局部任意阶增长的拟线性抛物方程解的存在性

数学物理学报 ›› 2015, Vol. 35 ›› Issue (5): 833-844.

• 论文 • 下一篇

局部任意阶增长的拟线性抛物方程解的存在性

马文雅¹, 张乔夫^2,3, 崔俊芝⁴

1 河南农业大学信息与管理科学学院郑州 450002;
2 浙江大学电气工程学院杭州 310027;
3 宁波鑫高益公司浙江余姚 315400;
4 中国科学院计算数学与科学工程计算研究所北京 100190

收稿日期:2014-02-07 修回日期:2015-04-20 出版日期:2015-10-25 发布日期:2015-10-25
作者简介:张乔夫,zhangqf@lsec.cc.ac.cn
基金资助:
国家重点基础研究发展计划(973计划)项目(2012CB025904)、国家自然科学基金(90916027)和NSFC-河南人才培养联合基金(U1404102)资助

Global Existence for Some Quasi-Linear Parabolic Equations with Locally Arbitrary Growth and Memory

Ma Wenya¹, Zhang Qiaofu^2,3, Cui Junzhi⁴

1 College of Information and Management Science, Henan Agricultural University, Zhengzhou 450002;
2 College of Electrical Engineering, Zhejiang University, Hangzhou 310027;
3 Ningbo Xingaoyi Co, Ltd, Zhejiang Yuyao 315400;
4 LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190

Received:2014-02-07 Revised:2015-04-20 Online:2015-10-25 Published:2015-10-25

摘要/Abstract

摘要：

该文运用Schauder不动点方法对一类具有局部任意阶增长、含有记忆项的拟线性抛物方程证明了全局弱解的存在性.具体地,通过固定系数及源项中的函数变量构造一个线性映射,其定义域取值范围是有界的,但可以局部任意阶增长.由极值原理知其值域包含在一个有界凸集中,又注意到解关于数据的连续依赖性,所以该映射是连续的,结合紧性得知存在不动点.证明中仅要求系数关于函数变量连续,关于时空变量可测即可.另外,对含记忆项的情形也进行了考察.

关键词: 拟线性抛物方程, 不动点, 存在性, 局部任意阶增长

Abstract:

Existence theory for a kind of quasi-linear parabolic equations is established by the Schauder fixed point method. Actually a linearized map is defined by fixing the function variables in the coefficients and the right-hand term. Its domain is chosen to be bounded but a locally arbitrary growth condition is considered. Therefore its range is contained in a closed convex set through the maximum principle. This map is continuous since the solution smoothly depends on the data. Compactness is deduced from the embedding theorem, so there exists a fixed point. The coefficients are just required to be continuous with respect to the function variables, but can be only measurable with respect to the space and time variables. Moreover, memory terms are also considered.

Key words: Existence theory, Quasi-linear parabolic equations, Fixed point, Locally arbitrary growth condition

中图分类号:

O175.29

马文雅, 张乔夫, 崔俊芝. 局部任意阶增长的拟线性抛物方程解的存在性[J]. 数学物理学报, 2015, 35(5): 833-844.

Ma Wenya, Zhang Qiaofu, Cui Junzhi. Global Existence for Some Quasi-Linear Parabolic Equations with Locally Arbitrary Growth and Memory[J]. Acta mathematica scientia,Series A, 2015, 35(5): 833-844.

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局部任意阶增长的拟线性抛物方程解的存在性

Global Existence for Some Quasi-Linear Parabolic Equations with Locally Arbitrary Growth and Memory

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