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

We consider the Hamilton-Jacobi equation $$H(x,Du)+\lambda (x)u=c,x\in M,$$ 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.