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K35 Initial-boundary value problems for higher-order parabolic equations
74N20 Dynamics of phase boundaries in solids
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