Boundedness in the higher-dimensional fully parabolic chemotaxis-competition system with loop
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Boundedness in the higher-dimensional fully parabolic chemotaxis-competition system with loop Xinyu Tu, Chunlai Mu, Shuyan Qiu and Li Yang Abstract. This paper deals with the two-species chemotaxis-competition system with loop ⎧ ∂t u1 = d1 Δu1 − χ11 ∇ · (u1 ∇v1 ) − χ12 ∇ · (u1 ∇v2 ) + μ1 u1 (1 − u1 − a1 u2 ), ⎪ ⎪ ⎪ ⎪ ⎨ ∂t u2 = d2 Δu2 − χ21 ∇ · (u2 ∇v1 ) − χ22 ∇ · (u2 ∇v2 ) + μ2 u2 (1 − u2 − a2 u1 ), ⎪ ∂t v1 = d3 Δv1 − λ1 v1 + α11 u1 + α12 u2 , ⎪ ⎪ ⎪ ⎩ ∂t v2 = d4 Δv2 − λ2 v2 + α21 u1 + α22 u2 , subject to homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R3 , where χij > 0, μi > 0, ai > 0, αij > 0, λi > 0, dk > 0 (i, j = 1, 2, k = 1, 2, 3, 4). Our main purpose is to extend the global boundedness result to the 3D setting. To address this issue, based on a new coupled function, by selecting sufficiently large μ1 and μ2 , we construct a Gronwall type inequality which directly renders the uniform boundedness of solutions. Mathematics Subject Classification. 35k51, 35K55, 35K57, 92C17. Keywords. Chemotaxis with loop, Lotka–Volterra-type competition, Global existence, Boundedness.
1. Introduction Chemotaxis is the influence of chemical substances on the movement of species, which plays an important role in disease progression, migration of neurons, tumor invasion, etc. A celebrated chemotaxis model was initially proposed by Keller–Segel [10] ut = Δu − χ∇ · (u∇v) + ru − μu2 , x ∈ Ω, t > 0, (1.1) vt = Δv − v + u, x ∈ Ω, t > 0. Until now, considerable efforts have been devoted to investigating the global boundedness, existence and blow-up of solutions to (1.1). For instance, when r = μ = 0, it is well known that the solution of system 4π (1.1) never blows up for n = 1 [24]; and the critical mass C (C = 8π χ in the radial setting or C = χ in the other setting) occurs for n = 2, one can refer to [6,23] for boundedness, and [7,8,18] for blow-up result; as for n ≥ 3, if Ω is a ball, Winkler proved that (1.1) possesses unbounded solutions for arbitrary small mass [38], later, he introduced a new method which led to the finite-time blow-up of solutions [40]. For the case r, μ = 0, it was found that the death term can rule out any collapse for any r ∈ R and arbitrary small μ > 0 when n = 2 [25], or for n ≥ 3 and μ is sufficiently large in the convex domain Ω [39], but in the latter case, the lower bound of μ is not explicitly known; then Yang et al. improved this result by establishing the global boundedness under the condition μ > θ0 χ with some θ0 > 0 [46]; and an explicit lower bound for μ when χ = 1, n = 3 was derived as a by-product by Tao and Winkler in [30], that is, μ ≥ 23 is enough to guarantee the global boundedness; later, replacing the logistic source in (1.1) by u − μuρ , ρ ≥ 2, Mu et 0123456789().: V,-vol
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al. [22] extended the condition in [30] to μ ρ−1 > 20χ when the domain Ω is convex; recently, their result was greatly improved by Xiang [44], he established √the global solvability of (1.1) for n =
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