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 (Barles and Souganidis, 2000; Barles, Ishii and
Mitake, 2012), under a refined "strict convexity" assumption on the
Hamiltonians. Not only such "strict convexity" conditions generalize the
corresponding requirements on the Hamiltonians in Barles and Souganidis (2000),
but also one of the most refined our conditions covers the situation studied in
Namah and Roquejoffre (1999)