一类具有退化的混合问题的局部解

Local solutions of a class of mixed problems with degeneracy

摘要

对于退化的抛物方程，抛物方程标准理论的可解性不再适用，因此本文通过逼近问题研究了一类退化的半线性混合问题的解。首先构建混合问题的逼近问题，利用 Schauder 不动点定理证明了逼近问题的局部解，并将得到的解进行延拓，证明逼近问题全局解的存在性，再建立了同时满足混合问题和逼近问题的比较原理，最后通过对混合问题的初值施加某些适当的限制条件，利用比较原理、抛物方程的内部 Schauder 估计和 Arzelà-Ascoli 定理等证明混合问题存在唯一的局部经典解。

Abstract

This paper studies the existence and uniqueness of local classical solutions to a class of degenerate semi-linear mixed problems. For degenerate parabolic equations, the standard solvability theory for parabolic equations does not apply. To address this issue, an approximation problem is constructed for the degenerate mixed problem, and solutions to the mixed problem are obtained via solutions to the approximation problem. First, the Schauder fixed point theorem is employed to obtain a local solution to the approximation problem. This solution is then extended to establish the existence of a global solution for the approximation problem. Subsequently, a comparison principle is established that simultaneously satisfies both the mixed problem and the approximating equation. Finally, by imposing certain appropriate constraints on the initial conditions of the mixed problem, the comparison principle, the internal Schauder estimate for the parabolic equation, and the Arzelà-Ascoli theorem are utilised to prove the existence of a unique local classical solution to the mixed problem.

关键词

退化 / 抛物方程 / 比较原理 / Schauder不动点定理 / 逼近问题

Key words

degeneracy / parabolic equations / comparison principle / Schauder fixed point theorem / approximation problems

作者贡献声明

赵亚双提出研究的问题、设计研究方案、确定理论框架、撰写初稿、修订内容和润色语言。许少鹏提供学术指导、把握研究方向、提出关键修改意见、调整论文逻辑和提供经费支持。

AI使用声明

本文的英文题名、摘要和关键词均采用DeepSeek翻译生成，并进行了部分修改。

利益冲突声明

作者声明，其不存在已知的可能影响本论文所报告工作的竞争性经济利益或个人关系。

伦理声明

不适用：本研究未涉及人类或动物实验。

数据可用性

不适用。

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