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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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