On the inhomogeneous discounted Hamilton-Jacobi equations
Abstract
In this paper, we study the family of inhomogeneous discounted Hamilton-Jacobi equations labelhjs1 λ(x)u+h(x,dx u)=c tag$∗$ on a closed manifold M with a non-identically vanishing discount factor λ(x). There is a critical value c0∈[−∞,∞) such that \eqref{hjs1} admits a viscosity solution if c>c0 and no solution if c<c0. Inspired by the recent development \cite{RWY} on the stability theory of viscosity solutions, we show that the equation admits an asymptotically stable solution if and only if c>c0. In this case, we determine the basin of the stable solution and investigate the long-time behavior of the solution semigroup associated with \eqref{hjs1}. In particular, we obtain a formula relating the lowest convergence rate of the solution semigroup near a stable solution to a minimizing problem concerning the integral of λ over Mather measures supporting on the $1$-graph of the stable solution. This formula has two direct consequences: 1) it yields a description of their asymptotic behavior as c tends to infinity; 2) it helps to classify the ergodic Mather measures and locate their distribution in the phase space. The second consequence leads to a dynamical necessary condition for c=c0.
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BibTeX
@misc{jin2026inhomogeneous,
title = {{On the inhomogeneous discounted Hamilton-Jacobi equations}},
author = {Liang Jin and Jun Yan and Kai Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.05670},
eprint = {2605.05670},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.05670},
}
RIS
TY - UNPB
TI - On the inhomogeneous discounted Hamilton-Jacobi equations
AU - Liang Jin
AU - Jun Yan
AU - Kai Zhao
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2605.05670
UR - https://arxiv.org/abs/2605.05670
AB - In this paper, we study the family of inhomogeneous discounted Hamilton-Jacobi equations labelhjs1 λ(x)u+h(x,d_x u)=c tag$∗$ on a closed manifold M with a non-identically vanishing discount factor λ(x). There is a critical value c_0∈[−∞,∞) such that \eqref{hjs1} admits a viscosity solution if c>c_0 and no solution if c<c_0. Inspired by the recent development \cite{RWY} on the stability theory of viscosity solutions, we show that the equation admits an asymptotically stable solution if and only if c>c_0. In this case, we determine the basin of the stable solution and investigate the long-time behavior of the solution semigroup associated with \eqref{hjs1}. In particular, we obtain a formula relating the lowest convergence rate of the solution semigroup near a stable solution to a minimizing problem concerning the integral of λ over Mather measures supporting on the $1$-graph of the stable solution. This formula has two direct consequences: 1) it yields a description of their asymptotic behavior as c tends to infinity; 2) it helps to classify the ergodic Mather measures and locate their distribution in the phase space. The second consequence leads to a dynamical necessary condition for c=c_0.
ER -