非线性扩散作用下一类四阶抛物方程解研究

The Research of a Class of Fourth Order Parabolic Equation Under the Action of Nonlinear Diffusion

主要研究相变理论及薄膜润滑理论中出现的一类四阶退化抛物方程,函数及二阶拉普拉斯算子作用下在边界上为0,初始时间为已知函数.通过对时间的半离散,依据椭圆型方程解的存在性,构造逼近解,进而获得相应的抛物方程解的存在性及唯一性.方法上,依赖于对逼近解做半离散迭代估计、能量估计以及紧性讨论.

This paper mainly studies a fourth-order degenerate parabolic equation in the field of phase transformation and thin film lubrication with the zero boundary conditions for the unknown function itself and the function under the Laplace operator. The initial condition is a known function. By the semi-discrete method with respect to the time variable,the existence of the elliptic equations and constructing approximate solutions,the corresponding existence and uniqueness of the parabolic equations are obtained. The method relies on semidiscrete iterative estimates,energy estimates and compactness arguments.