Abstract

The results obtained by Il’in and his school concerning the asymptotic behavior of solutions to the Cauchy problem for the quasi-linear parabolic equation with a small parameter multiplying the higher order derivative in the vicinity of singular points are presented. The equation under examination is of interest because it provides a model of the propagation of nonlinear waves in dissipative continuous media, and the importance of studying solutions in the vicinity of singular points is explained, in particular, by the fact that even though the singular events take a short time, they in many respects determine the subsequent evolution of the solutions. In this paper, we examine five types of singular points the emergence of which is caused by different initial data.

中文翻译：

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小参数抛物型方程奇异Cauchy问题的奇点与渐近性

摘要

提出了Il'in及其学校关于拟线性抛物方程的Cauchy问题解的渐近行为的结果，该拟线性抛物方程的小参数乘以奇点附近的高阶导数。所研究的方程式很有趣，因为它提供了非线性波在耗散连续介质中传播的模型，并且特别说明了即使奇异事件的发生也可以解释研究奇异点附近解的重要性。在短时间内，他们在许多方面决定了解决方案的后续发展。在本文中，我们检查了五种类型的奇异点，这些奇异点的出现是由不同的初始数据引起的。