Cherfils, L.; Miranville, A. Generalized Cahn-Hilliard equations with a logarithmic free energy. (English) Zbl 1002.35062 Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 94, No. 1, 19-32 (2000). 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. Cited in 9 Documents MSC: 35K55 Nonlinear parabolic equations 35B41 Attractors 35K35 Initial-boundary value problems for higher-order parabolic equations Keywords:existence and uniqueness; finite-dimensional attractors PDFBibTeX XMLCite \textit{L. Cherfils} and \textit{A. Miranville}, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 94, No. 1, 19--32 (2000; Zbl 1002.35062)