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Generalized Cahn-Hilliard equations with a logarithmic free energy. (English) Zbl 1002.35062

Summary: Our aim is to study some models of generalized Cahn-Hilliard equations of the type \[ \begin{cases} {\partial\over\partial t}\bigl( u-\beta\text{ div}(B_0 \nabla u)\bigr)+ \alpha \text{div}(B_0\nabla \Delta u)-\text{div}\bigl(B_0 \nabla f'(u)\bigr)+ \text{div}(B \nabla\gamma)=0,\\ u_{ |t=0}=u_0, \end{cases} \] where \(d>0\), \(\beta>0\). We obtain the existence and uniqueness of solutions and then study the existence of finite-dimensional attractors in the case of logarithmic free energy. We shall assume that the mobility tensor \(B_0\in {\mathcal M}_n(\mathbb{R})\) is symmetric, strictly positive and has constant coefficients. The equation is posed in \(\Omega\times \mathbb{R}^+\), where \(\Omega=\prod^n_{i=1}]0\), \(L_i[\), \(L_i>0\), \(n=1,2\), or 3. The function \(\gamma= \gamma(x)\) is assumed to be regular enough and \(\Omega\)-periodic. We consider periodic boundary conditions, i.e. \(u\) is \(\Omega\)-periodic.

MSC:

35K55 Nonlinear parabolic equations
35B41 Attractors
35K35 Initial-boundary value problems for higher-order parabolic equations
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