Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 26, pp. 1–20. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.26 GLOBAL WELL-POSEDNESS TO A MULTIDIMENSIONAL PARABOLIC-ELLIPTIC-ELLIPTIC ATTRACTION-REPULSION CHEMOTAXIS SYSTEM LING LIU Abstract. In this article we study the initial-boundary value problem for the attraction-repulsion chemotaxis system ut = ∆u− χ∇ · (u∇v) + ξ∇ · (u∇w), x ∈ Ω, ; t > 0, 0 = ∆v − βv + αu, x ∈ Ω, t > 0, 0 = ∆w − δw + γu, x ∈ Ω, t > 0, ∂u ∂ν = ∂v ∂ν = ∂w ∂ν = 0, x ∈ ∂Ω, t > 0, u(x, 0) = u0(x), x ∈ Ω, with homogenous Neumann boundary conditions in a multidimensional bounded domain Ω ⊂ RN (1 ≤ N ≤ 4) with smooth boundary, where χ, ξ, α, β, δ and γ are positive constants. We prove that under the assumption χα = ξγ the corresponding system possesses a unique global bounded classical solution in the cases N ≤ 3 or λ0γδξ∥u0∥10/7L1(Ω) < 1 CGN and N = 4. Moreover, the large time behavior of solutions is also investigated. Specially, when χα = ξγ, the solution of the system converges to (ū0, α β ū0, γ δ ū0) exponentially if ∥u0∥L∞(Ω) is small. 1. Introduction In this article, we study the global solvability, boundedness and asymptotic be- havior to the attraction-repulsion chemotaxis system ut = ∆u− χ∇ · (u∇v) + ξ∇ · (u∇w), x ∈ Ω, t > 0, 0 = ∆v − βv + αu, x ∈ Ω, t > 0, 0 = ∆w − δw + γu, x ∈ Ω, t > 0, (1.1) in a bounded domain in Ω ⊂ RN (N ≥ 1) with smooth boundary ∂Ω, where the parameters χ, ξ, α, β, γ, and δ are positive constants. Here u stands for the cell density, v denotes the concentration of an attracting signal, and w represents the concentration of a repulsive chemical. This model was proposed in [32] for describing the quorum effect in a chemotaxis process and in [28] for describing the aggregation of microglia in Alzheimer’s disease. 2020 Mathematics Subject Classification. 35K55, 35Q92, 35Q35, 92C17. Key words and phrases. Attraction-repulsion; well-posedness; asymptotic behavior; global solution; boundedness. ©2025. This work is licensed under a CC BY 4.0 license. Submitted January 5, 2025. Published March 11, 2025. 1 2 L. LIU EJDE-2025/26 Before going into our mathematical analysis, we recall some important progresses on system (1.1) and its variants. In the absence of chemorepulsive chemical (i.e. chemorepellent), namely ξ = 0, w is decoupled from the system (1.1) and the first two equations of (1.1) comprises a classical Keller-Segel model ut = ∆u− χ∇ · (u∇v), x ∈ Ω, t > 0, 0 = ∆v − βv + αu, x ∈ Ω, t > 0, (1.2) which has been widely investigated (see [1, 30, 31]). For instance, it is well-known that for large classes of initial data, solutions of system (1.2) blow up when either N ≥ 3, or N = 2 and the total mass of cells is large, while global bounded solutions can be constructed under appropriate smallness conditions on the initial data ([13, 39]). There is a large amount mathematical result of well-posedness and asymptotic behavior for system (1.2) and its variants. One can refer to [1, 4, 5, 10, 14, 15, 18, 21, 26, 27, 29, 36, 38, 39, 41, 47] and references therein. Unlike the classical Keller-Segel model (1.2), it appears to be difficult to find a Lyapunov functional for (1.1), and thus the mathematical analysis for is more challenging. However, in many biological processes, cells often interact with a combination of repulsive and attractive signaling chemicals to produce various in- teresting biological patterns [7, 32]. To describe such process of cells, Tao and Wang [35] proposed the following coupled attraction-repulsion chemotaxis system ut = ∆u− χ∇ · (u∇v) + ξ∇ · (u∇w), x ∈ Ω, t > 0, τvt = ∆v − βv + αu, x ∈ Ω, t > 0, τwt = ∆w − δw + γu, x ∈ Ω, t > 0, (1.3) where τ ∈ {0, 1}. In contrast to the Keller-Segel system, systems (1.3) describe an indirect signal production mechanism, that is, the chemoattractant is not produced by cells directly, but is controlled indirectly via parabolic equations or elliptic equa- tions. From their study, a large variety of mathematical analyses have been devoted, especially to the well-studied areas of global existence and blow-up of solutions in variants of (1.3) (see [1, 11, 12]). Fujie and Senba [8] proved that system (1.3) with homogeneous Neumann boundary conditions or mixed boundary conditions (no-flux for nand Dirichlet conditions for v and w) possesses a unique and global bounded classical solution for N ≤ 3, and showed the global boundedness of classical so- lution to (1.3) with homogeneous Neumann boundary conditions for N = 4 and∫ Ω u0 < (8π)2 χ in the radially symmetric setting (whereas this conclusion remains valid without radial symmetry to the mixed boundary value problem). In their later work [9], Fujie and Senba showed that the classical solution in will be blowing up in finite or infinite time if N = 4 and ∥u0∥L1(Ω) ∈ ( (8π) 2 χ ,∞) \ {j · (8π)2 χ |j ∈ N}. We point that the key ingredients for [8, 9] are a Lyapunov functional and an Adams-type inequality. However, unlike the τ = 1, it seems to be difficult to find an Adams-type inequality for (1.3) with the case τ = 0, and thus the mathematical analysis is more challenging. And therefore, the boundedness of the case τ = 0 or radially symmetric setting of the case τ = 1 of system (1.3) is still open. In [17], for any β > 0 and δ > 0, the large-time behavior of (1.3) was explored in the one-dimensional case. For a higher-dimensional case (N ≤ 3), [35] showed that each solution of (1.3) converges to a unique trivial stationary solution under the conditions that χα < ξγ and δ = β. Furthermore, similar results are also valid for the critical condition that χα = ξγ ([16]). To the best of our knowledge, for the EJDE-2025/26 ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 3 attraction-repulsion chemotaxis system (1.3), there is few rigorous mathematical results on large time behavior of the solutions under the condition N = 4. From this point of view, our results can be referred as an enrichment in this respect. Additionally, recent studies have shown that the solution behavior can be also impacted by the volume-filling or prevention of overcrowding (see [1, 2, 6, 48]), the nonlinear diffusion (see [3, 33, 42, 43, 47]), and the logistic damping (see [22, 24, 44]). In order to provide a more comprehensive description of the development of (1.3), it is necessary to add the following supplementary content, with specific references to [19, 20, 23, 25, 34, 45]. Inspired by the above works, we study system (1.2), and we will prove the global solvability, boundedness and asymptotic behavior of the system for various ranges of parameter values. For the sake of clearness, let us recall the Gagliardo-Nirenberg inequality in the four-dimensional case ∥ϕ∥3L3(Ω) ≤ CGN∥∇ϕ∥2L2(Ω)∥ϕ∥ 1/2 L2(Ω) + CGN,∗∥ϕ∥3L2(Ω) (1.4) for all ϕ ∈ W 1,2(Ω), where CGN and CGN,∗ are some positive constants only de- pending on Ω. We consider the elliptic system −∆w + δw = γg, x ∈ Ω, ∂w ∂ν = 0, x ∈ ∂Ω, where κ ∈ (1,+∞) and g ∈ Lκ(Ω). Then there exists a unique solution w ∈ W 2,κ(Ω). In addition, there exists a positive constant λ0 = λ0(Ω, δ) such that ∥v∥W 2,κ(Ω) ≤ λ0∥γg∥Lκ(Ω). (1.5) The aim of this study is to provide some further insights into the existence of global solutons as well as boundedness and large-time behavior for (1.1) in the case χα = ξγ and N = 4. To prepare a precise statement of our main results, let us fix the mathematical framework by considering (1.1) in a bounded domain Ω ⊂ RN (1 ≤ N ≤ 4) with smooth boundary, where χ, ξ, α, β, γ, and δ are positive constants. To state our results precisely, we specify the precise problem context by considering (1.1) along with the boundary conditions ∂u ∂ν = ∂v ∂ν = ∂w ∂ν = 0, x ∈ ∂Ω, t > 0, (1.6) and the initial conditions u(x, 0) = u0(x), x ∈ Ω. (1.7) We shall assume throughout this paper that the initial data satisfy u0 ∈ C0(Ω̄) with u0 ≥ 0 in Ω and u0 ̸≡ 0, x ∈ Ω̄. (1.8) Within the above framework, our main results concerning the existence and boundedness of global solutions to (1.1), (1.6), (1.7) read as follows. Theorem 1.1. Let Ω ⊂ RN be a bounded domain with smooth boundary. Suppose that the initial data satisfies (1.8). Then under the assumption χα = ξγ, we can prove that 4 L. LIU EJDE-2025/26 (i) if λ0γδξ∥u0∥10/7L1(Ω) < 1 CGN and N = 4, then system (1.1), (1.6), (1.7) pos- sesses a unique global bounded classical solution (u, v, w). Besides, there exists constant C > 0 independent of Υ(∥u0∥L∞(Ω)) such that ∥u(·, t)∥L∞(Ω) + ∥v(·, t)∥W 1,∞(Ω) + ∥w(·, t)∥W 1,∞(Ω) ≤ CΥ(∥u0∥L∞(Ω)) for all t > 0; (ii) if N ≤ 3, then system (1.1), (1.6), (1.7) admits a unique global bounded classical solution (u, v, w). Moreover, there exists C > 0 independent of Υ(∥u0∥L∞(Ω)) such that ∥u(·, t)∥L∞(Ω) + ∥v(·, t)∥W 1,∞(Ω) + ∥w(·, t)∥W 1,∞(Ω) ≤ CΥ(∥u0∥L∞(Ω)) for all t > 0. Here Υ is a continuous function which is non-decreasing respective to ∥u0∥L∞(Ω). Remark 1.2. (i) Since Υ(∥u0∥L∞(Ω)) ≥ 1 is non-decreasing with respective to ∥u0∥L∞(Ω), Theorem 1.1 implies that suitably small ∥u0∥L∞(Ω) in system (1.1), (1.6), (1.7) would provides the existence and boundedness of global solutions to system (1.1), (1.6), (1.7). (ii) The proof of Theorem 1.1 is inspired by [37, 22]. These results extend the previous work obtained in [22, 44] which require r = 3/2 with N = 4 and r > 2N−2 N for any N ≥ 1. (iii) From [9], we know that here the condition for system (1.1), (1.6), (1.7) is optimal. In light of these results, it seems natural and inevitable that our second result, ad- dressing asymptotic homogenization of all solution components, requires ∥u0∥L∞(Ω) to be appropriately small. We then show that the smallness assumption on u0 forces the corresponding solution in Theorem 1.1 to converge to (ū0, α β ū0, γ δ ū0) by using the ODE theory and some careful analysis. Indeed, based on the global existence, the solution has the following convergence property. Theorem 1.3. Let χα = ξγ and Ω ⊂ RN (1 ≤ N ≤ 4) be a bounded domain with a smooth boundary. Then for any u0 that satisfies (1.8), there exists ϵ0 > 0 such that if u0 satisfies ∥u0∥L∞(Ω) ≤ ϵ for some 0 < ϵ < ϵ0, then for any t > 0, there exists ρ1,∗ > 0 and C such that ∥u(·, t)− ū0∥L∞(Ω) ≤ Ce−ρ1,∗t, (1.9) ∥v(·, t)− α β ū0∥L∞(Ω) ≤ Ce−ρ1,∗t, (1.10) ∥w(·, t)− γ δ ū0∥L∞(Ω) ≤ Ce−ρ1,∗t, (1.11) where ū0 := 1 |Ω| ∫ Ω u0(x). Remark 1.4. (i) This result partly improves the previous work obtained in Li- Wang [22] and Xie-Zheng [44] which requires the logistic source f(u) = au − µur with r = 2− (2/N) or r > 2− (2/N) and N ≤ 4. (ii) To the best of our knowledge, these are the first results on boundedness of the system in four-dimensional space in the case τ = 0. (iii) From [9], we know that here the condition for system (1.1), (1.6), (1.7) is optimal. EJDE-2025/26 ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 5 (iv) Theorem 1.3 asserts that the solution of system (1.1), (1.6), (1.7) behaves asymptotically in a similar manner to the case where β = δ and N ≤ 3 in [35] (see also [24]), provided that the initial data u0 is sufficiently small in L∞(Ω). However, for N > 3 the large-time behavior of system (1.1), (1.6), (1.7) is given as an open problem. Hence, from this point of view, our results can be referred as an enrichment in this respect. 2. Preliminaries In this section, we present some basic properties of system (1.1), (1.6), (1.7). We start with the existence theory and extensibility of the local solution. To this end, by an adaptation of well-established fixed point arguments (see [40, Lemma 2.1] or [46]), we can establish the following local existence result for system (1.1), (1.6), (1.7). Lemma 2.1. Let Ω ⊂ RN (N ≥ 1) be a bounded domain with smooth boundary. Assume that the initial data satisfy (1.8). Then there exists a positive constant Tmax such that (1.1) has a unique non-negative classical solution (u, v, w) satisfying u ∈ C0(Ω̄× [0, Tmax)) ∩ C2,1(Ω̄× (0, Tmax)), v ∈ C0(Ω̄× [0, Tmax)) ∩ C2,0(Ω̄× (0, Tmax)), w ∈ C0(Ω̄× [0, Tmax)) ∩ C2,0(Ω̄× (0, Tmax)). Moreover, if Tmax < ∞, then ∥u(·, t)∥L∞(Ω) + ∥v(·, t)∥W 1,∞(Ω) + ∥w(·, t)∥W 1,∞(Ω) → ∞ as t ↗ Tmax. (2.1) The following well-known Gagliardo-Nirenberg inequality will be frequently used [46]. Lemma 2.2. ([46]) Let 0 < θ ≤ p < 2N (N−2)+ . There exists a positive constant CGN such that for all u ∈ W 1,2(Ω) ∩ Lθ(Ω), ∥u∥Lp(Ω) ≤ CGN (∥∇u∥aL2(Ω)∥u∥ 1−a Lθ(Ω) + ∥u∥Lθ(Ω)), is valid with a = N θ −N p 1−N 2 +N θ ∈ (0, 1). Some basic properties of the solution obtained in Lemma 2.1 can be derived as follows. Lemma 2.3. Under the assumption of local existence, we can obtain∫ Ω u = ∫ Ω u0 for all t ∈ (0, Tmax), (2.2) β α ∫ Ω v = δ γ ∫ Ω w = ∫ Ω u0 for all t ∈ (0, Tmax). (2.3) To deal with the repulsion mechanism in (1.1), inspired by [22] and [35], we define s(x, t) := χv(x, t)− ξw(x, t), (x, t) ∈ Ω× (0, Tmax). 6 L. LIU EJDE-2025/26 Then, recalling ξγ = χα, (1.1), (1.6), (1.7) can be rewritten as ut = ∆u−∇ · (u∇s), x ∈ Ω, t ∈ (0, Tmax), 0 = ∆s− δs+ ᾱv, x ∈ Ω, t ∈ (0, Tmax), 0 = ∆v − βv + αu, x ∈ Ω, t ∈ (0, Tmax), ∂u ∂ν = ∂s ∂ν = ∂w ∂ν = 0, x ∈ ∂Ω, t ∈ (0, Tmax), u(x, 0) = u0(x), x ∈ Ω, (2.4) where ᾱ = χ(δ − β). 3. Proof of Theorem 1.1 Notation: Sometimes, we will use C,Ci to denote uniform constants hat may be different on different lines. In this section, we focus on the global existence and boundedness of solutions. To this end, we shall establish a series of a priori estimates of solutions for system (1.1), (1.6), (1.7), which play an important role in proving Theorem 1.1. And the derivation of the uniform Lp bounds on u needs two cases. Firstly, we can obtain the boundedness of ∥u∥Lp(Ω) (for any p > 1) under the assumption N ≤ 3. Lemma 3.1. Let N ≤ 3. Then for any finite p > 2, there exists γ1 > 0 and Υ1(∥u0∥L∞(Ω)) such that ∥u(·, t)∥Lp(Ω) ≤ γ1Υ1(∥u0∥L∞(Ω)) for all t ∈ (0, Tmax). (3.1) Proof. According to the known results on elliptic boundary problem in L1(Ω) to- gether with Lemma 2.3, we can have that for any l ∈ (1, N (N−1)+ ), there exists C1(l,Ω) > 0 independent of u0 such that ∥w(·, t)∥W 1,l(Ω) ≤ C1∥u0∥L1(Ω) for all t ∈ (0, Tmax), which combining with the Sobolev embedding, W 1,l(Ω) ↪→ L2(Ω) by N ≤ 3 and l ∈ (1, N (N−1)+ ), derives that there exists C2 > 0 satisfying ∥w(·, t)∥L2(Ω) ≤ C2∥u0∥L1(Ω) for all t ∈ (0, Tmax) [37, Lemma 3.1]. Testing the first equation in (2.4) with up−1 and integrating by parts, we obtain 1 p d dt ∫ Ω up + 4(p− 1) p2 ∫ Ω |∇up/2|2 = (p− 1) ∫ Ω up−1∇u · ∇s = −p− 1 p ∫ Ω up∆s = p− 1 p ∫ Ω up(ᾱv − δs) = p− 1 p ∫ Ω up(χ(δ − β)v − δ(χv − ξw)) ≤ p− 1 p δξ ∫ Ω upw for all t ∈ (0, Tmax). EJDE-2025/26 ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 7 In light of the Hölder inequality and the Gagliardo-Nirenberg inequality (see Lemma 2.2), we can obtain some constant C3(p) > 0 such that p− 1 p δξ ∫ Ω upw ≤ p− 1 p δξ( ∫ Ω u2p)1/2 (∫ Ω w2 )1/2 ≤ p− 1 p δξ (∫ Ω u2p )1/2(∫ Ω w2 )1/2 ≤ p− 1 p δξC2∥u0∥L1(Ω) (∫ Ω u2p )1/2 = p− 1 p δξC2∥u0∥L1(Ω)∥up/2∥2L4(Ω) = p− 1 p δξC2C3(p)∥u0∥L1(Ω) [ ∥∇up/2∥ 2 Np 2 −N 4 1−N 2 + Np 2 L2(Ω) ∥up/2∥ 2−2 Np 2 −N 4 1−N 2 + Np 2 L 2 p (Ω) + ∥up/2∥2 L 2 p (Ω) ] = p− 1 p δξC2C3(p)∥u0∥L1(Ω) [ 4 p2 (∫ Ω |∇up/2|2 ) Np 2 −N 4 1−N 2 + Np 2 (∫ Ω u0 )p(1− Np 2 −N 4 1−N 2 + Np 2 ) + (∫ Ω u0 )p] for all t ∈ (0, Tmax). Applying the Young inequality, one has p− 1 p δξ ∫ Ω upw ≤ 2(p− 1) p ∫ Ω |∇up/2|2 + C4∥u0∥ p+ 1−N 4 1−N 2 + Np 2 L1(Ω) + C5∥u0∥p+1 L1(Ω) with C4 > 0, where C5 = p− 1 p δξC2C3(p). Using the Gagliardo-Nirenberg inequality (see Lemma 2.2), we can obtain a positive constant C6 > 0 such that ∥u∥p+ 2 N Lp+ 2 N (Ω) = ∥up/2∥ 2(p+ 2 N ) p L 2(p+ 2 N ) p (Ω) ≤ C6(∥∇up/2∥2L2(Ω)∥u p/2∥ 4 N L 2 p (Ω) + ∥u 2 p ∥ 2(p+ 2 N ) p L 2 p (Ω) ) = C6(∥∇up/2∥2L2(Ω)∥u∥ 2p N L1(Ω) + ∥u∥p+ 2 N L1(Ω)) = C6(∥∇up/2∥2L2(Ω)∥u0∥ 2p N L1(Ω) + ∥u0∥ p+ 2 N L1(Ω)), which implies that ∥∇up/2∥2L2(Ω) ≥ 1 C6∥u0∥ 2p N L1(Ω) ∥u∥p+ 2 N Lp+ 2 N (Ω) − ∥u0∥ pN+2−2p N L1(Ω) (3.2) 8 L. LIU EJDE-2025/26 for all t ∈ (0, Tmax). Thus, we 1 p d dt ∫ Ω up + 2(p− 1) p 1 C6∥u0∥ 2p N L1(Ω) ∥u∥p+ 2 N Lp+ 2 N (Ω) ≤ C7(∥u0∥L∞(Ω)) for all t ∈ (0, Tmax) with C7(∥u0∥L∞(Ω)) = 2(p− 1) p [∥u0∥L∞(Ω)|Ω|] pN+2−2p N + C4[∥u0∥L∞(Ω)|Ω|] p+ 1−N 4 1−N 2 + Np 2 + C5[∥u0∥L∞(Ω)|Ω|]p+1 (3.3) This and the Hölder inequality yields 1 p d dt ∫ Ω up + 2(p− 1) p 1 C6∥u0∥ 2p N L1(Ω) |Ω|− 2 pN (∫ Ω up ) p+ 2 N p ≤ C7(∥u0∥L∞(Ω)) Upon an ODE comparison, we have∫ Ω up(·, t) ≤ max{ ∫ Ω up 0, C(p)Λ(∥u0∥L1(Ω))} for all t ∈ (0, Tmax) (3.4) with C(p) = [ pC6 2(p− 1) |Ω| 2 pN ] p p+ 2 N , Λ(∥u0∥L∞(Ω)) = [C7(∥u0∥L∞(Ω))[∥u0∥L∞(Ω)|Ω|] 2p N ] p p+ 2 N . As a consequence of (3.3) and (3.4), (3.1) is valid by a choice of γ1 = [ pC6 2(p− 1) |Ω| 2 pN ] p p+ 2 N + |Ω|, Υ1(∥u0∥L∞(Ω)) = max{∥u0∥pL∞(Ω), [C7(∥u0∥L∞(Ω))[∥u0∥L∞(Ω)|Ω|] 2p N ] p p+ 2 N }. □ Lemma 3.2. Let N = 4 and λ0αδξ+∥u0∥10/7L1(Ω) < 1 CGN . Then for each finite p > 2, there exists γ2 and Υ2(∥u0∥L∞(Ω)) such that ∥u(·, t)∥Lp(Ω) ≤ γ2Υ2(∥u0∥L∞(Ω)) for all t ∈ (0, Tmax). (3.5) Proof. We will divide the proof into two steps. Step 1. A bound for u in L∞((0, Tmax);L 2(Ω)): Multiplying the first equation in (2.4) by u, integrating by parts and using the Hölder inequality, we have 1 2 d dt ∫ Ω u2 + ∫ Ω |∇u|2 = ∫ Ω u∇u · ∇s = − ∫ Ω u2∆s = ∫ Ω u2[−δs+ ᾱv] = ∫ Ω u2[−δ[χv − ξw] + χ(δ − β)v] ≤ δξ ∫ Ω u2w EJDE-2025/26 ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 9 ≤ δξ∥u∥2L5/2∥w∥L5(Ω) for all t ∈ (0, Tmax). (3.6) Then the Gagliardo-Nirenberg inequality (see Lemma 2.2) and Lemma 2.3 ensure that the constant CGN > 0 and CGN,∗ are such that ∥u∥5/2 L5/2(Ω) ≤ CGN∥∇u∥2L2(Ω)∥u∥ 1/2 L1(Ω) + CGN,∗∥u∥5/2L1(Ω) ≤ CGN∥∇u∥2L2(Ω)∥u0∥1/2L1(Ω) + CGN,∗∥u0∥5/2L1(Ω), which implies that ∥∇u∥2L2(Ω) ≥ 1 CGN∥u0∥1/2L1(Ω) ∥u∥5/2 L5/2(Ω) − CGN,∗ CGN ∥u0∥2L1(Ω) (3.7) In light of the Sobolev embedding theorem and elliptic Lp estimates, we can obtain a constant λ0= λ0(Ω, β) such that ∥w∥L5(Ω) ≤ λ0∥γu∥L10/7(Ω) for all t ∈ (0, Tmax). (3.8) Then from the Hölder inequality and Lemma 2.3 it follows that λ0∥γu∥L10/7(Ω) ≤ λ0γ( ∫ Ω u5/2) 1 5 (∫ Ω u )4/5 = λ0γ∥u∥1/2L5/2(Ω) ∥u0∥4/5L1(Ω) for all t ∈ (0, Tmax). Inserting the above inequality as well as (3.7) and (3.8) into (3.6), we derive that C4 > 0 such that 1 2 d dt ∫ Ω u2 + 1 CGN∥u0∥1/2L1(Ω) ∥u∥5/2 L5/2(Ω) ≤ λ0γδξ∥u0∥4/5L1(Ω)∥u∥ 5/2 L5/2(Ω) + CGN CGN,∗ ∥u0∥2L1(Ω) for all t ∈ (0, Tmax). This combined the Hölder inequality yields 1 2 d dt ∫ Ω u2 + ( 1 CGN∥u0∥1/2L1(Ω) − λ0γδξ∥u0∥4/5L1(Ω) ) |Ω|− 1 4 (∫ Ω u2 )5/4 ≤ CGN CGN,∗ ∥u0∥2L1(Ω) for all t ∈ (0, Tmax). Recalling the hypothesis λ0γδξ+∥u0∥10/7L1(Ω) < 1 2CGN , an ODE comparison implies that ∫ Ω u2 ≤ max{( CGN CGN,∗ ∥u0∥2L1(Ω)) 4/5∥u0∥2L1(Ω)|Ω| 1 5 , ∫ Ω u2 0} ≤ max{( CGN CGN,∗ ∥u0∥2L∞(Ω)|Ω| 2)4/5∥u0∥2L∞(Ω)|Ω| 11 5 , ∫ Ω u2 0} := C6(∥u0∥L∞(Ω)) for all t ∈ (0, Tmax). (3.9) 10 L. LIU EJDE-2025/26 Now, we can use the Sobolev embedding theorem and elliptic Lp estimates to deduce that ∥∇s∥L∞(Ω) ≤ C7∥s∥W 2,6(Ω) ≤ C8∥v∥L6(Ω) ≤ C9∥v∥ W 2, 3 2 (Ω) ≤ C10∥u∥ L 3 2 (Ω) ≤ C10∥u∥L2(Ω)|Ω| 3 4 ≤ C11C 1/2 6 (∥u0∥L∞(Ω)) for all t ∈ (0, Tmax) (3.10) with constants Ci > 0 (i = 7, 8, 9, 10, 11). Step 2. A bound for u in L∞((0, Tmax);L p(Ω)) for any p > 1: We test the first equation in (2.4) with up−1 and use the Young inequality and (3.10) to deduce that 1 p d dt ∫ Ω up + (p− 1) ∫ Ω up−2|∇u|2 = (p− 1) ∫ Ω up−1∇u · ∇s ≤ p− 1 2 ∫ Ω up−2|∇u|2 + p− 1 2 ∫ Ω up|∇s|2 ≤ p− 1 2 ∫ Ω up−2|∇u|2 + (p− 1)C2 11C6(∥u0∥L∞(Ω)) 2 ∫ Ω up (3.11) for all t ∈ (0, Tmax). Next, recalling the Gagliardo-Nirenberg inequality (see Lemma 2.2) and Lemma 2.3, we can obtain positive constants C12 and C13 such that (p− 1)C2 11C6(∥u0∥L∞(Ω)) 2 ∫ Ω up = (p− 1)C2 11C6(∥u0∥L∞(Ω)) 2 ∥up/2∥2L2(Ω) ≤ C12C6(∥u0∥L∞(Ω))(∥∇up/2∥2 Np−N 2−N+Np L2(Ω) ∥up/2∥2[1− Np−N 2−N+Np ] L2/p(Ω) + ∥up/2∥2L2/p(Ω)) = C12C6(∥u0∥L∞(Ω))(∥∇up/2∥2 Np−N 2−N+Np L2(Ω) ∥u0∥ p[1− Np−N 2−N+Np ] L2/p(Ω) + ∥u0∥pL1(Ω)) ≤ C13C6(∥u0∥L∞(Ω))([ p2 4 ] Np−N 2−N+Np ( ∫ Ω up−2|∇u|2) Np−N 2−N+Np ∥u0∥ p[1− Np−N 2−N+Np ] L2/p(Ω) + ∥u0∥pL1(Ω)) for all t ∈ (0, Tmax) (3.12) which with the Young inequality implies that for some C14 > 0 such that (p− 1)C2 11C6(∥u0∥L∞(Ω)) 2 ∫ Ω up ≤ p− 1 4 ∫ Ω up−2|∇u|2 + C14[C 2−N+Np 2 6 (∥u0∥L∞(Ω)) + 1][∥u0∥L∞(Ω)|Ω|]p (3.13) for all t ∈ (0, Tmax). Collecting (3.11) and (3.13), we have 1 p d dt ∫ Ω up + p− 1 4 ∫ Ω up−2|∇u|2 EJDE-2025/26 ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 11 ≤ C14(C 2−N+Np 2 6 (∥u0∥L∞(Ω)) + 1)[∥u0∥L∞(Ω)|Ω|]p for all t ∈ (0, Tmax) with C15 > 0. This combined with the Hölder inequality and (3.12) implies that 1 p d dt ∫ Ω up + (∫ Ω up ) 2−N+Np Np−N ≤ C15(C 2−N+Np 2 6 (∥u0∥L∞(Ω)) + 1)[∥u0∥L∞(Ω)|Ω|]p for all t ∈ (0, Tmax), with C15 > 0. Then, by an ODE comparison, we can obtain positive constants C16 and Λ2(∥u0∥L∞(Ω)) such that∫ Ω up ≤ max {∫ Ω up 0, C16Λ2(∥u0∥L∞(Ω)) } for all t ∈ (0, Tmax). (3.14) By choosing γ2 := C16 and Υ2(∥u0∥L∞(Ω)) := max{∥u0∥pL∞(Ω)|Ω|,Λ2(∥u0∥L∞(Ω))} in (3.14), we eventually obtain (3.5). □ In conjunction with the estimate for the estimate for u in Lp(Ω) provided by Lemmas 3.1–3.2, the latter entails boundedness of u as well as ∇v and ∇w in L∞(Ω). Lemma 3.3. Let N ≤ 4 and Υ(∥u0∥L∞(Ω)) = { Υ1(∥u0∥L∞(Ω)), if N ≤ 3, Υ2(∥u0∥L∞(Ω)), if N = 4. (3.15) where Υ1(∥u0∥L∞(Ω)) and Υ2(∥u0∥L∞(Ω)) are the same as Lemma 3.1 and Lemma 3.2, respectively. Then one can find ρ∗∗ > 1 independent of Υ(u0) such that the solution of (1.1) from Lemma 2.1 satisfies ∥u(·, t)∥L∞(Ω) ≤ ρ∗∗Υ(∥u0∥L∞(Ω)) for all t ∈ (0, Tmax), (3.16) ∥v(·, t)∥W 1,∞(Ω) ≤ ρ∗∗Υ(∥u0∥L∞(Ω)) for all t ∈ (0, Tmax), (3.17) ∥w(·, t)∥W 1,∞(Ω) ≤ ρ∗∗Υ(∥u0∥L∞(Ω)) for all t ∈ (0, Tmax). (3.18) Proof. In the following, we let κ∗∗,i(i ∈ N) denote some different constants, which are independent of ∥u0∥L∞(Ω), and if no special explanation, they may depend on Ω, α, β, γ, δ, ξ, χ. Now, applying the Lp estimate for the second and third equations of system (1.1), we derive that there exist positive constants κ∗∗,1, κ∗∗,2 as well as κ∗∗,3 and κ∗∗,4 independent of ∥u0∥L∞(Ω) such that ∥v(·, t)∥2NW 2,2N (Ω) ≤ κ∗∗,1∥αu(·, t)∥2NL2N (Ω) ≤ κ∗∗,2Υ(∥u0∥L∞(Ω)) for all t ∈ (0, Tmax), (3.19) ∥w(·, t)∥2NW 2,2N (Ω) ≤ κ∗∗,3∥γu(·, t)∥2NL2N (Ω) ≤ κ∗∗,4Υ(∥u0∥L∞(Ω)) for all t ∈ (0, Tmax), (3.20) where Υ(∥u0∥L∞(Ω)) is given by (3.15). Now, applying the Sobolev embedding theorems, we derive that W 2,2N (Ω) ↪→ W 1,∞(Ω), and therefore, we conclude from (3.19) that there exists a positive constants κ∗∗,5 independent of u0 such that ∥c(·, t)∥W 1,∞(Ω) + ∥w(·, t)∥W 1,∞(Ω) ≤ κ∗∗,5Υ 1 2N (∥u0∥L∞(Ω)) (3.21) 12 L. LIU EJDE-2025/26 for all t ∈ (0, Tmax). Now, let h̃ = χ∇v − ξ∇w. Then by (3.21), there exists a positive constant κ∗∗,6 > 0 such that ∥h̃(·, t)∥L∞(Ω) ≤ κ∗∗,6Υ 1 2N (∥u0∥L∞(Ω)) (3.22) for all t ∈ (0, Tmax). Next, by an associate variation-of-constants formula we can represent u(·, t) for each t ∈ (0, Tmax) according to u(·, t) = et∆u0(·)− ∫ t 0 e(t−s)∆∇ · (u(·, s)h̃(·, s))ds, t ∈ (0, Tmax). (3.23) The maximum principle implies that ∥et∆u0∥L∞(Ω) ≤ ∥u0∥L∞(Ω), (3.24) The last term on the right-hand side of (3.23) is estimated as follows. Invoking the known smoothing properties of the Neumann heat semigroup and the Hölder inequality to find κ∗∗,7 > 0 and κ∗∗,8 > 0 independent of ∥u0∥L∞(Ω) such that∫ t 0 ∥e(t−s)∆∇ · (u(·, s)h̃(·, s)∥L∞(Ω)ds ≤ κ∗∗,7 ∫ t 0 [1 + (t− s)− 1 2− N 4N ]e−λ(t−s)∥u(·, s)h̃(·, s)∥L2N (Ω)ds ≤ κ∗∗,7 ∫ t 0 [1 + (t− s)− 1 2− N 4N ]e−λ(t−s)∥u(·, s)∥L2N (Ω)∥h̃(·, s)∥L∞(Ω)ds ≤ κ∗∗,8Υ 1 N (∥u0∥L∞(Ω)) for all t ∈ (0, Tmax) (3.25) by using Lemmas 3.1 and 3.2 and (3.21). Thus the proof is complete. □ Combining (2.1) with Lemma 3.3, we obtain that system (1.1), (1.6), (1.7) are indeed global in time. Proposition 3.4. Let 1 ≤ N ≤ 4. Then the solution of (1.1) is global on [0,∞). Moreover, one can find independent of u0 such that the solution of (1.1) satisfies ∥u(·, t)∥L∞(Ω) + ∥v(·, t)∥W 1,∞(Ω) + ∥w(·, t)∥W 1,∞(Ω) ≤ λ∗Υ(∥u0∥L∞(Ω)) (3.26) for all t ∈ (0,∞). Proof. Firstly, relying on (2.1) and Lemma 3.3, we find λ1,∗ > 0 independent of u0 with the property that ∥u(·, t)∥L∞(Ω) + ∥v(·, t)∥W 1,∞(Ω) + ∥w(·, t)∥W 1,∞(Ω) ≤ λ∗Υ(∥u0∥L∞(Ω)) (3.27) for all t ∈ (0, Tmax) with Υ(∥u0∥L∞(Ω)) is the same as Lemma 3.3. In view of the extensibility criterion (2.1), we thus infer that Tmax = ∞, i.e., the solution (u, v, w) is global in time. Moreover, again based on Lemma 3.3, we can deduce that (3.26) holds. This completes the proof. □ 4. Asymptotic behavior In this section, we address the large time behavior of solution obtained above for system (1.1), (1.6), (1.7) with χα = ξγ and some appropriately small for ∥u0∥L∞(Ω). The crucial idea of the proof of large time behavior of solution for system (1.1), (1.6), (1.7) is to show a Lyapunov functional for system (1.1), (1.6), (1.7) under suitably small for ∥u0∥L∞(Ω). By means of an analysis of the corresponding energy EJDE-2025/26 ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 13 inequality, we can first establish the mere convergence of (u, v, w) to system (1.1), (1.6), (1.7) in L2(Ω) (see Corollary 4.29 and 4.3). We can thereupon make use of Lp estimate for the second and third equations of (1.1) and Lp-Lq estimates asso- ciated with the heat semigroup to show on the basis of additional higher regularity properties (Lemmas 4.4 and 4.5) that this convergence actually takes place at an exponential rate (Lemma 4.6). To implement our approach, we denote U(x, t) := u(x, t)− ū0, S(x, t) := s(x, t)− (χ α β − ξ γ δ )ū0, V (x, t) := v(x, t)− α β ū0 (4.1) for all x ∈ Ω and t > 0. Then we obtain from (4.1) and (2.4) that a triple (U, S, V ) satisfies Ut = ∆U −∇ · (U∇S), x ∈ Ω, t > 0, 0 = ∆S − δS + χ(δ − β)V, x ∈ Ω, t > 0, 0 = ∆V − βV + αU, x ∈ Ω, t > 0, ∂U ∂ν = ∂S ∂ν = ∂V ∂ν = 0, x ∈ ∂Ω, t > 0, U(x, 0) = u0(x)− ū0. (4.2) Having dealt with issues of boundedness so far, in the following, we next turn our attention to the claimed asymptotic behavior of solutions in (1.1). And we will show that in the large time limit, the classical global solution of (1.1) converges to (ū0, α β ū0, γ δ ū0) exponentially if ∥u0∥L∞(Ω) is smaller. To this end, as a preparation for the proof of Theorem 1.3, let us refine the argument from Proposition 3.4 to derive the following energy functional, which plays a crucial role in obtaining large time behavior of global solutions to system (1.1), (1.6), (1.7). Lemma 4.1. Suppose that λ∗Υ(∥u0∥L1(Ω)) < 2 √ CN , (4.3) where CN is the best Poincaré constant. Then there exists B > 0 B 2 d dt ∥u(·, t)− ū0∥2L2(Ω) + (BCN 2 − 1 4 ) ∫ Ω |u− ū|2 + ( 1− Bλ2 ∗Υ 2(∥u0∥L∞(Ω)) 2 ) ∫ Ω |∇v|2 ≤ 0 for all t > 0, (4.4) where ū0 = 1 |Ω| ∫ Ω u0. (4.5) 14 L. LIU EJDE-2025/26 Proof. Firstly, we use the first equation in (4.2), Theorem 1.1, and the Young inequality to compute 1 2 d dt ∥u(·, t)− ū∥2L2(Ω) = ∫ Ω (u− ū)[∆u−∇ · (u∇S)] ≤ − ∫ Ω |∇u|2 + ∫ Ω |u∥∇u∥∇S| ≤ −1 2 ∫ Ω |∇u|2 + 1 2 ∫ Ω u2|∇S|2 ≤ −1 2 ∫ Ω |∇u|2 + supt>0 ∥u(·, t)∥2L∞(Ω) 2 ∫ Ω |∇S|2 ≤ −1 2 ∫ Ω |∇u|2 + λ2 ∗Υ 2(∥u0∥L∞(Ω)) 2 ∫ Ω |∇S|2 for all t > 0, (4.6) where ū = 1 |Ω| ∫ Ω u. (4.7) Here λ∗ and ϱ(∥u0∥L∞(Ω)) are the same as in Proposition 3.4. We note from the Poincaré inequality that there is CN > 0 such that ∥φ− 1 |Ω| ∫ Ω φ∥2L2(Ω) ≤ CN ∫ Ω |∇φ|2 for all φ ∈ W 1,2(Ω). (4.8) This combined with (4.6) yields 1 2 d dt ∥u(·, t)− ū0∥2L2(Ω) ≤ −CN 2 ∫ Ω |u− ū0|2 + λ2 ∗Υ 2(∥u0∥L∞(Ω)) 2 ∫ Ω |∇S|2 (4.9) for all t > 0. Next, by the testing procedure, we may derive from the Young inequality that 0 = ∫ Ω S[∆S − δS + χ(δ − β)V ] ≤ −δ ∫ Ω |∇S|2 + χ2(δ − β)2 4δ ∫ Ω V 2 for all t > 0 (4.10) and 0 = ∫ Ω V [∆V − βV + αU ] ≤ − ∫ Ω |∇V |2 − β 2 ∫ Ω V 2 + α2 2β ∫ Ω (u− ū0) 2 for all t > 0, (4.11) where we have used that ∂S ∂ν = ∂V ∂ν = 0, x ∈ ∂Ω, t > 0. In view of (4.3), we can choose B > 0 such that A−B λ2 ∗Υ 2(∥u0∥L∞(Ω)) 2 > 0, (4.12) B 2 CN − α2 2β > 0, (4.13) EJDE-2025/26 ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 15 β 2 − χ2(δ − β)2A 4δ > 0. (4.14) Collecting (4.9)–(4.14), we infer that B 2 d dt ∥u(·, t)− ū0∥2L2(Ω) + ( BCN 2 − α2 2β ) ∫ Ω |u− ū0|2 + ∫ Ω |∇V |2 + (A− Bλ2 ∗Υ 2(∥u0∥L∞(Ω)) 2 ) ∫ Ω |∇S|2 + ( β 2 − χ2(δ − β)2A 4δ ) ∫ Ω V 2 ≤ 0 for all t > 0, (4.15) which completes the proof. □ The above Lemma entails exponential convergence for u(·, t) − ū0 at least with respect to the norm in L2(Ω). Corollary 4.2. Under the assumptions of Lemma 4.1, for each t > 0, there exists ρ1,∗ > 0 such that ∥u(·, t)− ū0∥2L2(Ω) ≤ e−ρ1,∗t[∥u0(·, t)− ū0∥2L2(Ω)], (4.16) and there exists C∗,1 > 0 such that∫ ∞ 0 ∫ Ω |∇V |2 + ∫ ∞ 0 ∫ Ω |∇S|2 + ∫ ∞ 0 ∫ Ω V 2 ≤ C∗,1, (4.17) where ū0 is given by (4.5). Proof. Let y(t) = B 2 d dt∥u(·, t)− ū0∥2L2(Ω). Then by (4.4), one can conclude that y(t) + (CN − α2 Bβ )y(t) ≤ 0, so that, integrating the above inequality and (4.4) in time, we can obtain (1.9) and (4.17) by using (4.6). □ As a application the above Lemma, we can derive the following stabilization property of V,∇V as well as S and ∇S which will be used in Lemma 4.6 below. Lemma 4.3. Under the assumptions of Lemma 4.1, for each t > 0, there exists ρ2,∗ > 0 such that ∥∇V (·, t)∥2L2(Ω) + ∥V (·, t)∥2L2(Ω) ≤ e−ρ2,∗t∥u0(·, t)− ū0∥2L2(Ω), (4.18) ∥∇S(·, t)∥2L2(Ω) + ∥S(·, t)∥2L2(Ω) ≤ e−ρ2,∗t∥u0(·, t)− ū0∥2L2(Ω), (4.19) where ū0 is given by (4.5). Proof. First, in view of the testing procedure, we derive from the Young inequality that 0 = ∫ Ω S[∆S − δS + χ(δ − β)V ] ≤ − ∫ Ω |∇S|2 − δ 2 ∫ Ω S2 + χ2(δ − β)2 2δ ∫ Ω V 2 for all t > 0 (4.20) 16 L. LIU EJDE-2025/26 and 0 = ∫ Ω V [∆V − βV + αU ] ≤ − ∫ Ω |∇V |2 − β 2 ∫ Ω V 2 + α2 2β ∫ Ω U2 for all t > 0, (4.21) which together with (1.9) implies that∫ Ω |∇V |2 + β 2 ∫ Ω V 2 ≤ α2 2β ∫ Ω U2 ≤ α2 2β e−ρ1,∗t∥u0(·, t)− ū0∥2L2(Ω) for all t > 0, (4.22) and therefore,∫ Ω |∇S|2 + δ 2 ∫ Ω S2 ≤ χ2(δ − β)2 2δ ∫ Ω V 2 ≤ χ2(δ − β)2 2δ α2 β2 e−ρ1,∗t∥u0(·, t)− ū0∥2L2(Ω) for all t > 0, (4.23) where ρ1,∗ and ū0 are given by (1.9) and (4.5), respectively. Hence, (1.10) and (4.19) holds by some basic analysis. □ Having found uniform bounds on u, v and w in the previous Proposition 3.4, also v and w share this regularity and these bounds. Lemma 4.4. Let N ≤ 4. Then for any p > 2, there exists a positive constant C∗,2 such that ∥v(·, t)∥W 2,p(Ω̄) + ∥w(·, t)∥W 2,p(Ω̄) ≤ C∗,2 for all t > 0. (4.24) Proof. Applying the Lp estimate for the second and third equations of (1.1), we derive from Proposition 3.4 that there exist positive constants κ∗∗∗,1, κ̃∗∗∗,1 as well as κ∗∗∗,2, κ̃∗∗∗,2 independent of u0 such that ∥v(·, t)∥pW 2,p(Ω) ≤ κ∗∗∗,1∥αu(·, t)∥pLp(Ω) ≤ κ∗∗∗,2Υ(∥u0∥L∞(Ω)) for all t > 0 (4.25) and ∥w(·, t)∥pW 2,p(Ω) ≤ κ̃∗∗∗,1∥γu(·, t)∥pLp(Ω) ≤ κ̃∗∗∗,2Υ(∥u0∥L∞(Ω)) for all t > 0. (4.26) □ To prepare our arguments concerning the large time behavior of solutions, we still need the following regularity estimates for u. Lemma 4.5. Assume that the conditions in Theorem 1.3 are satisfied. Then there exists a positive constant C∗,3 such that ∥u(·, t)∥W 1,∞(Ω) ≤ C∗,3 for all t > 0. (4.27) EJDE-2025/26 ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 17 Proof. Based on the regularity of u and v, one can readily obtain constants κ∗∗∗∗,1 > 0 such that ∥v(·, t)∥W 2,p(Ω) + ∥u(·, t)∥L∞(Ω) + ∥v(·, t)∥W 1,∞(Ω) ≤ κ∗∗∗∗,1 for all t > 0. (4.28) Next, we can rewrite the first equation of (1.1) as ut −∆u = a(u, v), (4.29) where a(x, t) = a(u(x, t), v(x, t), w(x, t)) = −χ∇ · (u∇v) + ξ∇ · (u∇w) = −χ∇u · ∇v − χu∆v + ξ∇u · ∇v + ξu∆w. To prove the boundedness of ∥∇u(·, t)∥L∞(Ω) on t > 0, by Duhamel’s principle, we see that the solution of (4.29) can be expressed as u(·, t) = e−t∆u0 + ∫ t 0 e−t∆a(·, τ) dτ for all t > 0. Next, for any T ∈ (0,∞), we let M(T ) := supt∈(0,T ) ∥∇u(·, t)∥L∞(Ω). By (4.28), there exists κ∗∗∗∗,4 > 0 such that ∥a(·, t)∥L2N (Ω) ≤ ∥ − χ∇u · ∇v − χu∆v + ξ∇u · ∇v + ξu∆w∥L2N (Ω) ≤ κ∗∗∗∗,4(∥∇u(·, t)∥L2N (Ω) + 1) for all t > 0. (4.30) Hence, in view of Lp-Lq estimates associated with the heat semigroup as well as (4.30), we derive that there exist positive constants λ, κ∗∗∗∗,5, κ∗∗∗∗,6, κ∗∗∗∗,7, κ∗∗∗∗,8, and κ∗∗∗∗,9 such that ∥u(·, t)∥W 1,∞(Ω) ≤ κ∗∗∗∗,5∥∇e−t∆u0(x) +∇ ∫ t 0 e−t∆a(x, τ)dτ∥L∞(Ω) ≤ κ∗∗∗∗,6e −λt∥u0∥L∞(Ω) + κ∗∗∗∗,6 ∫ t 0 [1 + (t− s)− 1 2− N 2 ( 1 2N − 1 ∞ )]e−λ(t−s)∥a(·, s)∥L2N (Ω)ds ≤ κ∗∗∗∗,8 + κ∗∗∗∗,7 ∫ t 0 [1 + (t− s)− 3 4 ]e−λ(t−s)(∥∇u(·, s)∥L2N (Ω) + 1)ds ≤ κ∗∗∗∗,9 + κ∗∗∗∗,7 ∫ t 0 [1 + (t− s)− 3 4 ]e−λ(t−s)∥∇u(·, s)∥L2N (Ω)ds (4.31) for all t ∈ (0, T ). Here, according to the Gagliardo-Nirenberg inequality (see Lemma 2.2), the boundedness of u in Ω × (0,∞) (see (3.22)), and the definition of M(T ) we can find κ∗∗∗∗,10 > 0 and κ∗∗∗∗,11 > 0 satisfying ∥∇u(·, t)∥L2N (Ω) ≤ κ∗∗∗∗,10∥∇u(·, t)∥1/2L∞(Ω)∥u(·, t)∥ 1/2 L∞(Ω) ≤ κ∗∗∗∗,11(M 1/2(T ) + 1) for all t ∈ (0, T ). (4.32) From (4.31), we obtain a positive constant κ∗∗∗∗,12 such that M(T ) ≤ κ∗∗∗∗,12 + κ∗∗∗∗,12M 1/2(T ) for all T ∈ (0,∞), (4.33) 18 L. LIU EJDE-2025/26 which in view of an elementary argument entails that for some κ∗∗∗∗,13 such that ∥∇u(·, t)∥L∞(Ω) ≤ κ∗∗∗∗,13 for all t ∈ (0,∞), (4.34) and thereby proves (4.27) by using (4.28). □ An immediate consequence of Corollary 4.2 and Lemmas 4.3 and 4.5, and Propo- sition 3.4 is that both u, v and w decay exponentially with respect to the norm in L∞(Ω). Lemma 4.6. Assume the hypothesis of Theorem 1.1 holds. Then one can find γ > 0 and C > 0 such that the global classical solution (u, v, w) of (1.1) satisfies ∥u(·, t)− ū0∥L∞(Ω) ≤ Ce−γt, for all t > 0, (4.35) ∥v(·, t)− α β ū0∥L∞(Ω) ≤ Ce−γt, for all t > 0, (4.36) ∥w(·, t)− γ δ ū0∥L∞(Ω) ≤ Ce−γt, for all t > 0, (4.37) where ū0 = 1 |Ω| ∫ Ω u0. Proof. We apply Corollary 4.2 and Lemmas 4.3 to find positive constants C1 and γ1 such that ∥u(·, t)− ū0∥L2(Ω) ≤ C1e −γ1t, for all t > 0, (4.38) ∥v(·, t)− α β ū0∥L2(Ω) ≤ C1e −γ1t, for all t > 0, (4.39) ∥w(·, t)− γ δ ū0∥L2(Ω) ≤ C1e −γ1t, for all t > 0. (4.40) Here we can use Proposition 3.4 and Lemma 4.5 to find C2 > 0 satisfying ∥u(·, t)−ū0∥W 1,∞(Ω)+∥v(·, t)−α β ū0∥W 1,∞(Ω)+∥w(·, t)− γ δ ū0∥W 1,∞(Ω) ≤ C2 (4.41) for all t > 0. Since an interpolation by the Gagliardo–Nirenberg inequality provides C3 > 0, C4 > 0, and C5 > 0 such that ∥u(·, t)− ū0∥L∞(Ω) ≤ C3(∥u(·, t)− ū0∥ N N+2 W 1,∞(Ω)∥u(·, t)− ū0∥ 2 N+2 L2(Ω) + ∥u(·, t)− ū0∥L2(Ω)) ≤ C4∥u(·, t)− ū0∥1/2L2(Ω) ≤ C5e −γt for all t > 0, (4.42) where γ = γ1 2 . Likewise, (4.36) and (4.37) can be obtained by combining the exponential con- vergence statement for v and w in Lemma 4.3 with the uniform higher order bound asserted by Lemma 3.3. □ Acknowledgments. 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Zheng; A new result for the global existence (and boundedness) and regularity of a three- dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization, J. Diff. Eqns., 272 (2021), 164–202. [49] J. Zheng; Eventual smoothness and stabilization in a three-dimensional Keller-Segel-Navier- Stokes system with rotational flux, Calc. Var. Partial Differ. Equ., 61 (2022), 1–34. Ling Liu Department of Basic Science, Jilin Jianzhu University, Changchun 130118, China Email address: liuling2004@sohu.com 1. Introduction 2. Preliminaries 3. Proof of Theorem 1.1 4. Asymptotic behavior Acknowledgments References