In this article, we consider the wave equation on hyperbolic spaces \(\mathbb {H}^n(n\ge 2)\) with nonlinear locally distributed damping as follow:

$$\begin{aligned} {\left\{ \begin{array}{ll}u_{tt}-\Delta _g u+a(x)g(u_t)=0\qquad (x,t)\in \mathbb {H}^n\times (0,+\infty ), \\ u(x,0)=u_0(x),\quad u_0(x,0)=u_1(x)\qquad x\in \mathbb {H}^n. \end{array}\right. } \end{aligned}$$(1)

It is well-known that the energy of the system (1) is of polynomial decay in the Euclidean space. However, on hyperbolic spaces, owing to the following inequality

$$\begin{aligned} \int _{\mathbb {H}^n} u^2 dx_g \le C \int _{\mathbb {H}^n} |\nabla _g u|_g^2dx_g , \quad for \quad u\in H^1(\mathbb {H}^n), \end{aligned}$$(2)

we prove the exponential stabilization of the wave equation by multiplier methods and compactness-uniqueness arguments.

中文翻译：

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具有非线性局部阻尼的双曲空间上波动方程的指数镇定。

在本文中，我们考虑具有非线性局部分布阻尼的双曲空间\（\ mathbb {H} ^ n（n \ ge 2）\）上的波动方程如下：

$$ \ begin {aligned} {\ left \ {\ begin {array} {ll} u_ {tt}-\ Delta _g u + a（x）g（u_t）= 0 \ qquad（x，t）\ in \ mathbb {H} ^ n \ times（0，+ \ infty），\\ u（x，0）= u_0（x），\ quad u_0（x，0）= u_1（x）\ qquad x \ in \ mathbb {H} ^ n。\ end {array} \ right。} \ end {aligned} $$（1）

众所周知，系统（1）的能量在欧几里得空间中具有多项式衰减。但是，在双曲空间上，由于以下不等式

$$ \ begin {aligned} \ int _ {\ mathbb {H} ^ n} u ^ 2 dx_g \ le C \ int _ {\ mathbb {H} ^ n} | \ nabla _g u | _g ^ 2dx_g，\ quad对于\ quad u \ in H ^ 1（\ mathbb {H} ^ n），\ end {aligned} $$（2）

我们通过乘数法和紧唯一性论证证明了波动方程的指数稳定性。