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

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