arXiv 1905.045132019-05-11math.AP

具环境依赖Logistic源的二维Keller-Segel系统的瞬时爆破

Finite-time blow-up in a two-dimensional Keller--Segel system with an environmental dependent logistic source

Mario Fuest

备注：: 16 pages
DOI：: 10.1016/j.nonrwa.2019.103022

摘要：

在球$Ω= B_R(0) \subset \mathbb R^2$，$R \gt 0$ for $p \ge 1$和充分光滑函数$κ, μ: [0, R] \rightarrow [0, \infty)$中研究了趋化系统\begin{align} \label{prob:abstract} \tag{$\star$} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v) + κ(|x|) u - μ(|x|) u^p, \\ 0 = Δv - \frac{m(t)}{|Ω|} + u, \quad m(t) := \int_Ωu(\cdot, t) \end{cases} \end{align}的Neumann初边值问题.我们证明了，当$μ', -κ' \ge 0$以及$μ(s) \le μ_1 s^{2p-2}$为所有$s \in [0, R]$和一些$μ_1 \gt 0$，那么对于所有$m_0 \gt 8 π$存在$u_0 \in C^0(\overline Ω)$与$\int_Ωu_0 = m_0$和一个解决方案$(u, v)$到\eqref{prob:abstract}与初始数据$u_0$爆破在有限时间。如果加上$κ\equiv 0$，那么所有初始质量小于$8 π$的解在时间上都是全局的，显示出一定的临界质量现象。另一方面，如果$p \gt 2$，我们证明了对于所有$μ$满足$μ(s) \ge μ_1 s^{p-2-\varepsilon}$，对于所有$s \in [0, R]$和某些$μ_1, \varepsilon \gt 0$，系统\eqref{prob:abstract}对每个初始数据$0 \le u_0 \in C^0(\overline Ω)$都有一个整体经典解

英文摘要：

The Neumann initial-boundary problem for the chemotaxis system \begin{align} \label{prob:abstract} \tag{$\star$} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v) + κ(|x|) u - μ(|x|) u^p, \\ 0 = Δv - \frac{m(t)}{|Ω|} + u, \quad m(t) := \int_Ωu(\cdot, t) \end{cases} \end{align} is studied in a ball $Ω= B_R(0) \subset \mathbb R^2$, $R \gt 0$ for $p \ge 1$ and sufficiently smooth functions $κ, μ: [0, R] \rightarrow [0, \infty)$. We prove that whenever $μ', -κ' \ge 0$ as well as $μ(s) \le μ_1 s^{2p-2}$ for all $s \in [0, R]$ and some $μ_1 \gt 0$ then for all $m_0 \gt 8 π$ there exists $u_0 \in C^0(\overline Ω)$ with $\int_Ωu_0 = m_0$ and a solution $(u, v)$ to \eqref{prob:abstract} with initial datum $u_0$ blowing up in finite time. If in addition $κ\equiv 0$ then all solutions with initial mass smaller than $8 π$ are global in time, displaying a certain critical mass phenomenon. On the other hand, if $p \gt 2$, we show that for all $μ$ satisfying $μ(s) \ge μ_1 s^{p-2-\varepsilon}$ for all $s \in [0, R]$ and some $μ_1, \varepsilon \gt 0$ the system \eqref{prob:abstract} admits a global classical solution for each initial datum $0 \le u_0 \in C^0(\overline Ω)$

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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