Hamilton–Jacobi equations with their Hamiltonians depending Lipschitz continuously on the unknown

We study the Hamilton-Jacobi equations $H(x,Du,u)=0$ in $M$ and $\partial u/\partial t +H(x,D_xu,u)=0$ $M\times(0,\infty)$, where Hamiltonian $H=H(x,p,u)$ depends Lipschitz continuously on variable $u$. In framework of semicontinuous viscosity solutions due to Barron-Jensen, we establish comparison principle, existence theorem, representation formula as value functions for extended real-valued, lower Cauchy problem. also some results long-time behavior problem classification stationary