演化折现Hamilton-Jacobi方程粘性解收敛问题的一个反例  

A counterexample on the convergence of the viscosity solutions of the evolutionary discounted Hamilton-Jacobi equation

作　　者：李霞 陈苏婷 LI Xia;CHEN Suting(School of Mathematical Sciences,SUST,Suzhou 215009,China)

机构地区：[1]苏州科技大学数学科学学院,江苏苏州215009

出　　处：《苏州科技大学学报（自然科学版）》2021年第4期10-15,共6页Journal of Suzhou University of Science and Technology（Natural Science Edition）

基　　金：国家自然科学基金资助项目(11971344)。

摘　　要：对演化Hamilton-Jacobi方程粘性解长时间渐近行为的研究是粘性解的一个重要方向。对于此类问题的研究,有基于变分法的弱KAM方法和PDE方法。当底空间紧时,Tonelli框架下演化Hamilton-Jacobi方程粘性解和演化接触Hamilton-Jacobi方程粘性解在t→+∞时是收敛的。首先,用PDE方法在较弱的条件下给出了非紧空间上演化折现Hamilton-Jacobi方程粘性解的一个表达式;然后,以此为基础给出了Tonelli框架下非紧空间上演化折现Hamilton-Jacob方程粘性解在t→+∞时收敛性的一个反例,此反例说明了空间的紧性对接触Hamilton-Jacobi方程粘性解在t→+∞时收敛性的影响。It’s an important research direction to study the long-time asymptotic behavior of the viscosity solution of the Hamilton-Jacobi equation.There are two methods for this study,the weak KAM method based on variational method and the PDE approach.When the domain was compact,the viscosity solutions of the evolutionary Hamilton-Jacobi equation and evolutionary contact Hamilton-Jacobi equation were convergent as t→+∞in the Tonelli framework.In this paper,we first gave an expression of the viscosity solution of the evolutionary Hamilton-Jacobi equation on non-compact space under weaker conditions with PDE approach.Then we provided a counterexample about the convergence of the viscosity solution of the evolutionary discounted Hamilton-Jacobi equation.This counterexample shows that the viscosity solution isn’t convergent in the non-compact domain even in the Tonelli framework.

关 键 词：折现Hamilton-Jacobi方程 粘性解 渐近行为 

分 类 号：O193[理学—数学]