The carrying capacity to chemotaxis system with two species and competitive kinetics in N dimensions
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
The carrying capacity to chemotaxis system with two species and competitive kinetics in N dimensions Guangyu Xu Abstract. This paper deals with the solution of two-species chemotaxis system ⎧ ⎨ ut = d1 Δu − χ1 ∇ · (u∇w) + μ1 u(1 − u − a1 v), x ∈ Ω, vt = d2 Δv − χ2 ∇ · (v∇w) + μ2 v(1 − a2 u − v), x ∈ Ω, ⎩ 0 = d3 Δw − γw + αu + βv x ∈ Ω,
t > 0, t > 0, t>0
(0.1)
in a smooth bounded domain Ω ⊂ RN , N ≥ 1. When d1 = d2 = 0, we first establish the local well-posedness of corresponding hyperbolic–hyperbolic–elliptic problem with the help of some compactness arguments and then obtain a blowup in finite time result for this problem. Using this blow-up conclusion, we further consider model (0.1) with small d1 , d2 > 0, and we then get that for any given M > 0 and T > 0, one can find suitable large, radially symmetric initial data and some appropriate parameters such that the corresponding classical solution of (0.1) satisfies u(x, t) + v(x, t) > M, with some x ∈ Ω and t ∈ (0, T ). Mathematics Subject Classification. 35B44, 35K55, 92C17, 35B51. Keywords. Multi-species chemotaxis system, Local existence, Blowup, Competitive source.
1. Introduction Keller and Segel heuristically derived the celebrated model in [17,18], which is the so-called KS model and describes growth phenomena mediated by a chemoattractant, that is, the aggregation of Dictyostelium discoideum due to an attractive chemical substance. The original model has been modified by various authors with the aim of improving its consistency with biological reality. For example, the following single-species chemotaxis–growth model ut = Δu − χ∇ · (u∇v) + ru − μu2 , x ∈ Ω, t > 0, (1.1) x ∈ Ω, t > 0 κvt = Δv − v + u, was first introduced by Mimura and Tsujikawa in [27] to study aggregating patterns based on the chemotaxis, diffusion and growth of bacteria. When κ = 0, if either N ≤ 2 and μ > 0 is arbitrary, or if N ≥ 3 and μ suitable larger, then for all reasonably regular initial data model (1.1) possesses a globally bounded classical solution [37]. Moreover, the effect of logistic damping is stronger than that of chemotactic aggregation when μ ≥ NN−2 χ ( [16,37,46]). For the case of κ = 1, Winkler [44] showed that if μ is sufficiently large, then problem (1.1) possesses a unique bounded solution. Then, a 3D chemotaxis–fluid system with This work is supported by graduate scientific research and innovation foundation of Chongqing, China (Grant No. CYB19070). 0123456789().: V,-vol
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G. Xu
ZAMP
logistic source has been considered in [36], and the result therein applied to model (1.1) with χ = 1 shows that μ ≥ 23 is enough to rule out blowup. This conclusion was further improved in [24] where the authors showed that logistic source wins over chemotactic aggregation if μ > 20χ. Xiang [45] recently proved that 9 χ. blowup does not occur when μ > √10−2 It is an open challenging problem whether or not blowup occurs for small μ > 0 to model (1.1). When κ = 0, μ ∈ (0, 1), Winkler [42] considered dynamical beha
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