On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations Associated with Nonlinear Boundary Conditions

Abstract

In this article, we study the large time behavior of solutions of first-order Hamilton-Jacobi Equations, set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy-Neumann problems by using two fairly different methods : the first one relies only on partial differential equations methods, which provides results even when the Hamiltonians are not convex, and the second one is an optimal control/dynamical system approach, named the "weak KAM approach" which requires the convexity of Hamiltonians and gives formulas for asymptotic solutions based on Aubry-Mather sets.

References in corpus (2)

• Weak KAM aspects of convex Hamilton-Jacobi equations with Neumann type boundary conditions
• Long-time asymptotic solutions of convex Hamilton-Jacobi equations with Neumann type boundary conditions

Cited by in corpus (3)

• Large Time Behavior of Periodic Viscosity Solutions for Uniformly Elliptic Integro-Differential Equations
• Large Time Behavior of Solutions to Hamilton-Jacobi Equations on Networks
• On Existence and Uniqueness of Viscosity Solutions for Second Order Fully Nonlinear PDEs with Caputo time fractional derivatives