Large Time Behavior of Solutions to Hamilton-Jacobi Equations on Networks

Abstract: Starting from Namah and Roquejoffre (Commun. Partial Differ. Equations, 1999) and Fathi (C. R. Acad. Sci., Paris, S\'er. I, Math., 1998), the large time asymptotic behavior of solutions to Hamilton-Jacobi equations has been extensively investigated by many authors, mostly on smooth compact manifolds and the flat torus. They all prove that such solutions converge to solutions to a corresponding static problem. We extend this study to the case where the ambient space is a network. The presence of a "flux limiter", that is the choice of appropriate constants on each vertex of the network necessary for the well-posedness of time-dependent problems on networks, enables a richer statement for the convergence compared to the classical setting. We indeed observe that solutions converge to subsolutions to a corresponding static problem depending on the value of the flux limiter. A finite time convergence is also established.