Boundedness and Stability in a Chemotaxis-Growth Model with Indirect Attractant Production and Signal-Dependent Sensitiv
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Boundedness and Stability in a Chemotaxis-Growth Model with Indirect Attractant Production and Signal-Dependent Sensitivity Shuyan Qiu1 · Chunlai Mu1 · Yafeng Li1
Received: 3 January 2019 / Accepted: 14 November 2019 © Springer Nature B.V. 2019
Abstract We study the chemotaxis-growth system with signal-dependent sensitivity function and logistic source ⎧ ⎨ ut = u − ∇ · uχ (v)∇v + μu(1 − u), x ∈ Ω, t > 0, x ∈ Ω, t > 0, vt = dv + h(v, w), ⎩ x ∈ Ω, t > 0, τ wt = −δw + u, under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ Rn (n ≥ 1), where the parameters μ, τ, δ > 0 and d ≥ 0, the functions χ (v), h(v, w) satisfying some conditions represent the chemotactic sensitivity and the balance between the production and degradation of the chemical signal which relies explicitly on the living organisms, respectively. In the case that χ (v) ≡ 1, d = 1 and h(v, w) = −v + w, Hu and Tao (Math. Models Methods Appl. Sci. 26:2111–2128, 2016) asserted global existence of bounded solutions for arbitrary μ > 0 and established asymptotic behavior of solutions to the mentioned system under the condition μ > 8δ12 in the three dimensional space. The purpose of the present paper is to investigate the global existence and boundedness of classical solutions and to improve the condition assumed in Hu and Tao (Math. Models Methods Appl. Sci. 26:2111–2128, 2016) by extending the previous method for obtaining asymptotic stability. Consequently, the range of μ is extended in the present paper. Keywords Chemotaxis · Boundedness · Indirect attractant production · Logistic growth · Asymptotic behavior Mathematics Subject Classification 35A01 · 92C17 · 35B45 · 35B40 · 35K57 · 35Q92
B S. Qiu
[email protected] C. Mu [email protected] Y. Li [email protected]
1
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, P.R. China
S. Qiu et al.
1 Introduction Chemotaxis, the directed movement of the cells as a response to gradients of the concentration of the chemical signal substance, plays a significant role in coordinating cell migration in numerous biological systems such as tumor invasion, wound healing, pattern formation and bacteria aggregation [5, 6, 21, 34]. A renowned chemotaxis model was initially proposed by Keller and Segel in 1970 [18], and a variety of analytic problems for this model as well as its variants have been extensively studied with regard to biological implications [2, 14, 39]. In contrast to the well-studied paradigmatic case, the chemotaxis signal substance is not produced directly by cells themselves in some realistic situations. Namely, the signal generation undergos intermediate stages, and such mechanisms of indirect signal production may lead to new properties. Considering more complex biological situations, we shall be concerned with the chemotaxis-growth model with indirect signal production and signal-dependent sensitivity ⎧ x ∈ Ω, t > 0, ut = u − ∇ · uχ (v)∇v + μu(1 − u), ⎪ ⎪ ⎪ ⎪ = dv + h(v, w), x ∈ Ω, t > 0, v ⎪ ⎪ ⎨ t x ∈ Ω, t > 0, τ wt = −δw + u,
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