On the inhomogeneous discounted Hamilton-Jacobi equations

Abstract

In this paper, we study the family of inhomogeneous discounted Hamilton-Jacobi equations \begin{equation}\label{hjs1} λ(x)u+h(x,d_x u)=c \quad \tag{$\ast$} \end{equation} on a closed manifold $M$ with a non-identically vanishing discount factor $\lambda (x)$. There is a critical value $c_0\in [-\infty ,\infty )$ 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, for ${\max }_M\lambda >0$ , 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 to \eqref{hjs1}. In particular, we relate the lowest convergence rate to the integral of $\lambda$ over Mather measures, which leads to an asymptotic behavior of Mather measures when $c$ goes to infinity. Assuming $c⩾c_0$ and the equation admits a solution, we classify ergodic Mather measures and locate their distribution in the phase space.