论文标题
逻辑型抛物线型涡旋keller-segel Systems中趋化性崩溃的放松参数条件
Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic-elliptic Keller-Segel systems
论文作者
论文摘要
我们研究了具有非线性扩散和逻辑源的抛物线 - 纤维化凯勒 - 塞格系统的两个变体中的有限时间爆炸。在$ n $二维球中，我们考虑\ begin {align*} \ begin {cases} u_t = \ nabla \ cdot（（（U+1）^{m -1）^{m -1} \ nabla u -u \ u \ nabla v） \int_Ω ΔV -v + u，\ end {case} \ tag {pe} \ end {align*}，其中$λ$和$μ$在空间上给出了radial radial非负功能，$ m，κ> 0 $被赋予受到进一步条件的参数。在统一的治疗中，我们在上述系统中的两个系统中的溶液上点上的较高估计值之间建立了一座桥梁，然后使用这种新发现的连接，使用了$ m，κ$的扩展参数范围，从而提供了扩展的参数范围。特别是，对于常数$λ，μ> 0 $，我们发现有一些初始数据导致（Jl）如果\ begin {alignat*} {2} {2} {2} 0 \ leqκ＆<\ min \ min \ min \ left \ left \ left \ weft \ {\ frac {1} \ right \} && \ qquad \ text {if} m \ in \ left [\ frac {\ frac {2} {n}，\ frac {2n-2} {n} {n} {n} \ right） κ＆<\ min \ left \ {\ frac {1} {2}，\ frac {n-1} n- \ frac {m} 2 \ right \} && \ qquad \ qquad \ qquad \ text {if text {if} m \ in \ in \ in \ weft（0，\ frac {2} n，\ frac {2}如果$ m \在[1，\ frac {2n-2} {n}）$和\ begin {align*} 0 \ leqκ<\ min \ left \ left \ left \ {\ frac {（m-1）n + 1}} {2（n-1）{2（n-1）} {2（n-1）}，\ frac \ frac {n-2-（n-2-（m-2）n} n} n-n} n}（n} n} { \ end {align*}
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*}
