SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 5155-5195, January 2020.
We consider a class of sixth-order Cahn--Hilliard-type equations with logarithmic potential. This system is closely connected to some important phase-field models relevant in different applications, for instance, the functionalized Cahn--Hilliard equation that describes phase separation in mixtures of amphiphilic molecules in solvent, and the Willmore regularization of the Cahn--Hilliard equation for anisotropic crystal and epitaxial growth. The singularity of the configuration potential guarantees that the solution always stays in the physically relevant domain [-1,1]. Meanwhile, the resulting system is characterized by some highly singular diffusion terms that make the mathematical analysis more involved. We prove existence and uniqueness of global weak solutions and show their parabolic regularization property for any positive time. In addition, we investigate long-time behavior of the system, proving existence of the global attractor for the associated dynamical process in a suitable complete metric space.

中文翻译：

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一类具有对数势的六阶Cahn-Hilliard型方程

SIAM数学分析杂志，第52卷，第5期，第5155-5195页，2020年1月。
我们考虑一类具有对数电势的六阶Cahn-Hilliard型方程。该系统与一些与不同应用相关的重要相场模型紧密相连，例如，描述溶剂中两亲分子混合物中相分离的功能化Cahn-Hilliard方程，以及Cahn-Hilliard方程的Willmore正则化用于各向异性晶体和外延生长。配置电位的奇异性确保解决方案始终停留在与物理相关的域中[-1,1]。同时，生成的系统的特征是一些高度奇异的扩散项，使数学分析更加复杂。我们证明了全局弱解的存在性和唯一性，并在任何积极的时间展示了它们的抛物线正则化性质。