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 , D u) = 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 − λ ≤ ∂ F ∂ u < 0 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. [ABSTRACT FROM AUTHOR]