Convergence of viscosity solutions of generalized contact Hamilton-Jacobi equations

For any compact connected manifold $M$, we consider the generalized contact Hamiltonian $H(x,p,u)$ defined on $T^*M\times\mathbb R$ which is conex in $p$ and monotonically increasing in $u$. Let $u_\epsilon^-:M\rightarrow\mathbb R$ be the viscosity solution of the parametrized contact Hamilton-Jacobi equation \[ H(x,\partial_x u_\epsilon^-(x),\epsilon u_\epsilon^-(x))=c(H) \] with $c(H)$ being the Ma\~n\'e Critical Value. We prove that $u_\epsilon^-$ converges uniformly, as $\epsilon\rightarrow 0_+$, to a specfic viscosity solution $u_0^-$ of the critical equation \[ H(x,\partial_x u_0^-(x),0)=c(H) \] which can be characterized as a minimal combination of associated Peierls barrier functions.