How unstable is spatial homogeneity in Keller-Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic-elliptic cases
How unstable is spatial homogeneity in Keller-Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic-elliptic cases
复制标题
DOI：
10.1007/s00208-018-1722-8
复制
发表时间：
2018-07
影响因子：
1.4
通讯作者：
M. Winkler
中科院分区：
文献类型：
--
作者：
M. Winkler
抛物-椭圆型Keller-Segel方程组{u_t= Δ u-μ_v（u_v），&0 = Δ v-μ+ u，& quad μ：= 1| Ω|_ Ω u，.\ qquad\qquad（μ）ut= Δ u-μ·（u v），0= Δ v-μ+ u，μ：= 1| Ω|在齐次Neumann边界条件下，考虑了球Ω= BR（0）<$Rn中的Ω u，（？）主要目的是揭示在径向对称解的背景下，该问题表现出一种显然新颖的临界质量现象：即对任意n ≥ 2 n≥ 2和R> 0 R> 0，存在一个正数m_c= m_c（n，R）mc= mc（n，R），具有下列性质：当m> m_cm> mc时，对于任意非常数非负径向初始数据u_0u_0，满足μ_ Ω u_0= m μ_Ω u_0 = m，在适当定义的意义下，比由u_Ω m确定的空间齐次平衡更集中| Ω|乌木| Ω|特别地，这意味着任何非常数且径向非增的初始数据u_0 u 0满足m_ Ω u_0> m_c <$Ω u_0> mc，都会使（？）若m< m_c m< mc，则存在无穷多个非负径向函数u_0 u 0满足m_ Ω u_0= m <$Ω u 0= m，且比u <$m集中| Ω|乌木| Ω|，但仍允许从u_0 u 0发出的（？）因此，恰好在m_c mc以上的质量水平上，（）的恒定稳态具有排斥任意浓度增加扰动的极端不稳定性，其程度如此剧烈，以至于相应的轨迹在有限时间内崩溃。
Abstract The parabolic-elliptic Keller-Segel system {u_t= Δ u-∇ ⋅ (u ∇ v), &\0= Δ v-μ+ u, &\quad μ:= 1| Ω| ∫ _ Ω u,.\qquad\qquad (⋆) ut= Δ u-∇·(u∇ v), 0= Δ v-μ+ u, μ:= 1| Ω|∫ Ω u,(⋆) is considered under homogeneous Neumann boundary conditions in the ball Ω= B_R (0) ⊂ R^ n Ω= BR (0)⊂ R n. The main objective is to reveal that in the context of radially symmetric solutions, this problem exhibits an apparently novel type of critical mass phenomenon: It is shown, namely, that for any choice of n ≥ 2 n≥ 2 and R> 0 R> 0 there exists a positive number m_c= m_c (n, R) mc= mc (n, R) with the following properties: Whenever m> m_c m> mc, for any nonconstant nonnegative radial initial data u_0 u 0 with ∫ _ Ω u_0= m∫ Ω u 0= m which are, in an appropriately defined sense, more concentrated than the associated spatially homogeneous equilibrium determined by u ≡ m| Ω| u≡ m| Ω|, the corresponding initial-value problem for (⋆⋆) admits a solution blowing up in finite time; in particular, this implies that any nonconstant and radially nonincreasing initial data u_0 u 0 with ∫ _ Ω u_0> m_c∫ Ω u 0> mc enforce blow-up in (⋆⋆). If m< m_c m< mc, however, then there exist infinitely many nonnegative radial functions u_0 u 0 which satisfy ∫ _ Ω u_0= m∫ Ω u 0= m and which are more concentrated than u ≡ m| Ω| u≡ m| Ω|, but which yet allow for global bounded solutions to (⋆⋆) emanating from u_0 u 0. In consequence, precisely at mass levels above m_c mc the constant steady states of (⋆⋆) possess the extreme instability property of repelling arbitrary concentration-increasing perturbations in such a drastic sense that corresponding trajectories collapse in finite time.