Large Time Behavior of Solutions to a 3D Keller-Segel-Stokes System Involving a Tensor-valued Sensitivity with Saturation

IF 0.9 4区 数学 Q3 MATHEMATICS, APPLIED
Yuan-yuan Ke, Jia-Shan Zheng
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引用次数: 0

Abstract

In this paper we deal with the initial-boundary value problem for the coupled Keller-Segel-Stokes system with rotational flux, which is corresponding to the case that the chemical is produced instead of consumed,

subject to the boundary conditions (∇nnS(x, n, c)∇c) · ν = ∇c · ν = 0 and u = 0, and suitably regular initial data (n0(x), c0(x), u0(x)), where Ω ⊂ ℝ3 is a bounded domain with smooth boundary Ω. Here S is a chemotactic sensitivity satisfying ∣S(x, n, c)∣ ≤ CS(1 + n)α with some CS > 0 and α > 0. The greatest contribution of this paper is to consider the large time behavior of solutions for the system (KSS), which is still open even in the 2D case. We can prove that the corresponding solution of the system (KSS) decays to (\({1 \over {|\Omega |}}\int_\Omega {{n_0}} \), \({1 \over {|\Omega |}}\int_\Omega {{n_0}} \), 0) exponentially, if the coefficient of chemotactic sensitivity is appropriately small. As a precondition to consider the asymptotic behavior, we also show the global existence and boundedness of the corresponding initial-boundary problem KSS with a simplified method. We find a new phenomenon that the suitably small coefficient CS of chemotactic sensitivity could benefit the global existence and boundedness of solutions to the model KSS.

Abstract Image

具有饱和张量值灵敏度的三维Keller-Segel-Stocks系统解的大时间行为
在本文中,我们讨论了具有旋转通量的耦合Keller-Segel-Stocks系统的初始边值问题,该问题对应于化学物质产生而不是消耗的情况,受边界条件(Şn−nS(x,n,c)Şc)·Γ=Γc·Γ=0和u=0,以及适当的正则初始数据(n0(x),c0(xℝ3是一个有界域,其光滑边界为?Ω。这里,S是一种趋化敏感性,满足|S(x,n,c)|≤CS(1+n)-α,其中一些CS>;0和α>;0。本文最大的贡献是考虑了系统(KSS)解的大时间行为,即使在2D情况下,它仍然是开放的。我们可以证明,如果趋化敏感性系数适当小,系统的相应解(KSS)以指数形式衰变为(\({1\over{|\Omega|}}\int_\Omega{n_0}}),\。作为考虑渐近行为的先决条件,我们还用一个简化的方法证明了相应的初始边界问题KSS的全局存在性和有界性。我们发现了一个新的现象,即趋化敏感性系数CS适当小,有利于模型KSS解的全局存在性和有界性。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
70
审稿时长
3.0 months
期刊介绍: Acta Mathematicae Applicatae Sinica (English Series) is a quarterly journal established by the Chinese Mathematical Society. The journal publishes high quality research papers from all branches of applied mathematics, and particularly welcomes those from partial differential equations, computational mathematics, applied probability, mathematical finance, statistics, dynamical systems, optimization and management science.
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