为研究四阶退化抛物方程解的存在性问题,需构建相应的半离散问题.研究与其相关的Dirichlet边界条件下,定态薄膜方程解的存在性,方法上,需将原有问题转化成二阶椭圆型方程组,利用Lax—Milgram定理,获得构造的不动点算子的可行性.再以紧嵌入定理为基础,应用Leray—Schauder不动点定理证得解的存在性.
In order to study the existence of solutions to the degenerated parabolic equation, it is necessary to construct a semi-discrete problem. The existence of a steady state thin film equation with Dirichlet boundary is studied, and the original problem is transformed into the system of second order elliptic equations. By using Lax-Milgram theorem, the fixed point operator is well defined. Moreover, the existence is obtained by applying the compact embedding theorem and Leray-Schauder' s fixed point theorem.