We consider the decay rate of energy of the 1D damped original nonlinear wave equation. We first construct a new energy function. Then, employing the perturbed energy method and the generalized Young’s inequality, we prove that, with a general growth assumption on the nonlinear damping force near the origin, the decay rate of energy is governed by a dissipative ordinary differential equation. This allows us to recover the classical exponential, polynomial, or logarithmic decay rate for the linear, polynomial or exponentially degenerating damping force near the origin, respectively. Unlike the linear wave equation, the exponential decay rate constant depends on the initial data, due to the nonlinearity.

我们考虑一维阻尼原始非线性波动方程的能量衰减率。我们首先构造一个新的能量函数。然后，采用扰动能量法和广义杨氏不等式，证明了在原点附近非线性阻尼力的一般增长假设下，能量衰减率受耗散常微分方程控制。这使我们能够分别恢复原点附近线性、多项式或指数退化阻尼力的经典指数、多项式或对数衰减率。与线性波动方程不同，由于非线性，指数衰减率常数取决于初始数据。