四阶退化抛物方程若干问题

The research is devoted to studying some fourth order degenerate parabolic equations relating to the thin film equations and Cahn-Hilliard models, which have important applications in thin film lubrication theory, fluid mechanics, semiconductor and biology. For the research method, it is necessary to overcome the difficulties that the maximum principle and comparison principle do not work directly, and solve the intrinsical difficulties of the degenerate term and singular term as well. The contents and methods contain the following several aspects. 1. In order to study the existence, uniqueness, long time behavior, positivity and asymptotic limit of the fourth order degenerate parabolic equations with singular terms, the research will establish Bernis-type inequality to get the critical energy estimates of the singular terms and then obtain the existence and asymptotic property by constructing entropy functional。 2. The existence, uniqueness, long time behavior, positivity, finite speed of propagation, waiting time phenomenon and small parameter limit of solutions of the viscous or nonviscous fourth order degenerate parabolic equation will be studied by applying the entropy functional method to get necessary a prior estimates。 3.Use the semi-discrete technique, iteration method and compactness argument to study the existence, uniqueness, long time behavior and asymptotic stability of solutions of the nonlinear fourth order parabolic equations near some positive steady state solution.

本项目主要研究与薄膜模型、Cahn-Hilliard模型相关的几类四阶退化抛物型方程问题，其在薄膜润滑理论、流体力学、半导体以及生物学等方面具有重要应用。研究方法上，需克服最大值原理和比较原理不再直接有效，以及退化项和奇性项带来的本质困难。拟研究内容及方法包括：(1)研究具有奇性低阶项的四阶退化抛物方程解的存在唯一性、长时间行为、正性及渐近极限等问题，拟建立Bernis型不等式来解决关键的奇性项能量估计，进而通过构造熵泛函来获得存在性及渐近性质；(2)研究具幂函数形式退化项粘性或非粘性四阶退化抛物方程解的存在唯一性、长时间行为、正性、有限传播、等待时间现象、支集性质以及小参数渐近极限等问题，拟引入恰当的熵泛函，以获得所需先验估计；(3)研究非线性四阶抛物方程解在定态解附近正解的存在性、唯一性、长时间行为和渐近稳定性，拟利用半离散方法、迭代技术及紧性讨论。

本项目主要研究了薄膜润滑理论、流体力学、半导体以及生物学等方面出现的几类四阶退化抛物型方程问题，主要包括薄膜方程以及Cahn-Hilliard型方程。项目进行过程中，解决了如下问题：（1）一类具有奇性低阶项的四阶退化抛物方程解的存在性及解的性质等问题，方法上，利用Bernis型不等式解决了关键的奇性项能量估计；(2)具幂函数退化项的薄膜方程及粘性薄膜方程解的存在唯一性、长时间行为及正性等问题，方法上，引入了适当的熵泛函，以获得所需先验估计；（3）几类非线性四阶抛物方程解在某定态解附近正解的存在性、唯一性及长时间行为等问题，主要采用了半离散方法、迭代技术及紧性讨论。