  Some properties of characteristic curves in connection with viscosity
solutions of Hamilton-Jacobi equations defined by Hopf-type formula are
studied. We investigate the points where the Hopf-type formula $u(t,x)$ is
differentiable, and the strip of the form $(0,t_0)\times \R^n$ of the domain
$\Omega$ where the viscosity solution $u(t,x)$ is continuously differentiable.
Moreover, we present the propagation of singularity in forward of $u(t,x).$
