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arXiv 2103.00420math.AP
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具有信号抑制运动的拟线性Keller-Segel系统的渐近行为

Asymptotic behavior of a quasilinear Keller--Segel system with signal-suppressed motility

Chi Xu, Yifu Wang

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摘要：

本文研究了密度抑制运动模型：光滑有界凸域$Ω\subset {\mathbb{R}}^2$中的$u_{t}=Δ(\displaystyle\frac{u^m}{v^α}) +βuf(w), v_{t}=DΔv-v+u, w_{t}=Δw-uf(w)$，其中$m>1$，$α>0, β>0$和$D>0$为参数，响应函数$f$满足$(0,\infty)$中的$f\in C^1([0,\infty)), f(0)=0, f(w)>0$.该系统描述了密度抑制运动的大肠杆菌细胞的时空模式的形成过程中，通过所谓的自陷机制。基于对偶性，证明了对于适当大的$D$，该问题至少存在一个全局弱解$(u,v,w)$，它将渐近收敛到空间一致平衡点$(\overline{u_0}+β\overline{w_0},\overline{u_0}+β\overline{w_0},0)$，且$\overline{u_0}=\frac1{|Ω|}\int_Ωu(x,0)dx $和$\overline{w_0}=\frac1{|Ω|}\int_Ωw(x,0)dx $在$L^\infty(Ω)$中.

英文摘要：

This paper is concerned with the density-suppressed motility model: $u_{t}=Δ(\displaystyle\frac{u^m}{v^α}) +βuf(w), v_{t}=DΔv-v+u, w_{t}=Δw-uf(w)$ in a smoothly bounded convex domain $Ω\subset {\mathbb{R}}^2$, where $m>1$, $α>0, β>0$ and $D>0$ are parameters, the response function $f$ satisfies $f\in C^1([0,\infty)), f(0)=0, f(w)>0$ in $(0,\infty)$. This system describes the density-suppressed motility of Eeshcrichia coli cells in process of spatio-temporal pattern formation via so-called self-trapping mechanisms. Based on the duality argument, it is shown that for suitable large $D$ the problem admits at least one global weak solution $(u,v,w)$ which will asymptotically converge to the spatially uniform equilibrium $(\overline{u_0}+β\overline{w_0},\overline{u_0}+β\overline{w_0},0)$ with $\overline{u_0}=\frac1{|Ω|}\int_Ωu(x,0)dx $ and $\overline{w_0}=\frac1{|Ω|}\int_Ωw(x,0)dx $ in $L^\infty(Ω)$.

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