When I deal with Energy decay rate estimates of wave equation$$u_{tt}-\Delta u=0\ in\ \Omega$$ with acoustic boundary condition$$z_{tt}+\varphi(z_{t})+z-g*z+u_{t}=0\ on\ \Gamma_{1},$$ $$\partial_{\nu}u=z_{t},on\ \Gamma_{1}$$ $$u=0,on\ \Gamma_{0},$$

I plan to use Multiplier method and higher order energy method.The higher order energy method needs to differentiate the system twice, but the higher order method forced us to differentiate $\varphi(z_{t})$ twice. First order derivative is $\frac{d}{dt}\varphi(z_{t})=\varphi'(z_{t})z_{tt}$ and the second order derivative is $\frac{d^{2}}{dt^{2}}\varphi(z_{t})=\varphi''(z_{t})z_{tt}^{2}+\varphi'(z_{t})z_{ttt}$. I want to know whether there are some papers deal with the wave equation with nonlinear damping using higher order method?

The appearance of $\varphi''(z_{t})z_{tt}^{2}$ is difficult to deal with, because of $\langle \varphi''(z_{t})z_{tt}^{2},z_{ttt}\rangle$.

Remark: $\Omega$ is a smooth domain in $\mathbb{R}^{3}$ and $\Gamma_{1}\cup\Gamma_{0}=\partial\Omega, \Gamma_{1}\cap\Gamma_{0}=\emptyset$. $\Gamma_{1}$ and $\Gamma_{0}$ are relatively both open sets.