A new PDE approach to the large time asymptotics of solutions of Hamilton-Jacobi equations

We introduce a new PDE approach to establishing the large time asymptotic behavior of solutions of Hamilton-Jacobi equations, which modifies and simplifies the previous ones (\cite{BS, BIM}), under a refined ''strict convexity'' assumption on the Hamiltonians. Not only such ''strict convexity'' conditions generalize the corresponding requirements on the Hamiltonians in \cite{BS}, but also one of the most refined our conditions covers the situation studied in \cite{NR}.