A Generalized Dynamical Approach to the Large Time Behavior of Solutions of Hamilton--Jacobi Equations

Abstract

We consider the Hamilton--Jacobi equation \[ \partial_t u+H(x,Du)=0\qquad \hbox{in $(0,+\infty)\times\T^{N}$}, \] 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.