A nonlinear semigroup approach to Hamilton-Jacobi equations--revisited

We consider the Hamilton-Jacobi equation $${H}(x,Du)+\lambda(x)u=c,\quad x\in M,$$ where $M$ is a connected, closed and smooth Riemannian manifold. The functions ${H}(x,p)$ and $\lambda(x)$ are continuous. ${H}(x,p)$ is convex, coercive with respect to $p$, and $\lambda(x)$ changes the signs. The first breakthrough to this model was achieved by Jin-Yan-Zhao \cite{JYZ} under the Tonelli conditions. In this paper, we consider more detailed structure of the viscosity solution set and large time behavior of the viscosity solution on the Cauchy problem.