Propagation of singularities for viscosity solutions of Hamilton-Jacobi equations in one space variable

In this note we study the generation and propagation of singularities (shock waves) of the solution of the Cauchy problem for Hamilton-Jacobi equations (P) $\{$ $\frac{\partial y}{\partial t}+H(t, x_{1,\ldots,n}x, \frac{\partial y}{\partial x_{1}}, \ldots, \frac{\partial y}{\partial x_{n}})=0