带源项的抛物-抛物高维 Keller-Segel 方程的全局解

数学物理学报 ›› 2025, Vol. 45 ›› Issue (4): 1100-1109.

带源项的抛物-抛物高维 Keller-Segel 方程的全局解

左文文(),周寿明^*()

重庆师范大学数学科学学院重庆 401331

收稿日期:2024-11-20 修回日期:2025-03-06 出版日期:2025-08-26 发布日期:2025-08-01
通讯作者: *E-mail: zhoushouming76@163.com
作者简介:E-mail:zuowenwen999@163.com
基金资助:
重庆市教育委员会科学技术研究项目(KJZD-M202200501);重庆市教育委员会科学技术研究项目(KJQN202300544);重庆师范大学(BWLJ2023005);重庆市自然科学基金(CSTB2024NSCQ-MSX0922);重庆市自然科学基金(CSTB2024NSCQ-MSX0801)

Global Solutions for the Parabolic-Parabolic Keller-Segel Equation with a Logistic Source Term in High Dimensions

Zuo Wenwen(),Zhou Shouming^*()

College of Mathematics Science, Chongqing Normal University, Chongqing 401331

Received:2024-11-20 Revised:2025-03-06 Online:2025-08-26 Published:2025-08-01
Supported by:
Science and Technology Research Program of Chongqing Municipal Educational Commission(KJZD-M202200501);Science and Technology Research Program of Chongqing Municipal Educational Commission(KJQN202300544);Chongqing Normal University(BWLJ2023005);Natural Science Foundation of Chongqing(CSTB2024NSCQ-MSX0922);Natural Science Foundation of Chongqing(CSTB2024NSCQ-MSX0801)

摘要/Abstract

摘要：

该文主要围绕带 Logistic 源的抛物-抛物 Keller-Segel 方程的 Cauchy 问题展开研究.

$\begin{cases}u_{t}=\Delta u-\nabla \cdot(u \nabla \varphi)+c u^{2}, & x \in \mathbb{R}^{d},\quad t>0, \\ \tau \varphi_{t}=\Delta \varphi+u, & x \in \mathbb{R}^{d},\quad t>0, \\ u(0)=u_{0}, \quad\varphi(0)=\varphi_{0}, & x \in \mathbb{R}^{d},\quad t=0,\end{cases}$

其中常数 $c \in \mathbb{R}$, $ \tau>0$, $ d \geq 2$, 初值 $u_{0} \in \mathcal{P M}^{d-2}\left(\mathbb{R}^{d}\right)$, $\varphi_{0} \in \mathcal{S}^{\prime}\left(\mathbb{R}^{d}\right)$. 当 $c=0$ 时, Biler-Boritchev-Brandolese 证明了在扩散参数 $\tau \gg 1$ 的情况下, 对任意大小的初值方程都存在全局解. 运用不动点引理和尺度不变性方法, 在初值 $u_{0}$ 和 $\varphi_{0}$ 满足一定限制条件 (该条件依赖于松弛参数 $\tau$), 得到带平方 Logistic 源的抛物-抛物 Keller-Segel 方程的温和解的全局存在性和唯一性.

关键词: Keller-Segel 方程, 抛物-抛物方程, 无粘极限

Abstract:

The research mainly focuses on the Cauchy problem of the parabolic-parabolic Keller-Segel equation with a Logistic source. The equation is as follows

$\begin{cases}u_{t}=\Delta u-\nabla \cdot(u \nabla \varphi)+c u^{2}, & x \in \mathbb{R}^{d},\quad t>0, \\ \tau \varphi_{t}=\Delta \varphi+u, & x \in \mathbb{R}^{d},\quad t>0, \\ u(0)=u_{0},\quad \varphi(0)=\varphi_{0}, & x \in \mathbb{R}^{d},\quad t=0.\end{cases}$

where the constants $c \in \mathbb{R}, \tau>0, d \geq 2$, initial values $u_{0} \in \mathcal{P} \mathcal{M}^{d-2}\left(\mathbb{R}^{d}\right), \varphi_{0} \in \mathcal{S}\left(\mathbb{R}^{d}\right)$. When $c=0$, Biler-Boritchev-Brandolese proved that for any initial value equation of arbitrary size, a global solution exists in the case of the diffusion parameter $\tau \gg 1$. Using the fixed point lemma and the method of scale invariance to obtain the existence and uniqueness of mild solutions for the parabolic-parabolic Keller-Segel equation with a square Logistic source, under initial conditions $u_{0}$ and $\varphi_{0}$ have certain restrictive conditions (which depend on the parameter $\tau$).

Key words: Keller-Segel equations, parabolic equations, inviscid limit

中图分类号:

O175.23

左文文, 周寿明. 带源项的抛物-抛物高维 Keller-Segel 方程的全局解[J]. 数学物理学报, 2025, 45(4): 1100-1109.

Zuo Wenwen, Zhou Shouming. Global Solutions for the Parabolic-Parabolic Keller-Segel Equation with a Logistic Source Term in High Dimensions[J]. Acta mathematica scientia,Series A, 2025, 45(4): 1100-1109.

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