一类抛物型Kirchhoff方程解的爆破性质

数学物理学报 ›› 2021, Vol. 41 ›› Issue (5): 1333-1346.

一类抛物型Kirchhoff方程解的爆破性质

杨慧*(),韩玉柱()

^{吉林大学数学学院长春 130012}

收稿日期:2020-04-17 出版日期:2021-10-26 发布日期:2021-10-08
通讯作者: 杨慧 E-mail:mathyh@126.com;yzhan@jlu.edu.cn
作者简介:韩玉柱, E-mail: yzhan@jlu.edu.cn
基金资助:
国家自然科学基金(11401252);吉林省教育厅基金(JJKH20190018KJ)

Blow-Up Properties of Solutions to a Class of Parabolic Type Kirchhoff Equations

Hui Yang*(),Yuzhu Han()

^{School of Mathematics, Jilin University, Changchun 130012}

Received:2020-04-17 Online:2021-10-26 Published:2021-10-08
Contact: Hui Yang E-mail:mathyh@126.com;yzhan@jlu.edu.cn
Supported by:
the NSFC(11401252);the Education Department of Jilin Province(JJKH20190018KJ)

摘要/Abstract

摘要：

该文研究一类抛物型Kirchhoff方程的初边值问题解的爆破性质.主要结果包含两个部分.第一部分中考虑了具一般扩散系数$M（\|\nabla u\|_2^2）$和一般非线性项$f（u）$的抛物型Kirchhoff方程.确立了新的有限时刻爆破准则，同时给出了爆破时刻的上下界估计.第二部分中研究了当$M（\|\nabla u\|_2^2）=a+b\|\nabla u\|_2^2$且$f（u）=|u|^{q-1}u$的情形，Han和Li就初始能量为次临界，临界和超临界的情形分别给出了当$q>3$时该问题解全局存在和有限时刻爆破的阈值结果^[11].该文补充了他们的结果，证明了$q=3$在某种意义上是该问题存在有限时刻爆破解的临界指数.

关键词: Kirchhoff方程, 一般非线性项, 爆破, 爆破时间, 临界指数

Abstract:

In this paper, blow-up properties of solutions to an initial-boundary value problem for a parabolic type Kirchhoff equation are studied. The main results contain two parts. In the first part, we consider this problem with a general diffusion coefficient $M(\|\nabla u\|_2^2)$ and general nonlinearity $f(u)$. A new finite time blow-up criterion is established, and the upper and lower bounds for the blow-up time are also derived. In the second part, we deal with the case that $M(\|\nabla u\|_2^2)=a+b\|\nabla u\|_2^2$ and $f(u)=|u|^{q-1}u$, which was considered in[Computers and Mathematics with Applications, 2018, 75:3283-3297] with $q>3$, where global existence and finite time blow-up of solutions were obtained for subcritical, critical and supercritical initial energy. Their results are complemented in this paper in the sense that $q=3$ will be shown to be critical for the existence of finite time blow-up solutions to this problem.

Key words: Kirchhoff equation, General nonlinearity, Blow-up, Blow-up time, Critical exponent

中图分类号:

杨慧,韩玉柱. 一类抛物型Kirchhoff方程解的爆破性质[J]. 数学物理学报, 2021, 41(5): 1333-1346.

Hui Yang,Yuzhu Han. Blow-Up Properties of Solutions to a Class of Parabolic Type Kirchhoff Equations[J]. Acta mathematica scientia,Series A, 2021, 41(5): 1333-1346.

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