In this contribution, a wave equation with a time-dependent variable-order fractional damping term and a nonlinear source is considered. Avoiding the circumstances of expressing the nonlinear variable-order fractional wave equations via closed-form expressions in terms of special functions, we investigate the existence and uniqueness of this problem with Rothe's method. First, the weak formulation for the considered wave problem is proposed. Then, the uniqueness of a solution is established by employing Gr\"onwall's lemma. The Rothe scheme's basic idea is to use Rothe functions to extend the solutions on single-time steps over the entire time frame. Inspired by that, we next introduce a uniform mesh time-discrete scheme based on a discrete convolution approximation in the backward sense. By applying some reasonable assumptions to the given data, we can predict a priori estimates for the time-discrete solution. Employing these estimates side by side with Rothe functions leads to proof of the solution's existence over the whole time interval. Finally, the full discretisation of the problem is introduced by invoking Galerkin spectral techniques in the spatial direction, and numerical examples are given.

翻译：本文研究了一类含有时间依赖变阶分数阶阻尼项和非线性源的波动方程。为避免通过特殊函数闭形式表达非线性变阶分数阶波动方程的复杂情形，我们采用Rothe方法探讨了该问题的解的存在唯一性。首先给出了考虑波动问题的弱形式，然后利用Grönwall引理证明了解的唯一性。Rothe方法的核心思想是通过Rothe函数将单时间步上的解延拓至整个时间区间。受此启发，我们基于后向离散卷积逼近构建了均匀网格时间离散格式。在给定数据满足合理假设的条件下，可预测时间离散解的先验估计。将这些估计与Rothe函数相结合，最终证明了整个时间区间上解的存在性。最后，通过引入空间方向上的Galerkin谱技术实现了问题的完全离散化，并给出了数值算例。