Critical mass for Keller–Segel systems with supercritical nonlinear sensitivity,Mathematical Models and Methods in Applied Sciences

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Critical mass for Keller–Segel systems with supercritical nonlinear sensitivity
Mathematical Models and Methods in Applied Sciences ( IF 3.6 ) Pub Date : 2023-07-14 , DOI: 10.1142/s0218202523400079
Xuan Mao ₁ , Yuxiang Li ₁

Affiliation

This paper is concerned with the following radially symmetric Keller–Segel systems with nonlinear sensitivity $u_{t} = Δ u - \nabla \cdot (u {(1 + u)}^{α - 1} \nabla v)$ and $0 = Δ v - ⨍_{Ω} u d x + u$ , posed on $Ω = {x \in ℝ^{n} : | x | < R}$ $(n \geq 2)$ and subjected to homogeneous Neumann boundary conditions. It is well-known that $\frac{2}{n}$ is the critical exponent of the systems in the sense that all solutions exist globally if $α < \frac{2}{n}$ and there exist finite-time blowup solutions if $α > \frac{2}{n}$ . Here we consider the supercritical case $α \geq \frac{2}{n}$ and show a critical mass phenomenon. Precisely, we prove that there exists a critical mass $m_{c} : = m_{c} (n, R, α)$ such that

(1)	for arbitrary nonincreasing nonnegative initial data $u_{0} (x) = u_{0} (\| x \|)$ with $\int_{Ω} u_{0} > m_{c}$ and $u_{0} ≢ ⨍_{Ω} u_{0}$ , the corresponding solution blows up in finite time if $α > \frac{2}{n}$ , and if $α = \frac{2}{n}$ we can only prove that the solution blows up in finite time or infinite time;
(2)	for some nonincreasing nonnegative initial data with $\int_{Ω} u_{0} < m_{c}$ , the corresponding solutions are globally bounded.

Our results extend that of Winkler’s paper [M. Winkler, How unstable is spatial homogeneity in Keller–Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic–elliptic cases, Math. Ann. 373 (2019) 1237–1282], where he proved similar results for the system with $α = 1$ .

中文翻译：

具有超临界非线性灵敏度的 Keller-Segel 系统的临界质量

本文涉及以下具有非线性灵敏度的径向对称 Keller-Segel 系统 $你_{t} = Δ 你 - \nabla \cdot （你 {（ 1 + 你）}^{α - 1} \nabla v ）$ 和 $0 = Δ v - ⨍_{Ω} 你 d X + 你$ ，摆在 $Ω = {X ε ℝ^{n} : | X | < 右}$ $（ n \geq 2 ）$ 并受到齐次诺依曼边界条件的影响。众所周知， $\frac{2}{n}$ 是系统的关键指数，因为所有解决方案都存在于全球范围内，如果 $α < \frac{2}{n}$ 并且存在有限时间爆炸解，如果 $α > \frac{2}{n}$ 。这里我们考虑超临界情况 $α \geq \frac{2}{n}$ 并表现出临界质量现象。准确地说，我们证明存在临界质量 $米_{C} : = 米_{C} （ n, 右, α ）$ 这样

(1)	对于任意非增非负初始数据 $你_{0} （ X ） = 你_{0} （ \| X \| ）$ 和 $\int_{Ω} 你_{0} > 米_{C}$ 和 $你_{0} ≢ ⨍_{Ω} 你_{0}$ ，相应的解在有限时间内爆炸，如果 $α > \frac{2}{n}$ ，而如果 $α = \frac{2}{n}$ 我们只能证明解在有限时间内或无限时间内爆炸；
(2)	对于一些非递增非负初始数据 $\int_{Ω} 你_{0} < 米_{C}$ ，相应的解是全局有界的。