  We consider the following evolutionary Hamilton-Jacobi equation with initial
condition: \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-Jacobi
equation. By introducing an implicitly defined solution semigroup, we extend
Fathi's weak KAM theory to certain more general cases, in which $H$ explicitly
depends on the unknown function $u$. As an application, we show the viscosity
solution of the evolutionary Hamilton-Jacobi equation with initial condition
tends asymptotically to the weak KAM solution of the following stationary
Hamilton-Jacobi equation: \begin{equation*} H(x,u(x),\partial_xu(x))=0.
\end{equation*}.
