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

TL;DR

This paper investigates the long-term behavior of solutions to Hamilton-Jacobi equations on networks, revealing convergence to subsolutions influenced by flux limiters and establishing finite-time convergence.

Contribution

It extends the analysis of Hamilton-Jacobi equations' asymptotics from smooth manifolds to networks, incorporating flux limiters for a more comprehensive convergence description.

Findings

Solutions converge to subsolutions depending on flux limiters

Finite time convergence is established

The study broadens understanding of Hamilton-Jacobi equations on complex structures

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.