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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.

##### MSC:
 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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