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依赖于时间的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式表达式表达非线性变阶分数阶波动方程，我们采用Rothe方法探讨了该问题的解的存在性和唯一性。首先，提出了所考虑波动问题的弱形式。然后，利用Grönwall引理建立了解的唯一性。Rothe方法的基本思想是利用Rothe函数将单时间步上的解扩展到整个时间域。受此启发，我们随后引入了一个基于后向离散卷积近似的均匀网格时间离散格式。通过对给定数据施加一些合理假设，我们可以预测时间离散解的先验估计。将这些估计与Rothe函数结合使用，便可证明解在整个时间区间上的存在性。最后，通过在空间方向上引入Galerkin谱技术，实现了问题的完全离散化，并给出了数值算例。