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.

参考文献

1	Brezis H . Functional Analysis. Sobolev Spaces and Partial Differential Equations. New York: Springer, 2010
2	Chipot M , Valente V , Caffarelli G V . Remarks on a nonlocal problem involving the Dirichlet energy. Rend Semin Mat U Pad, 2003, 110, 199- 220
3	Chipot M , Savitska T . Nonlocal p-Laplace equations depending on the $ L.p $ norm of the Gradient. Adv Differential Equ, 2014, 19, 997- 1020
4	D'Ancona P , Shibata Y . On global solvability of non-linear viscoelastic equation in the analytic category. Math Methods Appl Sci, 1994, 17, 477- 489
5	D'Ancona P , Spagnolo S . Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent Math, 1992, 108, 247- 262
6	Fu Y , Xiang M . Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent. Appl Anal, 2016, 95, 524- 544
7	Gazzola F , Weth T . Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level. Differ Integral Equ, 2005, 18, 961- 990
8	Ghisi M , Gobbino M . Hyperbolic-parabolic singular perturbation for middly degenerate Kirchhoff equations: time-decay estimates. J Differ Equ, 2008, 245, 2979- 3007
9	Han Y . Finite time blowup for a semilinear pseudo-parabolic equation with general nonlinearity. Appl Math Lett, 2020, 99, 1- 7
10	Han Y , Gao W , Sun Z , Li H . Upper and lower bounds of blow-up time to a parabolic type Kirchhoff equation with arbitrary initial energy. Comput Math Appl, 2018, 76, 2477- 2483
11	Han Y , Li Q . Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy. Comput Math Appl, 2018, 75, 3283- 3297
12	Levine H A . Some nonexistence and stability theorems for solutions of formally parabolic equations of the form $ Pu_t = -Au+F(u) $. Arch Ration Mech Anal, 1973, 51, 371- 386
13	Li J , Han Y . Global existence and finite time blow-up of solutions to a nonlocal $ p $-Laplace equation. Math Model Anal, 2019, 24, 195- 217
14	Liao M , Gao W . Blow-up phenomena for a nonlocal p-Laplace equation with Neumann boundary conditions. Arch Math, 2017, 108, 313- 324
15	Liu Y , Zhao J . On potential wells and applications to semilinear hyperbolic equations and parabolic equations. Nonlinear Anal TMA, 2006, 64, 2665- 2687
16	Nishihara K . On a global solution of some quasilinear hyperbolic equation. Tokyo J Math, 1984, 7, 437- 459
17	Payne L E , Sattinger D H . Saddle points and instability of nonlinear hyperbolic equtions. Israel J Math, 1975, 22, 273- 303
18	Philippin G A , Proytcheva V . Some remarks on the asymptotic behaviour of the solutions of a class of parabolic problems. Math Methods Appl Sci, 2006, 29, 297- 307
19	Xu R . Asymptotic behavior and blow up of solutions for semilinear parabolic equations at critical energy level. Math Comput Simulat, 2009, 80, 808- 813
20	Xu R , Su J . Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations. J Funct Anal, 2013, 264, 2732- 2763
21	Zheng S , MChipot M . Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms. Asymptotic Anal, 2005, 45, 301- 312

[1]	程钰淼, 方铭梓, 王友军. 一类局部和非局部椭圆型方程的 Brezis-Nirenberg 型问题[J]. 数学物理学报, 2025, 45(5): 1519-1534.
[2]	李倩, 邢艳元. 具有阻尼项的粘弹性波动方程解的高能爆破[J]. 数学物理学报, 2025, 45(3): 748-755.
[3]	李建军,李阳晨. 一类具有对数非线性源项的分数阶 $p$-Laplace 扩散方程解的存在性和爆破[J]. 数学物理学报, 2025, 45(2): 465-478.
[4]	石金诚, 刘炎. 带不同幂次型非线性项的半线性三阶发展方程整体解的存在性与爆破[J]. 数学物理学报, 2024, 44(6): 1550-1562.
[5]	高晓茹, 李建军, 徒君. 一类带有时变系数的分数阶扩散方程解的爆破性[J]. 数学物理学报, 2024, 44(5): 1230-1241.
[6]	王伟敏, 闫威. 修正Kawahara方程的收敛问题与色散爆破[J]. 数学物理学报, 2024, 44(3): 595-608.
[7]	段雪亮, 吴晓凡, 魏公明, 杨海涛. 带临界指数的Kirchhoff型线性耦合方程组正解的多重性[J]. 数学物理学报, 2024, 44(3): 699-716.
[8]	李锋杰, 李平. 伪抛物型 p-Kirchhoff 方程的爆破解[J]. 数学物理学报, 2024, 44(3): 717-736.
[9]	周英告, 李周欣. 环绕定理在退化的椭圆型方程上的应用[J]. 数学物理学报, 2023, 43(6): 1759-1773.
[10]	简慧, 龚敏, 王莉. 带部分调和势的非齐次非线性 Schrödinger 方程的爆破解[J]. 数学物理学报, 2023, 43(5): 1350-1372.
[11]	陈清方,廖家锋,元艳香. 一类带临界指数的 Schrödinger-Newton 系统正解的存在性[J]. 数学物理学报, 2023, 43(5): 1373-1381.
[12]	沈旭辉,丁俊堂. 具有梯度源项和非线性边界条件的多孔介质方程组解的爆破[J]. 数学物理学报, 2023, 43(5): 1417-1426.
[13]	蔡森林,周寿明,陈容. 大振幅浅水波模型的柯西问题研究[J]. 数学物理学报, 2023, 43(4): 1197-1120.
[14]	欧阳柏平. 非线性记忆项的Euler-Poisson-Darboux-Tricomi方程解的爆破[J]. 数学物理学报, 2023, 43(1): 169-180.
[15]	冯美强, 张学梅. Monge-Ampère方程边界爆破解的最优估计和不存在性[J]. 数学物理学报, 2023, 43(1): 181-202.