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Assume $M$ is a closed, connected and smooth Riemannian manifold. We considerthe 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)\inM\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 asolution semigroup, we provide a representation formula of the viscositysolution of the evolutionary equation. As its applications, we obtain anecessary and sufficient condition for the existence of the viscosity solutionsof the stationary equations. Moreover, we prove a new comparison theoremdepending on the neighborhood of the projected Aubry set essentially, which isdifferent from the one for the Hamilton-Jacobi equation independent of $u$.

A representation formula of the viscosity solution of the contact Hamilton-Jacobi equation and its applications