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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2017, Vol. 34 ›› Issue (6): 629-636.doi: 10.3969/j.issn.1005-3085.2017.06.006

• • 上一篇    下一篇

具非线性源的二阶非线性抛物方程的淬火现象

牛   屹1,2,   彭秀艳2,   张明有2,   沈继红3   

  1. 1- 山东师范大学信息科学与工程学院，济南  250001
    2- 哈尔滨工程大学自动化学院，哈尔滨  150001
    3- 哈尔滨工程大学理学院，哈尔滨  150001
  • 收稿日期:2015-10-30 接受日期:2016-09-21 出版日期:2017-12-15 发布日期:2018-02-15
  • 通讯作者: 彭秀艳 E-mail: pxygll@163.com
  • 基金资助:
    国家自然科学基金(61503091).

Quenching Phenomena for Second-order Nonlinear Parabolic Equation with Nonlinear Source

NIU Yi1,2,   PENG Xiu-yan2,   ZHANG Ming-you2,   SHEN Ji-hong3   

  1. 1- School of Information Science and Engineering, Shandong Normal University, Jinan 250001
    2- College of Automation, Harbin Engineering University, Harbin 150001
    3- College of Science, Harbin Engineering University, Harbin 150001
  • Received:2015-10-30 Accepted:2016-09-21 Online:2017-12-15 Published:2018-02-15
  • Contact: X. Peng. E-mail address: pxygll@163.com}
  • Supported by:
    The National Natural Science Foundation of China (61503091).

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摘要： 本文研究一类二阶非线性抛物型方程的柯西问题，在给出非线性源项的限定条件下得到了该问题的淬火现象．在针对更一般的非线性吸收源项时，发现非线性源项中指数的大小和确定的初值会影响问题的解淬火时间的早晚．在非线性吸收源项的结构发生变化时，二阶非线性抛物型方程柯西问题的淬火现象会消失．最后，利用仿真实验真实地描述了淬火现象的性态，并得出吸收源项的指数越大，发生淬火的时间就越小．本文所用主要研究方法是比较原理，极大值原理和特征函数法．

关键词: 抛物方程, 非线性源项, 淬火现象, 无界区域

Abstract: In this paper, we investigate a class of second-order nonlinear parabolic equations. Under some conditions about the nonlinear source term, we obtain the quenching phenomena of the Cauchy problem. It is shown that, with more generally nonlinear absorption, the solution quenches in finite time under some restrictions on the exponents of the source term and the initial data. When the structure of the nonlinear absorption is changed, the solution of the Cauchy problem for the second-order nonlinear parabolic equation may exist globally. In the end, we illustrate the behavior of quenching phenomena through simulation experiments. The larger the source term exponents are, the shorter the quenching time is. Our main tools are the comparison principle, the maximum principle and the eigenfunction method.

Key words: parabolic equation, nonlinear source term, quenching phenomena, unbounded domain

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