Researches On The Solution Of Two-Dimensional Variable Coefficient Allen-Cahn Equation

Partial differential equations have played a crucial role in mathematical studies.In this paper,we mainly study a crucial class of second-order elliptic partial differential equations:Allen-Cahn equation.It is a classical nonlinear equation originating from the study of phase transformation of alloys.It has a wide range of applications in practical problems such as image processing,mean curvature motion,and crystal growth.In recent years,many scholars have devoted themselves to the study of the Allen-Cahn equation and have obtained a large number of research findings.In this paper,we mainly study some properties of the entire solutions of the Allen-Cahn equation on a two-dimensional plane.First of all,we tell about some fundamental theories that are related to the Allen-Cahn equation and the research results at home and abroad.Then,it is proved that a special 2k-end solution of the Allen-Cahn equation on the plane is non-degenerate in the function space with special symmetry.Finally,we introduce some theoretical knowledges about the Toda system and the exoression of Laplacian in the Fermi coordinate.The Infinite-dimensional Lyapunov-Schmidt reduction method is used to prove that the two-dimensional variable coefficient Allen-Cahn equation exists Multiple-end solution on the plane.