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arXiv 2005.12089math.AP
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Logistic型抛物-椭圆型Keller-Segel系统趋化崩溃的松弛参数条件

Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic-elliptic Keller-Segel systems

Tobias Black, Mario Fuest, Johannes Lankeit

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21 pages
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摘要：

研究了具有非线性扩散和Logistic源的抛物-椭圆型Keller-Segel方程组的两个变形的有限时间爆破。在$n$ -维球中，我们考虑\begin{align*} \begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u - u\nabla v) + λu - μu^{1+κ}, \\ 0 = Δv - \frac1{|Ω|} \int_Ωu + u \end{cases} \tag{JL} \end{align*}和\begin{align*} \begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u - u\nabla v) + λu - μu^{1+κ}, \\ 0 = Δv - v + u, \end{cases}\tag{PE} \end{align*}，其中$λ$和$μ$是给定的空间径向非负函数，$m, κ> 0$是给定的参数，服从进一步的条件.在一个统一的治疗，我们建立了一个桥梁，以前采用的方法爆破检测和相对较新的结果点向上估计的解决方案，在上述两个系统，然后，利用这一新发现的连接，提供扩展的参数范围$m,κ$导致有限时间爆破解决方案的存在，在空间维度3及以上。特别地，对于常数$λ, μ> 0$，我们发现存在导致爆破的初始数据，如果是\begin{alignat*}{2} 0 \leq κ&< \min\left\{\frac{1}{2}, \frac{n - 2}{n} - (m-1)_+ \right\}&&\qquad\text{if } m\in\left[\frac{2}{n},\frac{2n-2}{n}\right)\\ \text{ or }\quad 0 \leq κ&<\min\left\{\frac{1}{2},\frac{n-1}n-\frac{m}2\right\} &&\qquad \text{if } m\in\left(0,\frac{2}{n}\right), \end{alignat*}，则在（JL）中，如果是$m \in [1, \frac{2n-2}{n})$和\begin{align*} 0 \leq κ< \min\left\{\frac{(m-1) n + 1}{2(n-1)}, \frac{n - 2 - (m-1) n}{n(n-1)} \right\}. \end{align*}，则在（PE）中

英文摘要：

We study the finite-time blow-up in two variants of the parabolic-elliptic Keller-Segel system with nonlinear diffusion and logistic source. In $n$-dimensional balls, we consider \begin{align*} \begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u - u\nabla v) + λu - μu^{1+κ}, \\ 0 = Δv - \frac1{|Ω|} \int_Ωu + u \end{cases} \tag{JL} \end{align*} and \begin{align*} \begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u - u\nabla v) + λu - μu^{1+κ}, \\ 0 = Δv - v + u, \end{cases}\tag{PE} \end{align*} where $λ$ and $μ$ are given spatially radial nonnegative functions and $m, κ> 0$ are given parameters subject to further conditions. In a unified treatment, we establish a bridge between previously employed methods on blow-up detection and relatively new results on pointwise upper estimates of solutions in both of the systems above and then, making use of this newly found connection, provide extended parameter ranges for $m,κ$ leading to the existence of finite-time blow-up solutions in space dimensions three and above. In particular, for constant $λ, μ> 0$, we find that there are initial data which lead to blow-up in (JL) if \begin{alignat*}{2} 0 \leq κ&< \min\left\{\frac{1}{2}, \frac{n - 2}{n} - (m-1)_+ \right\}&&\qquad\text{if } m\in\left[\frac{2}{n},\frac{2n-2}{n}\right)\\ \text{ or }\quad 0 \leq κ&<\min\left\{\frac{1}{2},\frac{n-1}n-\frac{m}2\right\} &&\qquad \text{if } m\in\left(0,\frac{2}{n}\right), \end{alignat*} and in (PE) if $m \in [1, \frac{2n-2}{n})$ and \begin{align*} 0 \leq κ< \min\left\{\frac{(m-1) n + 1}{2(n-1)}, \frac{n - 2 - (m-1) n}{n(n-1)} \right\}. \end{align*}

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