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

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.