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

In this paper, we investigate the non-autonomous Hamilton-Jacobi equation {partial derivative(t) u + H(t, x, partial derivative(x) u, u) = 0, u(x, t(0)) = phi(x), x epsilon M, 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-t0(t)phi(x) and show T-t0(t)phi(x) converges uniformly to a time-periodic viscosity solution u* (x, t) of partial derivative(t) u + H(t, x, partial derivative(x) u, u) = 0.

Automated summary

In this paper, we investigate the viscosity solutions of the non-autonomous Hamilton-Jacobi equation on a compact Riemannian manifold, showing convergence to a time-periodic viscosity solution.