Rockland 热方程的临界 Fujita 指数和爆破分析

数学物理学报 ›› 2023, Vol. 43 ›› Issue (3): 795-807.

Rockland 热方程的临界 Fujita 指数和爆破分析

杨志鹏()

云南师范大学数学学院昆明 650500

收稿日期:2021-11-05 修回日期:2022-10-19 出版日期:2023-06-26 发布日期:2023-06-01
通讯作者: 杨志鹏 E-mail:yangzhipeng326@163.com
基金资助:
国家自然科学基金(12261107);国家自然科学基金(12101546);云南省现代分析数学及其应用重点实验室

Critical Fujita Exponent and Blow-up Results for the Rockland Heat Equation

Yang Zhipeng()

Department of Mathematics, Yunnan Normal University, Kunming 650500

Received:2021-11-05 Revised:2022-10-19 Online:2023-06-26 Published:2023-06-01
Contact: Zhipeng Yang E-mail:yangzhipeng326@163.com
Supported by:
NSFC(12261107);NSFC(12101546);Yunnan Key Laboratory of Modern Analytical Mathematics and Applications

摘要/Abstract

摘要：

该文研究了如下非线性 Rockland 热方程柯西问题的 Fujita 指数和不存在结果

$ \left\{\begin{array}{ll} u_{t}(t,x)+{\cal R}_{x}u(t,x)=|u(t,x)|^{p}, &(t,x) \in (0,+\infty)\times{\Bbb G}:=\Omega, \\ u(0,x)=u_{0}(x), &x \in {\Bbb G}. \end{array}\right. $

考虑临界 Fujita 指数并利用 ODE 方法研究其爆破分析结果. 证明中主要用到 Rockland 算子的热核形式.

关键词: Rockland 算子, 抛物方程, Fujita 指数

Abstract:

We obtain the subcritical Fujita exponent and nonexistence result for the Cauchy problem of the nonlinear Rockland heat equation

$\begin{eqnarray*} \left\{\begin{array}{ll} u_{t}(t,x)+{\cal R}_{x}u(t,x)=|u(t,x)|^{p}, &(t,x) \in (0,+\infty)\times{\Bbb G}:=\Omega, \\ u(0,x)=u_{0}(x), & x \in {\Bbb G}. \end{array}\right. \end{eqnarray*}$

In this paper, we consider the critical Fujita exponent and obtain the blow-up result by an ODE method. Central to our proof is the heat kernel for Rockland operator.

Key words: Rockland operator, Parabolic equation, Fujita exponent

中图分类号:

O175

杨志鹏. Rockland 热方程的临界 Fujita 指数和爆破分析[J]. 数学物理学报, 2023, 43(3): 795-807.

Yang Zhipeng. Critical Fujita Exponent and Blow-up Results for the Rockland Heat Equation[J]. Acta mathematica scientia,Series A, 2023, 43(3): 795-807.

参考文献 11

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