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

Abstract

We study the long-time behavior of the unique viscosity solution $u$ of the viscous Hamilton-Jacobi Equation $u_t-\Delta u+|Du{|}^m=fin\; \Omega \times (0,+\infty )$ with inhomogeneous Dirichlet boundary conditions, where $\Omega$ is a bounded domain of ${\mathbb{R}}^N$. 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.