Termination time of characteristics of Hamilton-Jacobi equations

This paper concerns with Hamilton-Jacobi equations of n space variables, where the Hamiltonians are convex and the initial data are admitted to be unbounded. First, we study the characteristics for the general case of initial data being Lipschitz by using the Hopf formula. Sufficient and necessary conditions are established for guaranteeing a characteristic to start from y (0) at t = 0 with direction DH(P (0)) and for a characteristic never terminating on a singularity of the solution. Next, in the case of initial data being C (2), we prove that the set of singularities consists of at most countable path-connected components, which is an extension of (Zhao et al. in J Hyperbolic Differ Equ 5(3):663-680, 2008) and (Li in Sci Sinica 22(9):979-990, 1979).