Large time behavior of solutions of viscous Hamilton-Jacobi equations with superquadratic Hamiltonian

We study the long-time behavior of the unique viscosity solution u of the viscous Hamilton-Jacobi equation u(t) - Delta u + |Du|(m) = f in Omega x (0,+infinity) with inhomogeneous Dirichlet boundary conditions, where Omega is a bounded domain of R-N. We mainly focus on the superquadratic case (m > 2) and consider the Dirichlet conditions in the generalized viscosity sense. Under rather natural assumptions on f, the initial and boundary data, we connect the problem studied to its associated stationary generalized Dirichlet problem on one hand and to a stationary problem with a state constraint boundary condition on the other hand.