热传导方程初值问题解的最大模估计的教学探讨  

Some Teaching Discussions on the Supremum Norm Estimates of Solutions to the Initial Value Problem of the Heat Equation

作　　者：陈正争[1] 施敏加[1] CHEN Zhengzheng;SHI Minjia(School of Mathematical Sciences,Anhui University,Hefei 230601,China)

机构地区：[1]安徽大学数学科学学院,安徽合肥230601

出　　处：《安庆师范大学学报（自然科学版）》2022年第2期85-89,共5页Journal of Anqing Normal University(Natural Science Edition)

基　　金：国家自然科学基金面上项目(12071001);安徽省高等学校省级质量工程项目(2017jyxm0071);安徽省自然科学基金青年项目(2008085QA07);安徽省自然科学基金杰出青年项目(1808085J20)。

摘　　要：热传导方程初值问题解的最大模估计是“偏微分方程”课程教学的难点内容之一,目前多数教材都是通过构造辅助函数并结合初边值问题的极值原理来加以证明,然而形式特殊的辅助函数构造常使学生难以理解和掌握。与多数文献不同的是,本文不需要构造任何辅助函数以及利用初边值问题的极值原理,而仅用基本的分析方法给出了一类热传导方程初值问题解的最大模估计的一个简单证明,同时,探讨了具有一般形式的二阶线性抛物型方程初值问题解的最大模估计,丰富和改进了“偏微分方程”的教学内容和方法。The estimates on the supremum norm of solutions to the initial value problem of the heat equation is one of the difficult contents in the teaching of Partial Differential Equations.To derive it,the method of constructing special auxiliary functions together with the maximum principle of the initial-boundary value problem is widely used in most of the textbooks at present.However,the construction of auxiliary functions with special forms often makes it difficult for students to understand.Unlike most literatures,the supremum norm estimates of solutions to the initial value problem of one type of the heat equation is achieved by using only an elementary analysis,without the aid of any auxiliary functions and the maximum principle of the initial-boundary value problem in this paper.Meanwhile,the estimates on the supremum norm of solutions to the initial value problem of general second order linear parabolic equations is discussed,which enriches and improves the teaching contents and method of the course of Partial Differential Equations.

关 键 词：热传导方程 初值问题 极值原理 解的最大模估计 

分 类 号：O175.2[理学—数学]