Weak KAM solutions of Hamilton-Jacobi equations with decreasing dependence on unknown functions

We consider the Hamilton-Jacobi equation \[{H}(x,u,Du)=0,\quad x\in M, \] where $M$ is a connected, closed and smooth Riemannian manifold, ${H}(x,u,p)$ satisfies Tonelli conditions with respect to $p$ and certain decreasing condition with respect to $u$. Based on a dynamical approach developed in \cite{WWY,WWY1,WWY2}, we obtain a series of properties for weak KAM solutions (equivalently, viscosity solutions) of the stationary equation and the long time behavior of viscosity solutions of the evolutionary equation on the Cauchy problem \begin{equation*} \begin{cases} w_t+{H}(x,w,w_x)=0,\quad (x,t)\in M\times (0,+\infty),\\ w(x,0)=\varphi(x), \quad x\in M. \end{cases} \end{equation*}