带有线性项的超临界非线性热方程的临界范数爆破——献给邓引斌教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (4): 1320-1343.

带有线性项的超临界非线性热方程的临界范数爆破——献给邓引斌教授 70 寿辰

程婷(), 姜哲宇(), 王雨莹^*()

华中师范大学数学与统计学学院, 非线性分析教育部重点实验室, 数学物理湖北省重点实验室武汉 430079

收稿日期:2025-11-24 修回日期:2026-01-06 出版日期:2026-08-26 发布日期:2026-06-10
通讯作者: 王雨莹 E-mail:tcheng@ccnu.edu.cn;983535654@qq.com;yuyingwang@mails.ccnu.edu.cn
作者简介:程婷,E-mail: tcheng@ccnu.edu.cn;
姜哲宇, E-mail: 983535654@qq.com
基金资助:
国家自然科学基金(12271195);国家自然科学基金(12271196);国家自然科学基金(12271197)

Critical Norm Blow-Up Problem for Supercritical Nonlinear Heat Equation with a Linear Term

Ting Cheng(), Zheyu Jiang(), Yuying Wang^*()

School of Mathematics and Statistics, Key Laboratory of Nonlinear Analysis and Applications (Ministry of Education), Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079

Received:2025-11-24 Revised:2026-01-06 Online:2026-08-26 Published:2026-06-10
Contact: Yuying Wang E-mail:tcheng@ccnu.edu.cn;983535654@qq.com;yuyingwang@mails.ccnu.edu.cn
Supported by:
NSFC(12271195);NSFC(12271196);NSFC(12271197)

摘要/Abstract

摘要：

该文主要研究如下非线性热方程解的临界范数爆破问题

$\left\{ \begin{aligned}&{u_t} - \Delta u = {\left| u \right|^{p - 1}}u + au,\quad &&(x,t)\in {\mathbb{R}^{n}} \times (0,T), \hfill \\&u( \cdot,0)={{u}_0},\quad \quad \quad \quad \ \quad &&x\in {\mathbb{R}^{n}}, \hfill \\\end{aligned} \right.$

其中 $p>1, n\geqslant3, a \leqslant 0$. 对于 $a=0$, Miura H 和 Takahashi J [Miura H, Takahashi J. arXiv: 2310.09750] 证明了当 $p>p_S$ 时, 如果极大存在时间 $T$ 有限, 那么 $\mathop {\lim }\limits_{t \to T} \|u( \cdot,t)\|{_{{L^{{q_c}}}({\mathbb{R}^n})}} = \infty $, 其中 $q_c=n(p-1)/2$, $ p_{S} = (n+2)/(n-2).$ 而对于更一般的情形 $a \leqslant 0$, 该文将证明当 $p>p_S$ 时, 仍然有 $\mathop {\lim }\limits_{t \to T} \|u( \cdot,t)\|{_{{L^{{q_c}}}({\mathbb{R}^n})}} = \infty $.

关键词: 非线性热方程, 超临界, 临界范数爆破, 缺陷测度, $\varepsilon$-正则性

Abstract:

In this paper, we are concerned with the critical norm blow-up problem to the following nonlinear heat equation

where $p>1, n\geqslant3, a \leqslant 0$. For $a=0$, Miura H and Takahashi J [Miura H, Takahashi J. arXiv: 2310.09750] have proved that when $p>p_S$, if the maximal time $T$ is finite, then $\mathop {\lim }\limits_{t \to T} \|u( \cdot,t)\|{_{{L^{{q_c}}}({\mathbb{R}^n})}} = \infty $, where $q_c=n(p-1)/2$, $ p_{S} = (n+2)/(n-2).$ For general $a \leqslant 0$, we will prove that when $p>p_S$, we also have $\mathop {\lim }\limits_{t \to T} \|u( \cdot,t)\|{_{{L^{{q_c}}}({\mathbb{R}^n})}} = \infty $.

Key words: nonlinear heat equation, supcritical, critical norm blow-up, defect measure, $\varepsilon$-regularity

中图分类号:

O175.23

程婷, 姜哲宇, 王雨莹. 带有线性项的超临界非线性热方程的临界范数爆破——献给邓引斌教授 70 寿辰[J]. 数学物理学报, 2026, 46(4): 1320-1343.

Ting Cheng, Zheyu Jiang, Yuying Wang. Critical Norm Blow-Up Problem for Supercritical Nonlinear Heat Equation with a Linear Term[J]. Acta mathematica scientia,Series A, 2026, 46(4): 1320-1343.

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