  We investigate the properties of the set of singularities of semiconcave
solutions of Hamilton-Jacobi equations of the form \begin{equation*}
  u_t(t,x)+H(\nabla u(t,x))=0, \qquad\text{a.e. }(t,x)\in
(0,+\infty)\times\Omega\subset\mathbb{R}^{n+1}\,. \end{equation*} 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 the above equation cannot vanish in
a finite time.
