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On the Cahn-Hilliard equation with a logarithmic free energy. (English) Zbl 0831.35088
Let \(\Omega \subset \mathbb{R}^n\) be a bounded domain with smooth boundary, \(\nu > 0\) and \(0 < \theta < \theta_c < \infty\). The authors show that \[ \partial_t u - \Delta \left[ \nu \Delta u - \theta_c u + {\theta \over 2} \ln {1 + u \over 1 - u} \right] = 0 \quad \text{on } \Omega \times (0, \infty),\;\partial_nu \equiv 0 \quad \text{on } \partial \Omega \times (0, \infty) \] induces a solution semiflow on \(\{\gamma \in L^\infty (\Omega) : |\gamma |\leq 1\), \(|\int_\Omega \gamma |< \text{meas} (\Omega)\}\) equipped with the \(L^2\)- topology and on a corresponding \(H^1\)-set. They also establish that this semiflow has a global attractor and give estimates for the dimension of that attractor. The case of periodic boundary conditions is treated at the same time.
Reviewer: G.Hetzer (Auburn)

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
47H20 Semigroups of nonlinear operators
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References:
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