一类带非局部项的Allen-Cahn方程解的存在性

The Existence of Solutions for a Class of Allen-Cahn Equations with Nonlocal Term

从图像处理中的修复、去噪与分割等问题出发,结合两相流的数学理论,提出了一类具有非局部项的Allen-Cahn方程的初边值问题,该方程与经典的Allen-Cahn方程相比,最大的不同之处是主部含有非局部项,然后利用Schauder不动点定理得到了带非局部项的Allen-Cahn方程初边值问题解的存在性.

In this paper,inspiring from the study on the reconstruction,denoising and segmentation in image processing,and combining with the theory of two phase fluids,the initial-boundary value problem for a class of Allen-Cahn equations with nonlocal term is introduced.Compared with the classical Allen-Cahn equation,this equation has the biggest difference in the principal part including the non-partial item.By using the Schauder fixed point theorem,the existence of weak solutions to the initial boundary value problem for the nonlocal Allen-Cahn equation is proved.