LUO LIANG, LI XIA
The discounted Hamilton-Jacobi equation is a special form of the contact Hamilton-Jacobi equation. Hence,the study of the discounted Hamilton-Jacobi equation is more intuitionistic.In this paper,we mainly study the convergence of the viscosity solutions of time-periodic discounted Hamilton-Jacobi equation by variational method. For the evolutionary Hamilton-Jacobi equation $u_t+\lambda u+H\left(x, D_x u, t\right)=0$, under the assumptions that $H(x, p, t)$ is a Tonelli Hamiltonian,and $H(x, p, t)$ is 1-period in the variable $t$. we can get the 1-period solution $\bar{u}_\lambda(x, t)$ and the 1-periodic solution is unique under the given conditions. Furthermore,we prove $\lim _{n \rightarrow \infty} u_\lambda(x, t+n)=\bar{u}_\lambda(x, t)$, where $u_\lambda(x, t+n)$ is the viscosity solution of $u_t+\lambda u+H\left(x, D_x u, t\right)=0$. At last,we explain the conclusion with a specific example of Hamilton-Jacobi equation.
