  For each continuous initial data $\varphi(x)\in C(M,\mathbb{R})$, we obtain
the asymptotic Lipschitz regularity of the viscosity solution of the following
evolutionary Hamilton-Jacobi equation with convex and coercive Hamiltonians:
\partial_tu(x,t)+H(x,\partial_xu(x,t))=0, u(x,0)=\varphi(x).
