Ergodic type problems and large time behaviour of unbounded solutions of Hamilton-Jacobi Equations

Guy Barles (1) , Jean-Michel Roquejoffre (2)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 MIP - Mathématiques pour l'Industrie et la Physique

We study the large time behavior of Lipschitz continuous, possibly unbounded, viscosity solutions of Hamilton-Jacobi Equations in the whole space $\R^N$. The associated ergodic problem has Lipschitz continuous solutions if the analogue of the ergodic constant is larger than a minimal value $\lambda_{min}$. We obtain various large-time convergence and Liouville type theorems, some of them being of completely new type. We also provide examples showing that, in this unbounded framework, the ergodic behavior may fail, and that the asymptotic behavior may also be unstable with respect to the initial data.

Mots clés: Hamilton-Jacobi Equations; unbounded solutions; ergodic problems; large time behavior; geodesics.; geodesics

Domaines: Equations aux dérivées partielles [math.AP]

Guy Barles, Jean-Michel Roquejoffre. Ergodic type problems and large time behaviour of unbounded solutions of Hamilton-Jacobi Equations. Comm. Partial Differential Equations, 2006, 31 (7-9), pp.1209--1225. ⟨hal-00121921⟩