We prove that for a given strictly increasing initial datum in Ck, the solution of the initial value problem is piecewise Ck smooth except for flux functions of nonconvex conservation laws in a certain subset of C^k＋1 of first category, defined in the range of the initial datum.
We use Hopf-Lax formula to study local regularity of solution to Hamilton- Jacobi （HJ） equations of multi-dimensional space variables with convex Hamiltonian. Then we give the large time generic form of the solution to HJ equation, i.e. for most initial data there exists a constant T 〉 0, which depends only on the Hamiltonian and initial datum, for t 〉 T the solution of the IVP （1.1） is smooth except for ~ smooth n-dimensional hypersurface, across which Du（x, t） is discontinuous. And we show that the hypersurface 1 tends asymptotically to a given hypersurface with rate t-1/4.
