Persistence of mass in a chemotaxis system with logistic source
Persistence of mass in a chemotaxis system with logistic source
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DOI：
10.1016/j.jde.2015.07.019
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发表时间：
2015-12
影响因子：
2.4
通讯作者：
Youshan Tao;M. Winkler
中科院分区：
文献类型：
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作者：
Youshan Tao;M. Winkler
本文研究了趋化系统（-）{u t= Δ u−χ∇⋅(u∇v)+ r u−μ u 2, x∈Ω， t> 0, v t= Δ v−v+ u, x∈Ω， t> 0，在有界凸域Ω∧r n， n≥1，具有正常数χ， r， μ的齐次Neumann边界条件下的动力学性质。数值模拟和一些严格的证据表明，与χ= 0的情况相比，取决于r、μ和| Ω|的相对大小，这个问题可能表现出相当复杂的解行为，包括意想不到的影响，如在Ω的大子域中u的量的渐近衰减。目前的工作表明，任何这种消光现象，如果发生的话，必然是空间局部性质的。而人口总体上是持续的。更准确地说，证明了对于任意具有u <s:2> 0的（- -）的非负全局经典解（u, v），可以找到∫Ω u (x, t) dx≥m - - -对于所有的t - - - >。在这种情况下，证明是基于对泛函∫Ω ln (u)的新颖分析，通过适当地利用∫Ω ln (u)，∫Ω u和∫Ω v 2的合适线性组合的微分不等式，推导出这个量沿合适时间序列的下界。
This paper studies the dynamical properties of the chemotaxis system (⋆){u t= Δ u− χ∇⋅(u∇ v)+ r u− μ u 2, x∈ Ω, t> 0, v t= Δ v− v+ u, x∈ Ω, t> 0, under homogeneous Neumann boundary conditions in bounded convex domains Ω⊂ R n, n≥ 1, with positive constants χ, r and μ. Numerical simulations but also some rigorous evidence have shown that depending on the relative size of r, μ and| Ω|, in comparison to the well-understood case when χ= 0, this problem may exhibit quite a complex solution behavior, including unexpected effects such as asymptotic decay of the quantity u within large subdomains of Ω. The present work indicates that any such extinction phenomenon, if occurring at all, necessarily must be of spatially local nature, whereas the population as a whole always persists. More precisely, it is shown that for any nonnegative global classical solution (u, v) of (⋆) with u≢ 0 one can find m⋆> 0 such that∫ Ω u (x, t) d x≥ m⋆ for all t> 0. The proof is based on an, in this context, apparently novel analysis of the functional∫ Ω ln u, deriving a lower bound for this quantity along a suitable sequence of times by appropriately exploiting a differential inequality for a suitable linear combination of∫ Ω ln u,∫ Ω u and∫ Ω v 2.