Global boundedness in quasilinear attraction-repulsion chemotaxis system with logistic source

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• 作者：Miaoqing Tian^a ; ^b ; ^{npctian@126.com" class="auth_mail" title="E-mail the corresponding author} ; Xiao He^c ; ^{xiaohe@mail.dlut.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Sining Zheng^a ; ^{snzheng@dlut.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Attraction&ndash ; repulsion ; Quasilinear ; Chemotaxis ; Logistic source ; Boundedness
• 刊名：Nonlinear Analysis: Real World Applications
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：30
• 期：Complete
• 页码：1-15
• 全文大小：622 K

文摘

This paper studies the quasilinear attraction–repulsion chemotaxis system with logistic source u_t=∇⋅(D(u)∇u)−χ∇⋅(Φ(u)∇v)+ξ∇⋅(Ψ(u)∇w)+f(u)$u_t=\nabla \cdot \left(D\left(u\right)\nabla u\right)-\chi \nabla \cdot \left(\Phi \left(u\right)\nabla v\right)+\xi \nabla \cdot \left(\Psi \left(u\right)\nabla w\right)+f\left(u\right)$, τv_t=Δv+αu−βv$\tau v_t=\Delta v+\alpha u-\beta v$, τ∈{0,1}$\tau \in \{0,1\}$, 0=Δw+γu−δw$0=\Delta w+\gamma u-\delta w$, in bounded domain Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, N≥1$N\ge 1$, subject to the homogeneous Neumann boundary conditions, D,Φ,Ψ∈C²[0,+∞)$D,\Phi ,\Psi \in C^2[0,+\infty )$ nonnegative, with D(s)≥(s+1)^p$D\left(s\right)\ge {(s+1)}^p$ for s≥0$s\ge 0$, Φ(s)≤χs^q$\Phi \left(s\right)\le \chi s^q$, ξs^r≤Ψ(s)≤ζs^r$\xi s^r\le \Psi \left(s\right)\le \zeta s^r$ for s>1$s>1$, and f$f$ smooth satisfying f(s)≤μs(1−s^k)$f\left(s\right)\le \mu s(1-s^k)$ for s>0$s>0$, f(0)≥0$f\left(0\right)\ge 0$. It is proved that if the attraction is dominated by one of the other three mechanisms with $max\{r,k,p+\frac{2}{N}\}>q$, then the solutions are globally bounded. Under more interesting balance situations, the behavior of solutions depends on the coefficients involved, i.e., the upper bound coefficient χ$\chi$ for the attraction, the lower bound coefficient ξ$\xi$ for the repulsion, the logistic source coefficient μ$\mu$, as well as the constants α$\alpha$ and γ$\gamma$ describing the capacity of the cells u$u$ to produce chemoattractant and chemorepellent respectively. Three balance situations (attraction–repulsion balance, attraction–logistic source balance, and attraction–repulsion–logistic source balance) are considered to establish the boundedness of solutions for the parabolic–elliptic–elliptic case (with τ=0$\tau =0$) and the parabolic–parabolic–elliptic case (with τ=1$\tau =1$) respectively.