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

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

2006-12-22

arxiv.org/abs/math/0612704v1

Comm. Partial Differential Equations 31, 7-9 (2006) 1209--1225

Mathematics

Analysis of PDEs

Scientific paper

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.

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