Multidimensional stability of V-shaped traveling fronts in the Allen-Cahn equation  

作　　者：SHENG WeiJie LI WanTong WANG ZhiCheng 

机构地区：[1]Natural Science Research Center Harbin Institute of Technology [2]School of Mathematics and Statistics, Lanzhou University

出　　处：《Science China Mathematics》2013年第10期1969-1982,共14页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.11031003,11271172 and 11071105);the Fundamental Research Funds for the Central Universities(Grant No.HIT.NSRIF.2014063);China Postdoctoral Science Foundation Funded Project(Grant No.2012M520716);Heilongjiang Postdoctoral Fund(Grant No.LBH-Z12135);New Century Excellent Talents in University(Grant No.NCET-10-0470)

摘　　要：This paper is concerned with the multidimensional asymptotic stability of V-shaped traveling fronts in the Allen-Cahn equation under spatial decaying initial values. We first show that V-shaped traveling fronts are asymptotically stable under the perturbations that decay at infinity. Then we further show that there exists a solution that oscillates permanently between two V-shaped traveling fronts, which indicates that V-shaped traveling fronts are not always asymptotically stable under general bounded perturbations. Our main technique is the supersolutions and subsolutions method coupled with the comparison principle.This paper is concerned with the multidimensional asymptotic stability of V-shaped traveling fronts in the Allen-Cahn equation under spatial decaying initial values.We frst show that V-shaped traveling fronts are asymptotically stable under the perturbations that decay at infnity.Then we further show that there exists a solution that oscillates permanently between two V-shaped traveling fronts,which indicates that V-shaped traveling fronts are not always asymptotically stable under general bounded perturbations.Our main technique is the supersolutions and subsolutions method coupled with the comparison principle.

关 键 词：Allen-Cahn equation asymptotic stability MULTIDIMENSIONAL V-SHAPED traveling front 

分 类 号：O175[理学—数学]