Boundary stabilization for wave equations with damping only on the nonlinear Wentzell boundary

We are concerned with the problem of boundary stabilization for wave equations with damping only on the nonlinear Wentzell boundary, and the damping is also nonlinear. We obtain boundary stabilization results (explicit energy decay rates), as well as the wellposedness, for the systems. Our method consists in establishing a new type of non-uniform integral inequality of energy, and exploiting it to derive the decay rates. The results are set up in all space dimensions. Even for the one dimensional linear system, the result is new, and we also indicate that the obtained decay rate is optimal in this case, that is, the obtained decay rate for linear strings is optimal.