Weak KAM solutions of Hamilton-Jacobi equations with decreasing dependence on unknown functions

First, we provide a necessary and sufficient condition of the existence of viscosity solutions of the nonlinear first order PDE F(x,u,Du)=0,x∈M, under which we prove the compactness of the set of all viscosity solutions. Here, F(x,u,p) satisfies Tonelli conditions with respect to the argument p and [Formula presented] for some λ>0, and M is a compact manifold without boundary. Second, we study the long time behavior of viscosity solutions of the Cauchy problem for w_t+F(x,w,w_x)=0,(x,t)∈M×(0,+∞), from the weak KAM point of view. The dynamical methods developed in [13–15] play an essential role in this paper.