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

Article type: Research Article

Authors: Tabet Tchamba, Thierry

Affiliations: Laboratoire de Mathématiques et Physique Théorique, UMR CNRS 6083, Fédération Denis Poisson, FR 2964, CNRS, Université François Rabelais, Tours, France and Département de Mathématiques, Université de Yaoundé I, Yaoundé, Cameroun. E-mail: tabet@lmpt.univ-tours.fr

Abstract: We study the long-time behavior of the unique viscosity solution u of the viscous Hamilton–Jacobi equation ut−Δu+|Du|m=f in 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.

DOI: 10.3233/ASY-2009-0965

Journal: Asymptotic Analysis, vol. 66, no. 3-4, pp. 161-186, 2010