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 is an element of 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 -lambda <= partial derivative F/partial derivative u < 0 for some lambda > 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) is an element of M x (0, +infinity), from the weak KAM point of view. The dynamical methods developed in [13-15] play an essential role in this paper. (C) 2021 Elsevier Inc. All rights reserved.