The existence and stability of viscosity solutions to perturbed contact Hamilton-Jacobi equations

We consider a contact Hamiltonian $H(x,p,u)$ with certain dependence on the contact variable $u$. If $u_{-}$ is a viscosity solution of the contact Hamilton-Jacobi equation $$H(x,D_{x}u(x),u(x))=0,x\in M,$$ and $u_{-}$ is locally Lyapunov asymptotically stable, we will prove that the perturbed equation $$H(x,D_{x}u(x),u(x))+\epsilon P(x,D_{x}u(x),u(x))=0,x\in M,$$ does exist viscosity solution $u_{-}^{\epsilon }$ which converges uniformly to $u_{-}$, as perturbation parameter $\epsilon$ converges to 0. Moreover, we give a case that in a neighborhood of viscosity solution $u_{-}$, the perturbed equation has an unique viscosity solution $u_{-}^{\epsilon }$. Furthermore, $u_{-}^{\epsilon }$ keeps locally Lyapunov asymptotically stability.