带状区域中渐近周期曲率流方程的整体解

该文研究了具有渐近周期系数的曲率流方程的Neumann边值问题.首先,考虑一列初值问题及其相应的全局解,通过一致的先验估计取一个收敛子列,得到其极限就是一个整体解的结论.其次,向负无穷时间方向进行重整化,使用强极值原理证明了整体解的唯一性.最后,为了研究整体解的ω-和α-极限,再次使用重整化方法,通过构造拉回函数、进行一致的先验估计以及Cantor对角化方法取收敛子列,得到整体解的ω-和α-极限都是极限问题的整体解,即它们都是周期行波的结论.

Global Solutions of the Asymptotically Periodic Curvature Flow Equations in Band Domains

The curvature flow equations were studied with Neumann boundary conditions and asymptotically periodic coefficients. First, a series of initial value problems and corresponding global solutions were considered. By uniform prior estimates, a subsequence converging to the global solution was obtained. Second, the uniqueness of the global solution was proved with the renormalization method in the direction of negative infinite time and the strong maximum principle. Finally, to study the ω-limit and α-limit of the global solution, the renormalization method was used again. Through the construction of the pullback function, the uniform prior estimation and the convergent subsequence with the Cantor diagonalization method, it is shown that, the ω-limit and α-limit of global solutions are the global solutions of the corresponding limit problems, i.e., they both are periodic traveling waves.