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

Abstract

We study propagation of singularities for the Hamilton–Jacobi equation $$S_t+H(\nabla S) = 0,\quad (t,x) \in (0,T) \times \mathbb{R}^n,$$ where \({H(p)=\frac{1}{2} \langle p,Ap\rangle}\) 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.