Abstract

The article is devoted to the construction and investigation of exact solutions with free boundary to a second-order nonlinear parabolic equation. The solutions belong to the classes of generalized self-similar and generalized traveling waves. Their construction is reduced to Cauchy problems for second-order ordinary differential equations (ODE), for which we prove existence and uniqueness theorems for their solutions. A qualitative analysis of the ODE is carried out by passing to a dynamical system and constructing and studying its phase portrait. In addition, we present geometric illustrations.

中文翻译：

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含源非线性热方程自由边界精确解的构造与研究

摘要

本文致力于二阶非线性抛物方程具有自由边界的精确解的构造和研究。这些解属于广义自相似和广义行波类。将它们的构造简化为二阶常微分方程（ODE）的柯西问题，为此我们证明了它们的解的存在性和唯一性定理。对ODE进行定性分析，方法是将其传递给动力学系统，并构造和研究其相图。此外，我们还提供几何插图。