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

Abstract

Abstract:

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

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

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

CLC Number:

O175.23

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

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