Annales de l’Institut Henri Poincare. DOI 10.1016/j.anihpc.2013.10.005.
We introduce a new machinery to study the large time behavior for general classes of Hamilton--Jacobi type equations, which include degenerate parabolic equations and weakly coupled systems. We establish the convergence results by using the nonlinear adjoint method and identifying new long time averaging effects. These methods are robust and can easily be adapted to study the large time behavior of related problems.