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September  2017, 16(5): 1571-1585. doi: 10.3934/cpaa.2017075

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

 Division of Mathematical Sciences, Graduate School of Comparative Culture, Kurume University, Miimachi, Kurume, Fukuoka 839-8502, Japan

Received  September 2015 Revised  October 2015 Published  May 2017

Fund Project: To Shiho and Sarasa from Grandpapa. The author is partially supported by the Grant-in-Aid for Scientific Research (No.24540198) from Japan Society for the Promotion of Science.

Let
 $D\subset R^{d}$
be a bounded domain in the
 $d-$
dimensional Euclidian space
 $R^{d}$
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:
 $\left\{ \begin{gathered}u_{tt}(t)-ρ(t)Δ u(t)+b(x)u_{t}(t)=f(u(t)), \\ u(t)=0\ \ \text{on }Γ_{0}×(0,T), \\ \dfrac{\partial u(t)}{\partialν}+γ(u_{t}(t))=0\ \ \text{on }Γ _{1}×(0,T), \\ u(0)=u_{0},u_{t}(0)=u_{1},\end{gathered} \right.$
where
 $\left\| {{u_0}} \right\| < {\lambda _\beta },$
 $E(0) < d_{β},$
where
 $λ_{β},$
 $d_{β}$
are defined in (21), (22) and
 $Γ=Γ_{0}\cupΓ_{1}$
and
 $\bar{Γ}_{0}\cap\bar{Γ}_{1}=φ.$
Citation: Takeshi Taniguchi. Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1571-1585. doi: 10.3934/cpaa.2017075
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