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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*} Under some assumptions on $H(x,u,p)$ with respect to $p$ and$u$, we provide a variational principle on the evolutionary Hamilton-Jacobiequation. By introducing an implicitly defined solution semigroup, we extendFathi's weak KAM theory to certain more general cases, in which $H$ explicitlydepends on the unknown function $u$. As an application, we show the viscositysolution of the evolutionary Hamilton-Jacobi equation with initial conditiontends asymptotically to the weak KAM solution of the following stationaryHamilton-Jacobi equation: \begin{equation*} H(x,u(x),\partial_xu(x))=0.\end{equation*}.

Weak KAM theory for general Hamilton-Jacobi equations I: the solution semigroup under proper conditions