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

We study the Hamilton–Jacobi equations (Formula presented.) in M and (Formula presented.) in (Formula presented.) where the Hamiltonian (Formula presented.) 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.