Large time behavior of solutions to Hamilton–Jacobi equations on networks

Starting from Namah and Roquejoffre (Commun. Partial Differ. Equations, 1999) and Fathi (C. R. Acad. Sci., Paris, Sér. 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.