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On the Cahn-Hilliard equation with degenerate mobility. (English) Zbl 0856.35071
The authors consider the Cahn-Hilliard equation with a concentration dependent diffusional mobility \(B(u)\): \[ \partial u/\partial t= - \text{div } J,\;J= - B(u) \nabla w,\;w= -\gamma \Delta u+ \psi'(u),\;\gamma\in \mathbb{R}^+; \] the function \(\psi\) represents the homogeneous free energy. Under some assumptions on \(B(u)\) and \(\psi(u)\) it is proved the existence of a weak solution in \(\omega\times [0, T]\) (\(\omega\subset \mathbb{R}^n\) is a bounded domain), satisfying initial condition \(u(0)= u_0\in H^1(\omega)\) and boundary conditions \(\nu\cdot J= 0\), \(\nu\cdot \nabla u= 0\) on \(\partial\omega\times (0, T)\). The mobility is allowed to vanish for \(u= \pm 1\); it is proved that the solution is bounded in magnitude by one. Applications of the method to other degenerate forth-order parabolic equations are discussed.

35K65 Degenerate parabolic equations
35K35 Initial-boundary value problems for higher-order parabolic equations
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
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