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

Abstract

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

42 pages