Global Propagation of Singularities for Time Dependent Hamilton-Jacobi Equations

We investigate the properties of the set of singularities of semiconcave solutions of Hamilton-Jacobi equations. It is well known that the singularities of such solutions propagate locally along generalized characteristics. Special generalized characteristics, satisfying an energy condition, can be constructed, under some assumptions on the structure of the Hamiltonian H. In this paper, we provide estimates of the dissipative behavior of the energy along such curves. As an application, we prove that the singularities of any viscosity solution of such equations cannot vanish in a finite time.