Boundedness and stabilization in a two-species chemotaxis system with logistic source
- PDF / 400,929 Bytes
- 17 Pages / 547.087 x 737.008 pts Page_size
- 17 Downloads / 172 Views
Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Boundedness and stabilization in a two-species chemotaxis system with logistic source Guoqiang Ren Abstract. In this work, we consider the two-species chemotaxis system with logistic source in a two-dimensional bounded domain. We present the global existence of classical solutions under appropriate regularity assumptions on the initial data. In addition, the asymptotic behavior of the solutions is studied, and our results generalize and improve some well-known results in the literature. Mathematics Subject Classification. 35A01, 35B40, 35D30, 35Q92, 92C17. Keywords. Chemotaxis, Lotka–Volterra competitive kinetics, Boundedness, Stabilization.
1. Introduction In this paper, we consider the two-species chemotaxis system with Lotka–Volterra competitive kinetics: ⎧ ut = d1 Δu − ∇ · (uχ1 (w)∇w) + μ1 u(1 − u − a1 v), x ∈ Ω, t > 0, ⎪ ⎪ ⎪ ⎪ ⎨ vt = d2 Δv − ∇ · (vχ2 (w)∇w) + μ2 v(1 − v − a2 u), x ∈ Ω, t > 0, wt = d3 Δw − (αu + βv)w, x ∈ Ω, t > 0, (1.1) ⎪ ∂u ∂v ∂w ⎪ = = = 0, x ∈ ∂Ω, t > 0, ⎪ ⎪ ∂ν ∂ν ∂ν ⎩ u(x, 0) = u0 (x), v(x, 0) = v0 (x), w(x, 0) = w0 (x), x ∈ Ω, ∂ where Ω ⊂ R2 is a bounded domain with smooth boundary ∂Ω and ∂ν denotes the derivative with respect to the outer normal of ∂Ω, u and v represent the population densities of two species and w denotes the concentration of the oxygen. d1 , d2 , d3 , μ1 , μ2 , α, β are positive constants, a1 , a2 are nonnegative constants, initial data u0 , v0 , w0 are known functions satisfying
(u0 , v0 , w0 ) ∈ C 0 (Ω) × C 0 (Ω) × W 1,∞ (Ω) are nonnegative with u0 ≡ 0 and v0 ≡ 0.
(1.2)
χi (i = 1, 2) fulfilling χi ∈ C 2 ([0, ∞)) and χi > 0 (i = 1, 2).
(1.3)
Chemotaxis is the directed movement of cells or organisms in response to the gradients of concentration of the chemical stimuli. It plays fundamental roles in various biological processes including embryonic development, wound healing and disease progression. Chemotaxis is also crucial for many aspects of behavior, including locating food sources (such as the fruit fly Drosophila melanogaster navigates up gradients of attractive odors during food location), avoidance of predators and attracting mates (such as male moths follow pheromone gradients released by the female during mate location), slime mold formation, angiogenesis in tumor progression and primitive steak formation. The pioneering work of the This work was partially supported by NNSF of China (No. 12001214) and China Postdoctoral Science Foundation (Nos. 2020M672319, 2020TQ0111). 0123456789().: V,-vol
177
Page 2 of 17
G. Ren
ZAMP
chemotaxis model was introduced by Keller and Segel in [12], describing the aggregation of cellular slime mold toward a higher concentration of a chemical signal, which reads ut = Δu − ∇ · (u∇v), x ∈ Ω, t > 0, (1.4) vt = Δv − v + u, x ∈ Ω, t > 0. The mathematical analysis of (1.4) and the variants thereof mainly concentrates on the boundedness and blowup of the solutions [5,9,34,38]. In addition to the original model, a large number of variants of the classical form have also
Data Loading...
