  Assume $M$ is a closed, connected and smooth Riemannian manifold. We consider
the evolutionary Hamilton-Jacobi equation \begin{equation*}
  \left\{
  \begin{aligned}
  &\partial_t u(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\quad (x,t)\in
M\times(0,+\infty),
  \\ &u(x,0)=\varphi(x),
  \end{aligned}
  \right. \end{equation*} where $\varphi\in C(M)$ and the stationary one
\begin{equation*}
  H(x,u(x),\partial_x u(x))=0, \end{equation*} where $H(x,u,p)$ is continuous,
convex and coercive in $p$, uniformly Lipschitz in $u$. By introducing a
solution semigroup, we provide a representation formula of the viscosity
solution of the evolutionary equation. As its applications, we obtain a
necessary and sufficient condition for the existence of the viscosity solutions
of the stationary equations. Moreover, we prove a new comparison theorem
depending on the neighborhood of the projected Aubry set essentially, which is
different from the one for the Hamilton-Jacobi equation independent of $u$.
