Convergence of the viscosity solution of non-autonomous Hamilton-Jacobi equations

Abstract

In this paper, we investigate the non-autonomous Hamilton-Jacobi equation $$\left\{ {\begin{array}{*{20}{c}} {{\partial _t}u + H(t,x,{\partial _x}u,u) = 0,} \\ {\begin{array}{*{20}{c}} {u(x,{t_0}) = \phi (x),}&{x \in M,} \end{array}} \end{array}} \right.$$ where H is 1-periodic with respect to t and M is a compact Riemannian manifold without boundary. We obtain the viscosity solution denoted by $$T_{{t_0}}^t\phi (x)$$ and show $$T_{{t_0}}^t\phi (x)$$ converges uniformly to a time-periodic viscosity solution u* (x, t) of ∂tu + H(t, x, ∂xu, u) = 0.