The present paper takes up the underlying nonlinear initial value problem from a preceding author’s work about the dynamics of a single bubble in a highly viscous liquid medium under different pressure impacts. The arising ordinary differential equation is mainly based on the constitutive relation of a second-order liquid that in particular includes two non-Newtonian material constants. In this article, the significance of these coefficients is mathematically analyzed in detail by proving the existence of stable solutions of the named initial value problem. This is achieved by special transformations of the differential equation at hand and the introduction of appropriate Lyapunov functions. It particularly turns out that a combined condition of the non-Newtonian coefficients and diverse restrictions to the external pressure impact are decisive for the validity of the existence results. Furthermore, the convergence speed of solutions is investigated by considering the linearized equation associated with the present initial value problem and by applying a special variant of Gronwall’s lemma. The main theoretical result, being the prementioned strong condition for the non-Newtonian coefficients, is finally compared to real data sets.

中文翻译：

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单个气泡长期稳定行为的非牛顿系数条件：稳定解的存在和特征

本文从先前作者的工作中解决了潜在的非线性初始值问题，该问题涉及高压力液体介质中单个气泡在不同压力作用下的动力学。产生的常微分方程主要基于尤其包括两个非牛顿材料常数的二阶液体的本构关系。在本文中，通过证明存在所称初始值问题的稳定解，对这些系数的重要性进行了数学上的详细分析。这可以通过对微分方程进行特殊转换并引入适当的Lyapunov函数来实现。特别是，非牛顿系数的组合条件和对外部压力影响的各种限制对存在结果的有效性起决定性作用。此外，通过考虑与当前初始值问题相关的线性化方程并应用Gronwall引理的特殊变体来研究解的收敛速度。最后将作为非牛顿系数的前述强条件的主要理论结果与实际数据集进行比较。