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We consider the following evolutionary Hamilton-Jacobi equation with initialcondition: \begin{equation*} \begin{cases}\partial_tu(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\\ u(x,0)=\phi(x), \end{cases}\end{equation*} where $\phi(x)\in C(M,\mathbb{R})$. Under some assumptions onthe convexity of $H(x,u,p)$ with respect to $p$ and the uniform Lipschitz of$H(x,u,p)$ with respect to $u$, we establish a variational principle andprovide an intrinsic relation between viscosity solutions and certain minimalcharacteristics. By introducing an implicitly defined {\it fundamentalsolution}, we obtain a variational representation formula of the viscositysolution of the evolutionary Hamilton-Jacobi equation. Moreover, we discuss thelarge time behavior of the viscosity solution of the evolutionaryHamilton-Jacobi equation and provide a dynamical representation formula of theviscosity solution of the stationary Hamilton-Jacobi equation with strictlyincreasing $H(x,u,p)$ with respect to $u$.

Weak KAM theory for general Hamilton-Jacobi equations II: the fundamental solution under Lipschitz conditions