We study the long-time behavior of the unique viscosity solution u of the
viscous Hamilton-Jacobi Equation ut−Δu+∣Du∣m=fin Ω×(0,+∞) with inhomogeneous Dirichlet boundary conditions,
where Ω is a bounded domain of RN. 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