Global propagation of singularities for discounted Hamilton-Jacobi equations

The main purpose of this paper is to study the global propagation of singularities of viscosity solution to discounted Hamilton-Jacobi equation \begin{equation}\label{eq:discount 1}\tag{HJ${}_{\lambda }$} \lambda v(x)+H( x, Dv(x) )=0 , \quad x\in $\mathbb{R}$^n. \end{equation} We reduce the problem for equation \eqref{eq:discount 1} into that for a time-dependent evolutionary Hamilton-Jacobi equation. We proved that the singularities of the viscosity solution of \eqref{eq:discount 1} propagate along locally Lipschitz singular characteristics which can extend to $+\mathrm{\infty }$. We also obtained the homotopy equivalence between the singular set and the complement of associated the Aubry set with respect to the viscosity solution of equation \eqref{eq:discount 1}.