A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations

We consider the Hamilton–Jacobi equation ?_t u + H(x, Du) = 0 in (0, +?) × T^N , where T^N is the flat N -dimensional torus, and the Hamiltonian H(x, p) is assumed continuous in x and strictly convex and coercive in p. We study the large time behavior of solutions, and we identify the limit through a Lax-type formula. Some convergence results are also given for H solely convex. Our qualitative method is based on the analysis of the dynamical properties of the Aubry set, performed in the spirit of [A. Fathi and A. Siconolfi, Calc. Var. Partial Differential Equations, 22 (2005), pp. 185–228]. This can be viewed as a generalization of the techniques used in [A. Fathi, C. R. Acad. Sci. Paris Ser. I Math., 327 (1998), pp. 267–270] and [J. M. Roquejoffre, J. Math. Pures Appl. (9), 80 (2001), pp. 85–104]. Analogous results have been obtained in [G. Barles and P. E. Souganidis, SIAM J. Math. Anal., 31 (2000), pp. 925–939] using PDE methods.