Finite-time convergence of solutions of Hamilton-Jacobi equations

Abstract

Suppose that $H(x,u,p)$ is strictly decreasing in $u$ and satisfies Tonelli conditions in $p$. We show that each viscosity solution of $H(x,u,u_x)=0$ can be reached by many viscosity solutions of $$w_t+H(x,w,w_x)=0,$$ in a finite time.