Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source
Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source
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DOI：
10.1002/mma.2992
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发表时间：
2014-10
影响因子：
2.9
通讯作者：
Xinru Cao;Sining Zheng
中科院分区：
文献类型：
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作者：
Xinru Cao;Sining Zheng
研究了一类具Logistic型源项的拟线性抛物-椭圆型Keller-Segel方程组ut =（u）u）− χ（u v）+ g（u），− Δv = − v + u，Ω ×（0，T），在非负初值和齐次Neumann边界条件下，在光滑边界的有界区域Ω <$Rn中，n ≥ 1，χ > 0，当s ≥ s 0> 1时，g ≥ c1 sp，且当s > 0时，g（s）≤ as − μs2，其中a，g（0）≥ 0，μ > 0。趋化性模型包含三种非线性机制：非线性扩散、聚集和逻辑吸附。三重非线性之间的相互作用表明，与非线性扩散一起，逻辑吸收将主导聚集，使得系统的唯一经典解在时间上必须是全局的且有界的，无论初始数据如何，只要μ>χ1−2n（1−p）+，或等价地，p>1−2χn（χ−μ）+，这扩大了参数范围μ>n−2nχ，或p>1−2n，无Logistic源的拟线性K-S系统的全局有界解所要求的。版权所有© 2013约翰威利父子有限公司.
We study a quasilinear parabolic–elliptic Keller–Segel system involving a source term of logistic type ut = ∇ ⋅ (ϕ(u) ∇ u) − χ ∇ ⋅ (u ∇ v) + g(u), − Δv = − v + u in Ω × (0,T), subject to nonnegative initial data and the homogeneous Neumann boundary condition in a bounded domain Ω⊂Rn with smooth boundary, n ≥ 1, χ > 0, ϕ ≥ c1sp for s ≥ s0 > 1, and g(s) ≤ as − μs2 for s > 0 with a,g(0) ≥ 0, μ > 0. There are three nonlinear mechanisms included in the chemotaxis model: the nonlinear diffusion, aggregation and logistic absorption. The interaction among the triple nonlinearities shows that together with the nonlinear diffusion, the logistic absorption will dominate the aggregation such that the unique classical solution of the system has to be global in time and bounded, regardless of the initial data, whenever μ>χ1−2n(1−p)+ , or, equivalently, p>1−2χn(χ−μ)+ , which enlarge the parameter range μ>n−2nχ , or p>1−2n , required by globally bounded solutions of the quasilinear K‐S system without the logistic source. Copyright © 2013 John Wiley & Sons, Ltd.