We study the following wave equation \(u_{tt}-\Delta u+\alpha (t)\left| u_{t}\right| ^{m(\cdot )-2}u_{t}=0\) with a nonlinear damping having a variable exponent m(x) and a time-dependent coefficient \(\alpha (t)\). We use the multiplier method to establish energy decay results depending on both m and \(\alpha \). We also give four numerical tests to illustrate our theoretical results using the conservative Lax–Wendroff method scheme.

中文翻译：

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具有可变指数和时变非线性阻尼的波动方程的理论和计算结果

我们研究以下波动方程\（U_ {TT} - \德尔塔U + \α（t）的\左| U_ {吨} \右| ^ {M（\ CDOT）-2} U_ {吨} = 0 \）与具有可变指数m（x）和随时间变化的系数\（\ alpha（t）\）的非线性阻尼。我们使用乘数方法根据m和\（\ alpha \）建立能量衰减结果。我们还给出了四个数值测试，以说明使用保守的Lax-Wendroff方法方案的理论结果。