Perpetual propagation of singularities for evolutionary Hamilton–Jacobi equations

Abstract This paper studies viscosity solutions S ( t , x ) of the Hamilton–Jacobi equation $$ \partial S/\partial t+H(x,\nabla S)=0,\qquad (t,x)\in (0,\infty )\times {\mathbb {R}}^n. $$ ∂ S / ∂ t + H ( x , ∇ S ) = 0 , ( t , x ) ∈ ( 0 , ∞ ) × R n . The singular set $$\Sigma (S)$$ Σ ( S ) for a given solution S consists of all points where S is not differentiable. While it is well known that singularities propagate locally along certain curves, perpetual propagation has been established only in a few special cases. By means of intrinsic characteristics, we establish perpetual propagation of singularities in three different settings. More precisely, from every initial point $$(t_0,x_0)\in (0,\infty )\times {\mathbb {R}}^n$$ ( t 0 , x 0 ) ∈ ( 0 , ∞ ) × R n we construct a curve $$\varvec{x}(t)$$ x ( t ) , defined for $$t\in [t_0,\infty )$$ t ∈ [ t 0 , ∞ ) and with $$\varvec{x}(t_0)=x_0$$ x ( t 0 ) = x 0 , such that whenever $$t^\bullet \ge t_0$$ t ∙ ≥ t 0 and $$(t^\bullet ,\varvec{x}(t^\bullet ))\in \Sigma (S)$$ (