We study the Cauchy problem in the space of continuous functions for some nonlinear differential equation of Sobolev type that simulates longitudinal waves in an infinite viscoelastic rod. Under consideration are the conditions for the existence of the global classical solution and the blow-up of the solution to the Cauchy problem on a finite time interval.

中文翻译：

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粘弹性杆中非线性纵向波方程的柯西问题

我们研究了连续函数空间中的Cauchy问题，该问题是Sobolev类型的一些非线性微分方程的一种，它模拟无限粘弹性杆中的纵波。正在考虑的是存在全局经典解的条件以及在有限时间间隔上柯西问题的解的爆炸性条件。