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A Cahn-Hilliard model in a domain with non-permeable walls. (English) Zbl 1228.35051
Let $$\Omega$$ be a bounded domain of $$\mathbb R^3$$ with smooth boundary $$\Gamma$$ and consider the following Cahn-Hilliard equation
$\partial_t\rho - \Delta\mu = 0 , \quad \mu = -\Delta\rho + f(\rho) \quad\text{in }(0,\infty)\times \Omega$ with dynamic boundary conditions
$w\partial_t \rho - \delta \Delta_\Gamma \mu + \partial_n \mu = \partial_n\rho - \sigma\Delta_\Gamma \rho + g(\rho) - w\mu = 0 \quad \text{on } (0,\infty)\times\Gamma.$ Here $$\delta$$ and $$\sigma$$ are nonnegative real numbers, $$w$$ is a positive bounded function such that $$1/w$$ is bounded, $$\Delta_\Gamma$$ denotes the Laplace-Beltrami operator on $$\Gamma$$, $$f$$ is the derivative of a (possibly non-smooth) double-well potential and $$g$$ is a smooth function. The existence and uniqueness of a weak solution $$\rho$$ are proved for a rather general class of data $$f$$ and $$g$$. Regularity for positive times of the solution is investigated according to the growth of $$f$$ and $$g$$. Existence of a bounded absorbing set and a compact attractor is shown as well. Finally, if $$f$$ and $$g$$ are also assumed to be real analytic functions, convergence of the solution to a steady state is proved with the help of the now classical Łojasiewicz-Simon inequality.

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35K52 Initial-boundary value problems for higher-order parabolic systems 35B41 Attractors 80A22 Stefan problems, phase changes, etc.
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