A new proof of indefinite propagation of singularities for a Hamilton–Jacobi equation

We study propagation of singularities for the Hamilton–Jacobi equation S_t+H(∇ S) = 0, (t,x) ∈ (0,T) ×ℝ^n, where H(p)=1/2⟨ p,Ap⟩ is a positive definite quadratic form. Each viscosity solution S is semiconcave, and it is known that its singularities move along generalized characteristics. We give a new proof of the recent result by Cannarsa et al. (Discrete Contin Dyn Syst 35:4225–4239, 2015 ), namely that the singularities propagate along generalized characteristics indefinitely forward in time.