## Xifeng Su (Beijing Normal University, China)

### Weak KAM type solutions of evolutionary Hamilton-Jacobi equations and their asymptotic behaviors

We consider the following evolutionary Hamilton-Jacobi equation:

∂_{t}u(x, t) + H(x, u(x, t), ∂_{x}u(x, t)) = 0.

Under some assumptions on the function H(x,u,p) with respect to p and u, we associated such a system with a variational principle, which allows us to implicitly define a solution semigroup, and extend Fathi’s weak KAM theory to the cases where H explicitly depends on the unknown function u. As an application, we show that the viscosity solutions of this evolutionary Hamilton-Jacobi equation tend asymptotically to the weak KAM solutions of the stationary Hamilton-Jacobi equation

H(x,u(x),∂_{x}u(x)) = 0.