A nonlinear semigroup approach to Hamilton-Jacobi equations–revisited

We consider the Hamilton-Jacobi equation H(x,Du)+λ(x)u=c,x∈M, where M is a connected, closed and smooth Riemannian manifold. The functions H(x,p) and λ(x) are continuous. H(x,p) is convex, coercive with respect to p, and λ(x) changes the signs. The first breakthrough to this model was achieved by Jin-Yan-Zhao [11] 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. To the best of our knowledge, it is the first detailed description of the large time behavior of the HJ equations with non-monotone dependence on the unknown function.