高初始能量下半线性角退化抛物方程组解的整体存在性与有限时间爆破——献给陈化教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (2): 584-603.

高初始能量下半线性角退化抛物方程组解的整体存在性与有限时间爆破——献给陈化教授 70 寿辰

徐辉阳^*(), 李策策()

河南科技大学数学与统计学院河南洛阳 471023

收稿日期:2025-12-16 修回日期:2026-02-05 出版日期:2026-04-26 发布日期:2026-04-27
通讯作者: 徐辉阳 E-mail:xuhuiyang@haust.edu.cn;ceceli@haust.edu.cn
作者简介:李策策, Email：ceceli@haust.edu.cn
基金资助:
国家自然科学基金(12101194);河南省自然科学基金(262300421837);河南省自然科学基金(242300420243)

Finite Time Blow-Up and Global Solutions Under High Energy Initial Data for Coupled Semilinear Corner-Degenerate Parabolic Systems

Huiyang Xu^*(), Cece Li()

School of Mathematics and Statistics, Henan University of Science and Technology, Henan Luoyang 471023

Received:2025-12-16 Revised:2026-02-05 Online:2026-04-26 Published:2026-04-27
Contact: Huiyang Xu E-mail:xuhuiyang@haust.edu.cn;ceceli@haust.edu.cn
Supported by:
NSFC(12101194);Natural Science Foundation of Henan Province(262300421837);Natural Science Foundation of Henan Province(242300420243)

摘要/Abstract

摘要：

该文在角奇异流形上研究了带奇异位势项的耦合半线性抛物方程组的初边值问题.在高初始能量的条件下, 分别得到了一个充分条件来判别问题 (1.1) 解的整体存在性与有限时间爆破. 此外, 利用 Levine 的凹函数方法, 得到了任意初始能量下使得解有限时间爆破的充分条件以及相应的爆破时间估计.

关键词: 角退化抛物方程, 奇异位势, 高初始能量, 整体存在, 有限时间爆破

Abstract:

This paper investigates the initial-boundary value problem for a class of coupled semilinear degenerate parabolic equations with singular potential term on manifolds with corner singularities. Under the condition of high initial energy, a sufficient condition is established to describe the global existence and finite-time blow-up of solutions to problem (1.1), respectively. Furthermore, by employing Levine's concavity method, we prove that solutions blow up in finite time for any initial energy, and derive an upper bound for the blow-up time.

Key words: corner-degenerate parabolic equations, singular potentials, high initial energy, global existence, finite time blow-up

中图分类号:

O175

徐辉阳, 李策策. 高初始能量下半线性角退化抛物方程组解的整体存在性与有限时间爆破——献给陈化教授 70 寿辰[J]. 数学物理学报, 2026, 46(2): 584-603.

Huiyang Xu, Cece Li. Finite Time Blow-Up and Global Solutions Under High Energy Initial Data for Coupled Semilinear Corner-Degenerate Parabolic Systems[J]. Acta mathematica scientia,Series A, 2026, 46(2): 584-603.

参考文献 29

[1]	Alimohammady M, Kalleji M K. Existence result for a class of semilinear totally characteristic hypoelliptic equations with conical degeneration. J Funct Anal, 2013, 265: 2331-2356
[2]	Baras P, Goldstein J. The heat equation with a singular potential. Trans Amer Math Soc, 1984, 284: 121-139
[3]	Chen H, Liu N. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete Contin Dyn Syst, 2016, 36: 661-682
[4]	Chen H, Liu N. On the existence with exponential decay and the blow-up of solutions for coupled systems of semilinear corner-degenerate parabolic equations with singular potentials. Acta Math Sci, 2021, 41B(1): 257-282
[5]	Chen H, Liu X C, Wei Y W. Dirichlet problem for semilinear edge-degenerate elliptic equations with singular potential term. J Differ Equ, 2012, 252: 4289-4314
[6]	Chen H, Liu X C, Wei Y W. Multiple solutions for semi-linear corner degenerate elliptic equations. J Funct Anal, 2014, 266: 3815-3839
[7]	Chen H, Tian S Y, Wei Y W. Multiple sign changing solutions for semi-linear corner degenerate elliptic equations with singular potential. J Funct Anal, 2016, 270: 1602-1621
[8]	Chen H, Tian S Y, Wei Y W. Multiple solutions for semi-linear corner degenerate elliptic equations with singular potential term. Communications in Contemporary Mathematics, 2017, 19(4): Art 1650043
[9]	Chen H, Luo P, Liu G W. Global solution and blow up of a semilinear heat equation with logarithmic nonlinearity. J Math Anal Appl, 2015, 422: 84-98
[10]	Chen H, Tian S Y. Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity. J Differential Equations, 2015, 285: 4424-4442
[11]	Chen Y X, Radulescu 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
[12]	Chen H, Wang J, Xu H Y. Global existence, exponential decay and finite time blow-up for a class of finitely degenerate coupled parabolic systems. Methods Appl Anal, 2021, 28(2): 173-194
[13]	Chen H, Xu H Y. Global existence, exponential decay and blow-up in finite time for a class of finitely degenerate semilinear parabolic equations. Acta Math Sci, 2019, 39B(5): 1290-1308
[14]	Egorov J V, Schulze B W. Pseudo-Differential Operators, Singularities, Applications. Oper Theory Adv Appl, Basel: Birkhäuser Verlag, 1997
[15]	Escobedo M, Herrero M A. A semilinear parabolic system in a bounded domain. Annali di Matematica pura ed applicata, 1993, 165: 315-336
[16]	Escobedo M, Levine H A. Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations. Arch Rational Mech Anal, 1995, 129: 47-100
[17]	Gazzola F, Weth T. Finite time blow up and global solutions for semilinear parabolic equations with initial data at high energy level. Differential Integral Equations, 2005, 18: 961-990
[18]	Levine H A. Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_{t} =-Au+F(u)$. Arch Ration Mech Anal, 1973, 51: 371-386
[19]	Liu Y, Zhao J. On potential wells and applications to semilinear hyperbolic equations and parabolic equations. Nonlinear Anal, 2006, 64: 2665-2687
[20]	Mazzeo R. Elliptic theory of differential edge operators, I. Comm Partial Differ Equ, 1991, 16(10): 1616-1664
[21]	Melrose R B, Mendoza G A. Elliptic operators of totally characteristic type. MSRI Math Sci Res Institute, 1983: 47-83
[22]	Melrose R B, Piazza P. Analytic K-theory on manifolds with corners. Adv Math, 1992, 92(1): 1-26
[23]	Payne L E, Sattinger D H. Saddle points and instability of nonlinear hyperbolic equations. Israel Journal of Mathematics, 1975, 22: 273-303
[24]	Schulze B W. Mellin representations of pseudo-differential operators on manifolds with corners. Ann Global Anal Geom, 1990, 8(3): 261-297
[25]	Schulze B W. Boundary Value Problems and Singular Pseudo-Differential Operators. Chichester: John Wiley, 1998
[26]	Souplet P, Zhang Q S. Stability for semilinear parabolic equations with decaying potentials in $\mathbb{R}^n$ and dynamical approach to the existence of ground states. Ann Inst H Poincaré Anal Non Linéaire, 2002, 19: 683-703
[27]	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
[28]	Xu H Y. Finite time blow-up and global solutions under high energy initial data for finitely degenerate semilinear parabolic equations with singular potentials. Discrete and Continuous Dynamical Systems-Series S, 2025. DOI:10.3934/dcdss.2026014
[29]	Xu R Z, Lian W, Niu Y. Global well-posedness of coupled parabolic systems. Sci China Math, 2020, 63: 321-356