Weak KAM theory for Hamilton-Jacobi equations depending on unknown functions

We consider the evolutionary Hamilton-Jacobi equation depending on the unknown function with the continuous initial condition on a connected closed manifold. Under certain assumptions on H(x, u, p) with respect to u and p, we provide an implicit variational principle. By introducing an implicitly defined solution semigroup and an admissible value set C_H, we extend weak KAM theory to certain more general cases, in which H depends on the unknown function u explicitly. As an application, we show that for 0 ∉ C_H, as t → +∞, the viscosity solution of {∂_tu(x, t) + H(x, u(x, t), ∂_xu(x, t)) = 0, u(x, 0) = φ(x), diverges, otherwise for 0 ∈ C_H, it converges to a weak KAM solution of the stationary Hamilton-Jacobi equation H(x, u(x), ∂_xu(x)) = 0.