摘要
研究一类定态四阶退化薄膜方程Dirichlet边值条件下的弱解存在性.通过方程变形、构造逼近方程及先验估计的方法,得到非退化问题弱解存在性及正性结果.利用截断方法、Leray-Schauder不动点定理以及Sobolev空间紧性结果,得到退化模型弱解的存在性.由于最大值原理和比较原理对于薄膜方程并不成立,故将方程变形为二阶椭圆型方程系统,进行能量估计并利用二阶偏微分方程已有理论克服这一困难.
The existence of a steady-state thin film equation is studied in one space dimension with Dirichlet boundary.Through the transmutation,approximate equation and a priori estimate,the existence and positivity of the non-degenerative problem are obtained.Applying truncation method,Leray-Schauder fixed point theorem and some compactness results for Sobolev space,the existence of the degenerative problem is received.Since the maximum principle and comparison principle do not hold for the thin film equation,the fourth order problem is transformed into a second order elliptic system.Therefore,the classic elliptic theory can be used well in order to overcome the difficulty.
出处
《大连交通大学学报》
CAS
2011年第2期101-104,共4页
Journal of Dalian Jiaotong University
基金
国家自然科学基金数学天元基金资助项目(11026128)
辽宁省教育厅高等学校科研计划资助项目(L2010075)
关键词
薄膜方程
椭圆型
存在性
定态
thin film equation
elliptic
existence
steady-state
