Local strict singular characteristics: Cauchy problem with smooth initial data

Main purpose of this paper is to study the local propagation singularities viscosity solution contact type evolutionary Hamilton-Jacobi equationDtu(t,x)+H(t,x,Dxu(t,x),u(t,x))=0. An important issue topic existence, uniqueness and regularity strict singular characteristic. We apply recent existence results on Herglotz' variational problem aforementioned equation with smooth initial data. obtain some new structure cut set near non-conjugate points. Especially, we an result characteristic from point based superdifferential solution, which even in classical time-dependent case. also get a global for C1 support