Global attractor for a two-dimensional chemotaxis system with linear degradation and indirect signal production
Global attractor for a two-dimensional chemotaxis system with linear degradation and indirect signal production
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DOI：
10.1007/s13160-019-00376-0
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发表时间：
2020-01
影响因子：
0.9
通讯作者：
E. Nakaguchi;Kanako Noda;Koichi Osaki;Kenta Uemichi
中科院分区：
文献类型：
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作者：
E. Nakaguchi;Kanako Noda;Koichi Osaki;Kenta Uemichi
本文研究了Deneubbery提出的具有间接信号产生的趋化系统解的渐近性态（Insectes Sociaux 24：117-130，1977）：{u_t=\varDelta\; u-x\quad（u\quad w）+ 1-μ u\quad in\varOmega\; *（0，∞），\qquad\δ\，v_t=-v+ u\quad in\varOmega\; *（0，∞），\qquad\τ\，w_t=\varDelta\; w-w+ v &\quad in\varOmega\; *（0，∞）.\ qquad\。在Ω×（0，∞）中，ut= Δ u-χ·（u w）+ 1-μ u，δ vt=-v+ u，τ wt= Δ w-w+ v.这里，\varOmega\;<$R^ 2 Ω <$R 2是一个光滑有界域，其边界上具有齐次Neumann边界条件。系数都是正的常数。该系统模拟了社会性昆虫，特别是白蚁的自组织筑巢过程。我们首先证明了具有足够大的休息率μ μ的工蜂的趋化强度χ χ或初始总质量μ然后，我们定义的动力系统的解决方案，并构建整体吸引子。此外，当μ/χ μ/χ进一步增大时，我们还构造了唯一齐次平衡点（1/μ，1/μ，1/μ）（1/μ，1/μ，1/μ）的李雅普诺夫泛函，表明全局吸引子仅由平衡点组成.
We study the asymptotic behavior of solutions to a chemotaxis system with indirect signal production presented by Deneubourg (Insectes Sociaux 24: 117–130, 1977):{u_t=\varDelta\; u-χ ∇ ⋅ (u ∇ w)+ 1-μ u &\quad in\varOmega\; * (0, ∞),\qquad\δ\, v_t=-v+ u &\quad in\varOmega\; * (0, ∞),\qquad\τ\, w_t=\varDelta\; w-w+ v &\quad in\varOmega\; * (0, ∞).\qquad\. ut= Δ u-χ∇·(u∇ w)+ 1-μ u in Ω×(0,∞), δ vt=-v+ u in Ω×(0,∞), τ wt= Δ w-w+ v in Ω×(0,∞). Here,\varOmega\; ⊂ R^ 2 Ω⊂ R 2 is a smooth bounded domain with homogeneous Neumann boundary conditions imposed on its boundary. The coefficients are all positive constants. The system models the self-organized nest construction process of social insects, specifically, termites. We first show the global-in-time existence of solutions with some smallness conditions for chemotactic intensity χ χ or the initial total mass ‖ u_0 ‖ _ L_1‖ u 0‖ L 1 of worker insects with sufficiently large rest rate μ μ of working. We then define the dynamical system of solutions and construct the global attractor. In addition, for μ/χ μ/χ further large, we also construct a Lyapunov functional for the unique homogeneous equilibrium (1/μ, 1/μ, 1/μ)(1/μ, 1/μ, 1/μ), which indicates that the global attractor consists only of the equilibrium.