arXiv 2103.001592021-03math.AP

具Logistic源的拟线性抛物-椭圆型Keller-Segel方程组的爆破

Blow-up in a quasilinear parabolic-elliptic Keller-Segel system with logistic source

Yuya Tanaka

摘要：

本文研究具Logistic源的拟线性抛物-椭圆型Keller-Segel方程组\begin{align*} u_t=Δ(u+1)^m - χ\nabla \cdot (u(u+1)^{α- 1} \nabla v) + λ(|x|) u - μ(|x|) u^κ, \quad 0=Δv - v + u, \quad x\inΩ,\ t>0, \end{align*}，其中$Ω:=B_{R}(0)\subset\mathbb{R}^n\ (n\ge3)$是具有某些$R>0$的球，$m>0$，$χ>0$，$α>0$和$κ\ge1$，$λ$和$μ$是空间径向非负函数.关于这个问题，Winkler（Z。Angew.数学物理; 2018; 69; Art. 69，40）发现了$κ$的条件，使得当$m=α=1$。在$m=1$和$α\in(0,1)$以及$λ$和$μ$为常数的情况下，在先前的论文（Math. Methods Appl.Sci.; 2020; 43; 7372-7396）。此外，在$m\ge1$和$α=1$ Black的情况下，Fuest和Lankeit（arXiv：2005.12089[math.AP]）表明，存在初始数据，使得在$m$和$κ$的某些条件下，解在有限时间内爆破。本文的目的是给出$m\ge1$，$α>0$和$κ\ge1$解在有限时间内爆破的条件.

英文摘要：

This paper deals with the quasilinear parabolic-elliptic Keller-Segel system with logistic source, \begin{align*} u_t=Δ(u+1)^m - χ\nabla \cdot (u(u+1)^{α- 1} \nabla v) + λ(|x|) u - μ(|x|) u^κ, \quad 0=Δv - v + u, \quad x\inΩ,\ t>0, \end{align*} where $Ω:=B_{R}(0)\subset\mathbb{R}^n\ (n\ge3)$ is a ball with some $R>0$; $m>0$, $χ>0$, $α>0$ and $κ\ge1$; $λ$ and $μ$ are spatially radial nonnegative functions. About this problem, Winkler (Z. Angew. Math. Phys.; 2018; 69; Art. 69, 40) found the condition for $κ$ such that solutions blow up in finite time when $m=α=1$. In the case that $m=1$ and $α\in(0,1)$ as well as $λ$ and $μ$ are constant, some conditions for $α$ and $κ$ such that blow-up occurs were obtained in a previous paper (Math. Methods Appl. Sci.; 2020; 43; 7372-7396). Moreover, in the case that $m\ge1$ and $α=1$ Black, Fuest and Lankeit (arXiv:2005.12089[math.AP]) showed that there exists initial data such that solutions blow up in finite time under some conditions for $m$ and $κ$. The purpose of the present paper is to give conditions for $m\ge1$, $α>0$ and $κ\ge1$ such that solutions blow up in finite time.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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