Some regularity properties of viscosity solution defined by Hopf formula

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