The fast signal diffusion limit in a chemotaxis system with strong signal sensitivity
The fast signal diffusion limit in a chemotaxis system with strong signal sensitivity
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DOI：
10.1002/mana.201700270
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发表时间：
2017-11
影响因子：
1
通讯作者：
M. Mizukami
中科院分区：
文献类型：
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作者：
M. Mizukami
本文给出了在抛物线-抛物线信号依赖的趋化系统和它的抛物线-椭圆版本之间建立数学桥梁的见解。更精确地说，本文讨论了具有强信号灵敏度（λ）t=Δuλ -∇·（uλχ(vλ)∇vλ），λ(vλ)t=Δvλ - vλ+uλinΩ×（0,∞）的抛物-椭圆型趋化系统解的收敛性(t=Δu -∇·（uλχ(v)∇v），0=Δv - v+uinΩ×（0,∞），其中Ω是具有光滑边界的Rn （n∈n）中的有界域，λ >是常数，χ是推广χ(v)=χ0(1+v)k（χ0>0,k>1）的函数。在趋化系统中，抛物-椭圆型系统通常对抛物-抛物型系统的方法和结果具有一定的指导意义。然而，除了Ω=Rn的情况外，抛物-椭圆系统和抛物-抛物系统之间的关系尚未得到研究。也就是说，在Ω是有界域的情况下，还需要分析以下问题：抛物型-抛物型系统的解是否收敛于抛物型-椭圆型系统的解为λ d ？本文在具有强信号敏感性的趋化系统中给出了一些肯定的答案。
This paper gives an insight into making a mathematical bridge between the parabolic‐parabolic signal‐dependent chemotaxis system and its parabolic‐elliptic version. To be more precise, this paper deals with convergence of a solution for the parabolic‐parabolic chemotaxis system with strong signal sensitivity (uλ)t=Δuλ−∇·(uλχ(vλ)∇vλ),λ(vλ)t=Δvλ−vλ+uλinΩ×(0,∞)to that for the parabolic‐elliptic chemotaxis system ut=Δu−∇·(uχ(v)∇v),0=Δv−v+uinΩ×(0,∞),where Ω is a bounded domain in Rn ( n∈N ) with smooth boundary, λ>0 is a constant and χ is a function generalizing χ(v)=χ0(1+v)k(χ0>0,k>1).In chemotaxis systems parabolic‐elliptic systems often gave some guide to methods and results for parabolic‐parabolic systems. However, the relation between parabolic‐elliptic systems and parabolic‐parabolic systems has not been studied except for the case that Ω=Rn . Namely, in the case that Ω is a bounded domain, it still remains to analyze on the following question: Does a solution of the parabolic‐parabolic system converge to that of the parabolic‐elliptic system as λ↘0? This paper gives some positive answer in the chemotaxis system with strong signal sensitivity.