Finite-time convergence of solutions of Hamilton-Jacobi equations

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.

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