Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition

Let \begin{document}$ D\subset R^{d}$\end{document} be a bounded domain in the \begin{document}$d- $\end{document} dimensional Euclidian space \begin{document}$R^{d} $\end{document} with smooth boundary $Γ=\partial D.$ In this paper we consider exponential boundary stabilization for weak solutions to the wave equation with nonlinear boundary condition: \begin{document}$\left\{ \begin{gathered}u_{tt}(t)-ρ(t)Δ u(t)+b(x)u_{t}(t)=f(u(t)), \\ u(t)=0\ \ \text{on }Γ_{0}×(0,T), \\ \dfrac{\partial u(t)}{\partialν}+γ(u_{t}(t))=0\ \ \text{on }Γ _{1}×(0,T), \\ u(0)=u_{0},u_{t}(0)=u_{1},\end{gathered} \right.$ \end{document} where \begin{document}$\left\| {{u_0}} \right\| \begin{document}$ E(0) where \begin{document}$λ_{β}, $\end{document} \begin{document}$d_{β} $\end{document} are defined in (21), (22) and \begin{document}$Γ=Γ_{0}\cupΓ_{1} $\end{document} and \begin{document}$\bar{Γ}_{0}\cap\bar{Γ}_{1}=φ. $\end{document}