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Regularity of viscosity solutions defined by Hopf-type formula for Hamilton-Jacobi equations

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 $Ω$ where the viscosity solution $u(t,x)$ is continuously differentiable. Moreover, we present the propagation of singularity in forward of $u(t,x).$