We use Hopf-Lax formula to study local regularity of solution to HamiltonJacobi (H J) 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 a smooth n-dimensional hypersurface,across which Du(x,t) is discontinuous.And we show that the hypersurface tends asymptotically to a given hypersurface with rate t-(1/4).