Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions

Abstract

We consider the evolutionary Hamilton-Jacobi equation depending on the unknown function with the continuous initial condition on a connected closed manifold.Under certain assumptions on $H(x,u,p)$ with respect to $u$ and $p$, we provide an implicit variational principle. By introducing an implicitly defined solution semigroup and an admissible value set $\mathcal{C}_H$, we extend weak KAM theory to certain more general cases, in which $H$ depends on the unknown function $u$ explicitly. As an application, we show that for $0\notin \mathcal{C}_H$, as $t\rightarrow +\infty$, the viscosity solution of\begin{equation*}\begin{cases}\partial_tu(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\\u(x,0)=\varphi(x),\end{cases}\end{equation*}diverges, otherwise for $0\in \mathcal{C}_H$, it converges to a weak KAM solution of the stationary Hamilton-Jacobi equation\begin{equation*}H(x,u(x),\partial_xu(x))=0.\end{equation*}