伪抛物型 p-Kirchhoff 方程的爆破解

数学物理学报 ›› 2024, Vol. 44 ›› Issue (3): 717-736.

伪抛物型 p-Kirchhoff 方程的爆破解

李锋杰^*(),李平

中国石油大学(华东) 山东青岛 266580

收稿日期:2023-05-19 修回日期:2023-10-14 出版日期:2024-06-26 发布日期:2024-05-17
通讯作者: *李锋杰，E-mail:fjli@upc.edu.cn
基金资助:
山东省自然科学基金面上项目(ZR2021MA003)

Blow-up Solutions in a p-Kirchhoff Equation of Pseudo-Parabolic Type

Li Fengjie^*(),Li Ping

College of Science, China University of Petroleum, Shandong Qingdao 266580

Received:2023-05-19 Revised:2023-10-14 Online:2024-06-26 Published:2024-05-17
Supported by:
Shandong Provincial Natural Science Foundation of China(ZR2021MA003)

摘要/Abstract

摘要：

该文研究了一类具有变指数的伪抛物型 p-Kirchhoff 方程的齐次 Dirichlet 初边值问题. 首先, 采用改进的势阱法和辅助函数法, 对初始能量进行最优分类, 得到解的爆破和整体存在的判据. 其次, 分别得到了解的爆破时间估计和整体解的大时间性态.

关键词: 伪抛物型 p-Kirchhoff 方程, 大时间性态, 爆破时间, 初始能量, 变指数

Abstract:

This paper deals with a homogeneous Dirichlet initial-boundary value problem of the p-Kirchhoff pseudo-parabolic equation involving a variable exponent. Firstly, we give the optimal classification of initial energy by using modified potential well method and the auxiliary function method and obtain the criteria of the existence of blow-up and global existence of solutions. Secondly, we show asymptotic estimates about blow-up time for blow-up solutions and large time estimate for global solutions, respectively.

Key words: p-Kirchhoff pseudo-parabolic equation, Large time estimate, Blow-up time, Initial energy, Variable exponent

中图分类号:

O175.4

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.

Li Fengjie, Li Ping. Blow-up Solutions in a p-Kirchhoff Equation of Pseudo-Parabolic Type[J]. Acta mathematica scientia,Series A, 2024, 44(3): 717-736.

参考文献

[1]	Acerbi E, Mingione G. Regularity results for stationary electro-rheological fluids. Arch Ration Mech Anal, 2002, 164(3): 213-259
[2]	Antontsev S N, Rodrigues J F. On stationary thermo-rheological viscous flows. Ann Univ Ferrara Sez VII Sci Mat, 2006, 52(1): 19-36
[3]	Cao Y, Yin J X, Wang C P. Cauchy problems of semilinear pseudo-parabolic equations. J Differential Equations, 2009, 246(12): 4568-4590
[4]	Cao Y, Zhao Q T. Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations. Electron Res Arch, 2021, 29(6): 3833-3851
[5]	Chen Y, Levine S, Rao M. Variable exponent, linear growth functionals in image restoration. SIAM J Math Appl, 2006, 66(4): 1383-1406
[6]	Chen Y X, R?dulescu V D, Xu R Z. High energy blowup and blowup time for a class of semilinear parabolic equations with singular potential on manifolds with conical singularities. Commun Math Sci, 2023, 21(1): 25-63
[7]	Fu Y Q, Xiang M Q. Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent. Appl Anal, 2016, 95(3): 524-544
[8]	Han Y Z. Finite time blowup for a semilinear pseudo-parabolic equation with general nonlinearity. Appl Math Lett, 2020, 99(7): 105986
[9]	Han Y Z, Li J. Global existence and finite time blow-up of solutions to a nonlocal $ p $-laplace equation. Math Model Anal, 2019, 24(2): 195-217
[10]	Han Y Z, 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(9): 3283-3297
[11]	Khelghati A, Baghaei K. Blow-up phenomena for a nonlocal semilinear parabolic equation with positive initial energy. Comput Math Appl, 2015, 70(5): 896-902
[12]	Lian W, Wang J, Xu R Z. Global existence and blow up of solutions for pseudo-parabolic equation with singular potential. J Differential Equations, 2020, 269(6): 4914-4959
[13]	Lin Q, Tian X T, Xu R Z, Zhang M N. Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy. Discrete Contin Dyn Syst Ser S, 2020, 13(7): 2095-2107
[14]	Liu Y, Moon B, R?dulescu V D, et al. Qualitative properties of solution to a viscoelastic Kirchhoff-like plate equation. J Math Phys, 2013, 64: 051511
[15]	Pan N, Pucci P, Xu R Z, Zhang B L. Degenerate Kirchhoff-type wave problems involving the fractional Laplacian with nonlinear damping and source terms. J Evol Equ, 2019, 19(3): 615-643}
[16]	Payne L E, Sattinger D H. Saddle points and instability of nonlinear hyperbolic equations. Israel J Math, 1975, 22(3/4): 273-303
[17]	Qu C Y, Zhou W S. Asymptotic analysis for a pseudo-parabolic equation with nonstandard growth conditions. Appl Anal, 2022, 101(13): 4701-4720
[18]	Tuan N H, Au V V, Xu R Z. Semilinear Caputo time-fractional pseudo-parabolic equations. Commun Pure Appl Anal, 2021, 20(2): 583-621
[19]	Wang X C, Xu R Z. Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation. Adv Nonlinear Anal, 2021, 10(1): 261-288
[20]	Xu H Y. Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials. Commun Anal Mech, 2023, 15(2): 132-161
[21]	Xu R Z, Lian W, Niu Y. Global well-posedness of coupled parabolic systems. Sci China Math, 2020, 63(2): 321-356
[22]	Xu R Z, Su J. Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations. J Funct Anal, 2013, 264(12): 2732-2763