  Some properties of characteristic curves in connection with viscosity
solution of Hamilton-Jacobi equations $(H,\sigma)$ defined by Hopf formula
$u(t,x)=\max_{q\in\R^n}\{ \langle x,q\rangle -\sigma^*(q)-tH(q)\}$ are studied.
We are concerned with the points where the solution $u(t,x)$ is differentiable,
and the strip of the form $\mathcal R=(0,t_0)\times \R^n$ of the domain
$\Omega$ where $u(t,x)$ is of class $C^1(\mathcal R).$ Moreover, we investigate
the propagation of singularities in forward of this solution.
