Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 58, pp. 1–17. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu BOUNDEDNESS AND ASYMPTOTIC STABILITY IN A CHEMOTAXIS MODEL WITH INDIRECT SIGNAL PRODUCTION AND LOGISTIC SOURCE XIAOBING YE, LIANGCHEN WANG Abstract. This article concerns the chemotaxis-growth system with indirect signal production ut = ∆u−∇ · (u∇v) + µu(1− u), x ∈ Ω, t > 0, 0 = ∆v − v + w, x ∈ Ω, t > 0, wt = −δw + u, x ∈ Ω, t > 0, on a smooth bounded domain Ω ⊂ Rn (n ≥ 1) with homogeneous Neumann boundary condition, where the parameters µ, δ > 0. It is proved that if n ≤ 2 and µ > 0, for all suitably regular initial data, this model possesses a unique global classical solution which is uniformly-in-time bounded. While in the case n ≥ 3, we show that if µ is sufficiently large, this system possesses a global bounded solution. Furthermore, the large time behavior and rates of convergence have also been considered under some explicit conditions. 1. Introduction In this article, we consider the chemotaxis model with indirect signal production ut = ∆u−∇ · (u∇v) + µu(1− u), x ∈ Ω, t > 0, 0 = ∆v − v + w, x ∈ Ω, t > 0, wt = −δw + u, x ∈ Ω, t > 0, ∂u ∂ν = ∂v ∂ν = 0, x ∈ ∂Ω, t > 0, u(x, 0) = u0(x), w(x, 0) = w0(x), x ∈ Ω, (1.1) where Ω ⊂ Rn(n ≥ 1) is a bounded domain with smooth boundary ∂Ω and ∂/∂ν denotes the derivative with respect to the outer normal of ∂Ω, the parameters µ > 0 and δ > 0. System (1.1) describes the spread and aggregative behavior of the Mountain Pine Beetle (MPB) in a forest habitat, u represents the density of flying MPB, v denotes the concentration of beetle pheromone, w is the density of nesting MPB. We can refer to [28] for more details about the biological background. Chemotaxis is the biased movement of the cells as a response to gradients of the concentration of the chemical signal substance, which plays an important role in 2020 Mathematics Subject Classification. 92C17, 35K35, 35A01, 35B35. Key words and phrases. Chemotaxis; boundedness; asymptotic behavior; indirect signal production. ©2022. This work is licensed under a CC BY 4.0 license. Submitted February 15, 2021. Published August 2, 2022. 1 2 X. B. YE, L. C. WANG EJDE-2022/58 numerous biological fields such as cell sorting, wound healing, pattern formation, embryonic morphogenesis and bacteria aggregation [7, 10, 16, 28]. The renowned chemotaxis system was put forward by Keller and Segel in 1970 [17] to describe the collective behavior of cells. And since then, many mathematicians have widely studied different types chemotaxis system for a variety of chemotaxis processes [4], the prototypical chemotaxis system with logistic source under homogeneous Neumann boundary condition reads as follows: ut = ∆u− χ∇ · (u∇v) + κu− µuγ , x ∈ Ω, t > 0, τvt = ∆v − v + u, x ∈ Ω, t > 0, (1.2) where Ω ⊂ Rn(n ≥ 1) is a bounded domain with smooth boundary, χ > 0, γ > 1 and κ, µ, τ ≥ 0, u and v is the density of the cells and the concentration of the chemical signal substance, respectively. In the case τ = 0 and γ = 2, if either n ≤ 2 or µ > n−2 n χ, Tello and Winkler [33] proved that the model (1.2) possesses a global bounded classical solution. Moreover, the global existence and boundedness of weak solutions to (1.2) was proved in [33, 40]. When the second equation in (1.2) is replaced by 0 = v−m(t)+u with m(t) := 1 |Ω| ∫ Ω u(x, t)dx, Winkler [39] proved that the solution of (1.2) blows up in finite time if n ≥ 5 and γ < 3 2 + 1 2n−2 . On the other hand, in the case τ = 1 and γ = 2, if either n = 2 or n ≥ 3 and µ > 0 is suitably large, then the solution of system (1.2) is globally bounded [38]. Model (1.2) has also been extensively studied by many other authors [2, 5, 6, 18, 24, 31, 41, 42, 43]. In model (1.2), we know that the chemical signal is directly produced by cells themselves, however, in some realistic biological processes the chemical signal pro- duction through the intermediate stages, and such indirect signal production mech- anisms may have new properties [13]. Before stating our main results in the present paper, we consider the system ut = ∆u−∇ · (u∇v) + µ(u− uα), x ∈ Ω, t > 0, vt = ∆v − v + w, x ∈ Ω, t > 0, wt = −δw + u, x ∈ Ω, t > 0, (1.3) model (1.3) has been intensively studied by many authors. For instance, if µ = 0 and the second equation in (1.3) is replaced by 0 = ∆v− 1 |Ω| ∫ Ω w(x, t)dx+w, in the case n = 2, Tao and Winkler [32] proved that under the assumption ∫ Ω u0 < 8πδ with δ > 0 the solution remain uniformly bounded, if ∫ Ω u0 > 8πδ with δ ≥ 0, the corresponding solution blow-up in infinite time. Moreover, if n = 2, α > 1, the boundedness of solution in (1.3) was showed in [20]. If µ > 0, α = 2, Hu and Tao [15] obtained that the solution of (1.3) is uniformly-in-time bounded in three dimensional setting, while under the assumption that µ > 1 8δ2 the solution will exponentially converges to (1, 1 δ , 1 δ ) as t → ∞. Model (1.3) has also been intensively studied by many other authors [9, 12, 25, 26, 34, 35]. Inspired by the arguments in [15, 32], we mainly consider the boundedness and asymptotic behavior of solutions to (1.1). In this paper, we suppose that the initial data (u0, w0) satisfies u0 ∈ C0(Ω), u0 ≥ 0 (6≡ 0) in Ω, w0 ∈ C1(Ω), w0 ≥ 0 in Ω. (1.4) Our main results are stated as follows. The first of our results addresses the corresponding low-dimensional framework. EJDE-2022/58 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 3 Theorem 1.1. Let Ω ⊂ Rn (n ≤ 2) be a bounded domain with smooth boundary and µ, δ > 0. Suppose that (1.4) holds, then (1.1) possesses a unique global classical solution (u, v, w) which is uniformly bounded in the sense that ‖u(·, t)‖L∞(Ω) + ‖v(·, t)‖L∞(Ω) + ‖w(·, t)‖L∞(Ω) ≤ C for all t > 0 with some constant C > 0. Next, we consider the global boundedness of solution to (1.1) in the case n ≥ 3. Theorem 1.2. Let Ω ⊂ Rn (n ≥ 3) be a bounded domain with smooth boundary and δ > 0. Assume that µ is sufficiently large and (1.4) holds, then (1.1) possesses a unique global classical solution (u, v, w) which is uniformly bounded. Finally, motivated by the ideas in [15, 36], we shall investigate asymptotic be- havior of solutions to (1.1) under the condition µ > 1/(16δ2). Theorem 1.3. Let Ω ⊂ Rn(n ≥ 1) be a bounded domain with smooth boundary and δ > 0. Suppose that (u, v, w) is a global bounded classical solution of (1.1), and the initial data (u0, w0) satisfies (1.4). Assume that µ > 1 16δ2 , then there exist C > 0 and κ > 0 such that ‖u− 1‖L∞(Ω) + ‖v − 1 δ ‖L∞(Ω) + ‖w − 1 δ ‖L∞(Ω) ≤ Ce−κt for all t > 0. Remark 1.4. Under the assumption µ > 1/(8δ2), Hu and Tao [15] proved that the asymptotic behavior of the solutions to (1.3) in the case α = 2. However, for the simplification (1.1), this condition µ > 1/(8δ2) in [15] can be relaxed to µ > 1/(16δ2). The rest of this article is organized as follows. In the next section, we give the local existence of a solution to (1.1) and some important inequalities for later proofs. Section 3, we main consider the global boundedness of solution to (1.1) in the case n = 2. In the case when n = 1 and n ≥ 3, the global boundedness of solution will be shown in Section 4. Section 5 is concern with the asymptotic behavior of the solution to system (1.1). 2. Preliminaries We first state the following result related to local existence of a classical solution to (1.1), for the detail proof, we refer the reader to [27, 30]. Lemma 2.1. Let Ω ⊂ Rn(n ≥ 1) be a bounded domain with smooth boundary and µ, δ > 0. Assume that (1.4) holds, then there exist Tmax ∈ (0,∞] and a unique triple (u, v, w) of nonnegative functions: u ∈ C0 ( Ω× [0, Tmax) ) ∩ C2,1 ( Ω× (0, Tmax) ) , v ∈ C2,0 ( Ω× [0, Tmax) ) , w ∈ C0,1 ( Ω× [0, Tmax) ) , which solves (1.1) classically in Ω × (0, Tmax) and has the following extensibility property if Tmax <∞ then ‖u(·, t)‖L∞(Ω) →∞ as t↗ Tmax. (2.1) 4 X. B. YE, L. C. WANG EJDE-2022/58 The Gagliardo-Nirenberg inequality will be used in our later proofs. For details, we refer the reader to [11, 23]. Lemma 2.2. Let Ω ⊂ Rn(n ≥ 1) be a bounded domain with smooth boundary, and let p ≥ 1 and r ∈ (0, p). Then there exists a constant CGN > 0 such that ‖ϕ‖Lp(Ω) ≤ CGN ( ‖∇ϕ‖λL2(Ω)‖ϕ‖ 1−λ Lr(Ω) + ‖ϕ‖Lr(Ω) ) for all ϕ ∈W 1,2(Ω) ∩ Lr(Ω) holds with λ ∈ (0, 1) satisfies n p = λ (n 2 − 1 ) + n r (1− λ), that is, λ = n r − n p 1− n 2 + n r . Finally, we give the following property on the total mass of the first equation in (1.1) which can easily be achieved. Lemma 2.3. Let Ω ⊂ Rn(n ≥ 1) be a bounded domain with smooth boundary and µ, δ > 0. Then there exists a constant K > 0 such that∫ Ω u(·, t) ≤ K for all t ∈ (0, Tmax). (2.2) Proof. Integrating the first equation in (1.1) with respect to x ∈ Ω, we have d dt ∫ Ω u = µ ∫ Ω u− µ ∫ Ω u2 for all t ∈ (0, Tmax). Using Hölder’s inequality we can deduce that d dt ∫ Ω u ≤ µ ∫ Ω u− µ |Ω| (∫ Ω u )2 for all t ∈ (0, Tmax), which implies ‖u(·, t)‖L1(Ω) ≤ K := max{‖u0‖L1(Ω), |Ω|} for all t ∈ (0, Tmax). Hence, this proof can be completed. � 3. A priori estimate for n = 2 In this section, we consider the global boundedness of solutions to (1.1) with n = 2. Firstly, to prove Lemma 3.2, we shall need the following basic inequality. Lemma 3.1 ([15, Lemma 2.3]). Let µ > 0 and δ > 0. Then there exists M := M(µ, δ) > 0 such that µs+ (2 δ − µ ) s2 + (µ+ 1)s ln s− µs2 ln s ≤M for all s > 0. Next, we start to establish a bound for ∫ Ω w2. Lemma 3.2. Let µ, δ > 0 and suppose that (1.4) holds. Then there exists a constant C > 0 such that the classical solution of (1.1) satisfying∫ Ω w2 ≤ C for all t ∈ (0, Tmax). (3.1) EJDE-2022/58 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 5 Proof. Testing the first equation in (1.1) by lnu+ 1 and integrating we have d dt ∫ Ω u lnu = ∫ Ω (lnu+ 1)∆u− ∫ Ω (lnu+ 1)∇ · (u∇v) + µ ∫ Ω (lnu+ 1)u(1− u) = − ∫ Ω |∇u|2 u + ∫ Ω ∇u · ∇v + µ ∫ Ω (lnu+ 1)u(1− u) ≤ − ∫ Ω u∆v + µ ∫ Ω (lnu+ 1)u(1− u) ≤ ∫ Ω uw + µ ∫ Ω u+ µ ∫ Ω u lnu− µ ∫ Ω u2 − µ ∫ Ω u2 lnu (3.2) for all t ∈ (0, Tmax). Next, testing the third equation in (1.1) by w and integrating with respect to x ∈ Ω, we see that 1 2 d dt ∫ Ω w2 = −δ ∫ Ω w2 + ∫ Ω uw for all t ∈ (0, Tmax). (3.3) Combining (3.2) and (3.3) and adding ∫ Ω u lnu to both sides of this, we obtain d dt {∫ Ω u lnu+ 1 2 ∫ Ω w2 } + ∫ Ω u lnu ≤ 2 ∫ Ω uw − δ ∫ Ω w2 + µ ∫ Ω u+ (µ+ 1) ∫ Ω u lnu− µ ∫ Ω u2 − µ ∫ Ω u2 lnu (3.4) for all t ∈ (0, Tmax). We will deal with the first term on the right hand of (3.4). Applying Young’s inequality we obtain 2 ∫ Ω uw ≤ δ 2 ∫ Ω w2 + 2 δ ∫ Ω u2. (3.5) Therefore, combining (3.4) and (3.5) we have d dt {∫ Ω u lnu+ 1 2 ∫ Ω w2 } + {∫ Ω u lnu+ δ 2 ∫ Ω w2 } ≤ µ ∫ Ω u+ (µ+ 1) ∫ Ω u lnu+ (2 δ − µ ) ∫ Ω u2 − µ ∫ Ω u2 lnu (3.6) for all t ∈ (0, Tmax). Letting y(t) := ∫ Ω u lnu + 1 2 ∫ Ω w2 and using Lemma 3.1, we have y′(t) + C1y(t) ≤ C2 for all t ∈ (0, Tmax), where C1 := min{1, δ} and C2 := M |Ω| with M is defined in Lemma 3.1. Therefore, by the definition of y(t) and together with the basic inequality −s ln s ≤ 1/e for all s > 0, we obtain (3.1). � To obtain (3.8) below, we give the following important lemma. Lemma 3.3. Let n = 2 and suppose that (1.4) holds. Then for any 1 < q < ∞, one can find a constant C > 0 such that∫ Ω |∇v|q ≤ C for all t ∈ (0, Tmax). (3.7) 6 X. B. YE, L. C. WANG EJDE-2022/58 Proof. Noting that v satisfies −∆v + v = w, x ∈ Ω, t > 0, ∂v ∂ν = 0, x ∈ ∂Ω, t > 0. From (3.1) and elliptic equation theory, for some C1 > 0 and C2 > 0, we infer that ‖v‖W 2,2(Ω) ≤ C1‖w‖L2(Ω) ≤ C2 for all t ∈ (0, Tmax). Since n = 2 and according to Sobolev embedding theory, for any 1 < q < ∞, we see that (3.7). � Finally, our aim is to sutdy the boundedness of ∫ Ω up + ∫ Ω wp+1. Lemma 3.4. Let n = 2, µ, δ > 0, and suppose that (1.4) holds. Then for any p > 1, there exists a constant C > 0 such that∫ Ω up + ∫ Ω wp+1 ≤ C for all t ∈ (0, Tmax). (3.8) Proof. Testing the first equation in (1.1) by up−1, integrating by parts and using (3.7) and Young’s inequality, for some C1 > 0, we obtain 1 p d dt ∫ Ω up = −(p− 1) ∫ Ω up−2|∇u|2 + (p− 1) ∫ Ω up−1∇u · ∇v + µ ∫ Ω up − µ ∫ Ω up+1 ≤ −p− 1 2 ∫ Ω up−2|∇u|2 + p− 1 2 ∫ Ω up|∇v|2 + µ ∫ Ω up − µ ∫ Ω up+1 ≤ −2(p− 1) p2 ∫ Ω |∇up/2|2 + p− 1 2 ∫ Ω up+ 1 2 + p− 1 2 ∫ Ω |∇v|4p+2 + µ ∫ Ω up − µ ∫ Ω up+1 ≤ −2(p− 1) p2 ∫ Ω |∇up/2|2 + p− 1 2 ∫ Ω up+ 1 2 + µ ∫ Ω up − µ ∫ Ω up+1 + C1 (3.9) for all t ∈ (0, Tmax). Now we will cope with the second term on the right-hand side of (3.9). According to Lemma 2.2 and (2.2), there exist C2 > 0 and C3 > 0 such that ∫ Ω up+ 1 2 = ‖up/2‖ 2p+1 p L 2p+1 p (Ω) ≤ C2‖∇up/2‖ 2p+1 p ·θ L2(Ω) ‖u p/2‖ 2p+1 p ·(1−θ) L2/p(Ω) + C2‖up/2‖ 2p+1 p L2/p(Ω) ≤ C3‖∇up/2‖ 2p+1 p ·θ L2(Ω) + C3 (3.10) for all t ∈ (0, Tmax), where θ = 2p−1 2p+1 ∈ (0, 1), thus 2p+ 1 p θ = 2p− 1 p = 2− 1 p < 2. Applying Young’s inequality, for some C4 > 0, we can derive p− 1 2 ∫ Ω up+ 1 2 ≤ 2(p− 1) p2 ‖∇up/2‖2L2(Ω) + C4. (3.11) EJDE-2022/58 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 7 Combining (3.9) and (3.11), for some C5 > 0, we have 1 p d dt ∫ Ω up ≤ µ ∫ Ω up − µ ∫ Ω up+1 + C5 (3.12) for all t ∈ (0, Tmax). Adding ∫ Ω up to both sides of (3.12) we obtain d dt ∫ Ω up + ∫ Ω up ≤ (µp+ 1) ∫ Ω up − µp ∫ Ω up+1 + C5p (3.13) for all t ∈ (0, Tmax). Next applying Young’s inequality, for some C6 > 0, we have d dt ∫ Ω up + ∫ Ω up + µp 2 ∫ Ω up+1 ≤ C6 for all t ∈ (0, Tmax), which implies ‖u(·, t)‖Lp(Ω) ≤ C for all t ∈ (0, Tmax), (3.14)∫ t+τ t ∫ Ω up+1 ≤ C for all t ∈ (0, Tmax − τ) (3.15) where τ := min{1, 1 2Tmax}. Testing the third equation in (1.1) by wp and integrat- ing yields 1 p+ 1 d dt ∫ Ω wp+1 + δ ∫ Ω wp+1 = ∫ Ω uwp ≤ δ 2 ∫ Ω wp+1 + ( 2 δ )p ∫ Ω up+1 namely, d dt ∫ Ω wp+1 + δ(p+ 1) 2 ∫ Ω wp+1 ≤ (p+ 1) (2 δ )p ∫ Ω up+1 (3.16) for all t ∈ (0, Tmax), which implies ‖w(·, t)‖Lp+1(Ω) is bounded for all t ∈ (0, Tmax) because of (3.15). Hence, with (3.14), we have (3.8). � 4. A priori estimate for n = 1 and n ≥ 3 In this section, we establish the boundedness of ∫ Ω up+ ∫ Ω wp+1 in the case n = 1 and n ≥ 3. To obtain our goals, we first establish a differential inequality. Lemma 4.1. Let p > 1 and µ, δ > 0, and suppose that (1.4) holds. Then we have d dt {1 p ∫ Ω up + 1 p+ 1 ∫ Ω wp+1 } + δ ∫ Ω wp+1 + 4(p− 1) p2 ∫ Ω |∇up/2|2 ≤ p− 1 p ∫ Ω upw + ∫ Ω uwp + µ ∫ Ω up − µ ∫ Ω up+1 (4.1) for all t ∈ (0, Tmax). 8 X. B. YE, L. C. WANG EJDE-2022/58 Proof. Multiplying the first equation in (1.1) by up−1, integrating by parts and combining the second equation in (1.1), we obtain 1 p d dt ∫ Ω up = −(p− 1) ∫ Ω up−2|∇u|2 + (p− 1) ∫ Ω up−1∇u · ∇v + µ ∫ Ω up − µ ∫ Ω up+1 = −4(p− 1) p2 ∫ Ω |∇up/2|2 + p− 1 p ∫ Ω ∇up · ∇v + µ ∫ Ω up − µ ∫ Ω up+1 = −4(p− 1) p2 ∫ Ω |∇up/2|2 − p− 1 p ∫ Ω up ·∆v + µ ∫ Ω up − µ ∫ Ω up+1 ≤ −4(p− 1) p2 ∫ Ω |∇up/2|2 + p− 1 p ∫ Ω upw + µ ∫ Ω up − µ ∫ Ω up+1 (4.2) for all t ∈ (0, Tmax). Next, testing the third equation in (1.1) by wp and integrating, we have 1 p+ 1 d dt ∫ Ω wp+1 + δ ∫ Ω wp+1 = ∫ Ω uwp for all t ∈ (0, Tmax). (4.3) Therefore, combining (4.2) and (4.3) we obtain (4.1). � Next, we will establish a bound for ∫ Ω up + ∫ Ω wp+1 in the case n = 1. Lemma 4.2. Let n = 1 and µ, δ > 0, and suppose that (1.4) holds. Then for any p > 1, there exists a constant C > 0 such that∫ Ω up + ∫ Ω wp+1 ≤ C for all t ∈ (0, Tmax). (4.4) Proof. Using Young’s inequality, there exists C1 > 0 such that p− 1 p ApB +ABp ≤ δ 2 Bp+1 + C1A p+1 for all p > 1 and A,B ≥ 0. (4.5) Therefore, combining (4.1) and (4.5) and applying Young’s inequality, for some C2 > 0, C3 > 0, we can obtain d dt {1 p ∫ Ω up + 1 p+ 1 ∫ Ω wp+1 } + δ ∫ Ω wp+1 + 4(p− 1) p2 ∫ Ω |∇up/2|2 ≤ C2 ∫ Ω up+1 + δ 2 ∫ Ω wp+1 + C3, (4.6) that is, d dt {1 p ∫ Ω up + 1 p+ 1 ∫ Ω wp+1 } + δ 2 ∫ Ω wp+1 + 4(p− 1) p2 ∫ Ω |∇up/2|2 ≤ C2 ∫ Ω up+1 + C3 (4.7) for all t ∈ (0, Tmax). Now we deal with the term C2 ∫ Ω up+1 in (4.7). Since n = 1, using Lemma 2.2 there exists CGN > 0 such that∫ Ω up+1 = ‖up/2‖ 2(p+1) p L 2(p+1) p (Ω) ≤ CGN‖∇up/2‖ 2(p+1) p ·θ L2(Ω) ‖u p/2‖ 2(p+1) p ·(1−θ) L2/p(Ω) + CGN‖up/2‖ 2(p+1) p L2/p(Ω) , (4.8) EJDE-2022/58 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 9 where θ := p2 p2 + 2p+ 1 ∈ (0, 1). We can easily be checked that 2(p+ 1) p · θ = 2(p+ 1) p p2 (p+ 1)2 = 2p p+ 1 < 2. Therefore, using (2.2) and Young’s inequality, for some C4 > 0, we can derive C2 ∫ Ω up+1 ≤ 2(p− 1) p2 ‖∇up/2‖2L2(Ω) + C4. (4.9) Combining (4.7) and (4.9) we have d dt {1 p ∫ Ω up + 1 p+ 1 ∫ Ω wp+1 } + δ 2 ∫ Ω wp+1 + 2(p− 1) p2 ∫ Ω |∇up/2|2 ≤ C5 (4.10) for all t ∈ (0, Tmax), where C5 = C3 + C4. Applying the Poincaré inequality, for some C6 > 0, we obtain∫ Ω ϕ2 ≤ C6 ∫ Ω |∇ϕ|2 + C6 (∫ Ω |ϕ|p/2 )p for all ϕ ∈W 1,2(Ω). (4.11) Therefore, combining (4.11) and (2.2) there exists C7 > 0 such that 2(p− 1) p2 ∫ Ω |∇up/2|2 ≥ 2(p− 1) C7p2 ∫ Ω up − 2(p− 1) p2 Kp for all t ∈ (0, Tmax). (4.12) From (4.10) and (4.12) we have d dt {1 p ∫ Ω up + 1 p+ 1 ∫ Ω wp+1 } + {2(p− 1) C7p2 ∫ Ω up + δ 2 ∫ Ω wp+1 } ≤ C8 (4.13) for all t ∈ (0, Tmax), where C8 = C5 + 2(p−1) p2 Kp. Letting y(t) := 1 p ∫ Ω up + 1 p+1 ∫ Ω wp+1 and C9 = min{ 2(p−1) C7p , δ(p+1) 2 } we obtain y′(t) + C9y(t) ≤ C8 for all t ∈ (0, Tmax). By the definition of y(t), for some C10 > 0 we can derive∫ Ω up ≤ C10 and ∫ Ω wp+1 ≤ C10 for all t ∈ (0, Tmax). (4.14) The proof is complete � Next, we consider the boundedness of ∫ Ω up + ∫ Ω wp+1 in the case n ≥ 3. Lemma 4.3. Let Ω ⊂ Rn(n ≥ 3) be a bounded domain with smooth boundary and δ > 0, and suppose that (1.4) holds. Then for p > 1 and µ sufficiently large, there exists a constant C > 0 such that∫ Ω up + ∫ Ω wp+1 ≤ C for all t ∈ (0, Tmax). (4.15) 10 X. B. YE, L. C. WANG EJDE-2022/58 Proof. According to (4.1) and (4.5) and using Young’s inequality, for some C1 > 0, C2 > 0, we have d dt {1 p ∫ Ω up + 1 p+ 1 ∫ Ω wp+1 } + δ ∫ Ω wp+1 + 4(p− 1) p2 ∫ Ω |∇up/2|2 ≤ p− 1 p ∫ Ω upw + ∫ Ω uwp + µ ∫ Ω up − µ ∫ Ω up+1 ≤ δ 2 ∫ Ω wp+1 − (µ 2 − C1 ) ∫ Ω up+1 + C2 (4.16) for all t ∈ (0, Tmax). Since µ is sufficiently large, we can derive µ 2 − C1 > 0. Thus, we know that d dt {1 p ∫ Ω up + 1 p+ 1 ∫ Ω wp+1 } + δ 2 ∫ Ω wp+1 + 4(p− 1) p2 ∫ Ω |∇up/2|2 ≤ C2 (4.17) for all t ∈ (0, Tmax). Using the Poincaré inequality and (2.2), there exists C3 > 0 such that 4(p− 1) p2 ∫ Ω |∇up/2|2 ≥ 4(p− 1) C3p2 ∫ Ω up − 4(p− 1) p2 Kp for all t ∈ (0, Tmax). (4.18) Thus, combining (4.17) and (4.18) we obtain d dt {1 p ∫ Ω up + 1 p+ 1 ∫ Ω wp+1 } + {4(p− 1) C3p2 ∫ Ω up + δ 2 ∫ Ω wp+1 } ≤ C4, (4.19) where C4 = C2 + 4(p−1) p2 Kp. Letting y(t) := 1 p ∫ Ω up + 1 p+1 ∫ Ω wp+1 we obtain y′(t) + C5y(t) ≤ C4 for all t ∈ (0, Tmax), where C5 = min { 4(p−1) C3p , δ(p+1) 2 } . By an ODE comparison argument and the defi- nition of y(t), we can derive (4.15). � Finally, we will give a boundedness criterion for the solutions of (1.1). Lemma 4.4. Let Ω ⊂ Rn(n ≥ 1) be a bounded domain with smooth boundary and (1.4) hold. If there exist p ≥ 2 such that p > n− 1 and that∫ Ω up + ∫ Ω wp+1 ≤ C(p) for all t ∈ (0, Tmax) (4.20) with a constant C(p) > 0, then Tmax =∞ and ‖u(·, t)‖L∞(Ω) <∞ for all t > 0. (4.21) Proof. Noting that v satisfies −∆v + v = w, x ∈ Ω, t > 0, ∂v ∂ν = 0, x ∈ ∂Ω, t > 0. According to (4.20) and elliptic regularity theory, one can find c1 > 0 such that ‖v(·, t)‖W 2,p+1(Ω) ≤ c1 for all t ∈ (0, Tmax). Applying Sobolev embedding theory [14], fix p > n−1, for some c2 > 0 independent of p, we can derive ‖∇v(·, t)‖L∞(Ω) ≤ c2 for all t ∈ (0, Tmax). (4.22) EJDE-2022/58 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 11 From the well-known Moser-Alikakos iteration [1, 8], testing the first equation of (1.1) by up−1(p ≥ 2) and integrating by parts over Ω, using (4.22) and Young’s inequality we have 1 p d dt ∫ Ω up + (p− 1) ∫ Ω up−2|∇u|2 + µ ∫ Ω up+1 = (p− 1) ∫ Ω up−1∇u · ∇v + µ ∫ Ω up ≤ c2(p− 1) ∫ Ω up−1|∇u|+ µ(p− 1) ∫ Ω up ≤ p− 1 2 ∫ Ω up−2|∇u|2 + (c22 2 + µ ) (p− 1) ∫ Ω up (4.23) for all t ∈ (0, Tmax). Using the following interpolation inequality [29], we can find c3 > 0 independent p such that ‖ϕ‖2L2(Ω) ≤ ε‖∇ϕ‖ 2 L2(Ω) + c3 ( 1 + ε− n 2 ) ‖ϕ‖2L1(Ω) for all ε > 0. (4.24) Letting c4 = c22 2 + µ+ 1, combining (4.23) and (4.24) we conclude d dt ∫ Ω up + p(p− 1) ∫ Ω up + 2(p− 1) p ∫ Ω |∇up/2|2 ≤ c4p(p− 1) ∫ Ω up ≤ 2(p− 1) p ∫ Ω |∇up/2|2 + c5p(p− 1)(1 + pn) (∫ Ω up/2 )2 (4.25) for all t ∈ (0, Tmax), where c5 = c3c4 max{1, (c4/2)n/2}. Therefore, the inequality (4.25) can be rewritten as d dt ∫ Ω up + p(p− 1) ∫ Ω up ≤ c5p(p− 1)(1 + pn) (∫ Ω up/2 )2 for all t ∈ (0, Tmax), together with the fact (1 + pn) ≤ (1 + p)n we have d dt ∫ Ω up + p(p− 1) ∫ Ω up ≤ c5p(p− 1)(1 + p)n (∫ Ω up/2 )2 for all t ∈ (0, Tmax). Thus, d dt [ ep(p−1)t ∫ Ω up ] ≤ c5ep(p−1)tp(p− 1)(1 + p)n (∫ Ω up/2 )2 (4.26) for all t ∈ (0, Tmax). Integrating (4.26) over [0, t] for 0 < t < Tmax, we obtain∫ Ω up(x, t) ≤ ∫ Ω up0(x) + c5(1 + p)n sup 0≤t≤Tmax (∫ Ω up/2(x, t) )2 . (4.27) We define G(p) := max { ‖u0‖L∞(Ω), sup 0≤t≤Tmax (∫ Ω up(x, t) )1/p} . (4.28) Then, from (4.27) and (4.28) we obtain G(p) ≤ [c6(1 + p)n]1/pG( p 2 ) for all p ≥ 2, where c6 = |Ω|+ c5. Taking p = 2j , j = 1, 2, · · · , we obtain G(2j) ≤ c2 −j 6 ( 1 + 2j )2−jn G ( 2j−1 ) ≤ · · · 12 X. B. YE, L. C. WANG EJDE-2022/58 ≤ c2 −j+···+2−1 6 ( 1 + 2j )2−jn · · · (1 + 2)2−1nG(1) ≤ c6[2j2 −jn(2−j + 1)2−jn] · · · [22−1n(2−1 + 1)2−1n]G(1) ≤ c62[j2−j+(j−1)2−(j−1)+···+2−1]n · 2[2−j+2−(j−1)+···+2−1]nG(1) ≤ c623nG(1). Letting j →∞ and using (2.2) we infer that ‖u(·, t)‖L∞ ≤ c623nG(1) ≤ C := c623n max{‖u0‖L∞(Ω), ‖u0‖L1(Ω)} (4.29) for all t ∈ (0, Tmax). Therefore, using Lemma 2.1, (4.29), and the extensibility criterion we can obtain (4.21). � Proof of Theorems 1.1 and 1.2. The results follow by using Lemmata 3.4, 4.2 and 4.3 and the boundedness criterion of Lemma 4.4. � 5. Asymptotic behavior of solutions In this section, we consider the asymptotic behavior of solutions to (1.1) under the assumption that µ is positive and properly large. Then the solution (u, v, w) will exponentially converges to the constants stationary solution (1, 1/δ, 1/δ). Firstly, we give the following lemma which plays an important role for later proofs. Lemma 5.1 ([3, Lemma 3.1]). Suppose that f : (1,∞) is a uniformly continuous nonnegative function such that ∫∞ 1 f(t)dt <∞. Then f(t)→ 0 as t→∞. To proof of Theorem 1.3, we shall need the following basic property. For a detail proof, we refer the reader to [22, 36]. Lemma 5.2. Let a1, a2, a3, a4, a5 ∈ R satisfy a1 > 0, a2 > 0, a1a2a3 − a1a 2 5 4 − a2a 2 4 4 > 0. (5.1) Then there exists ε > 0 such that a1x 2 + a2y 2 + a3z 2 + a4xz + a5yz ≥ ε(x2 + y2 + z2) (5.2) for any x, y, z ∈ R. Now, we use an energy functional to establish the asymptotic behavior of solu- tions to system (1.1), and an idea from [15, 37, 44]. Lemma 5.3. Let δ > 0 and (u, v, w) be a non-negative global bounded classical solution of (1.1). Assume that the initial data (u0, w0) satisfies (1.4). If µ > 1 16δ2 , (5.3) then there exist ε > 0 and β > 0, such that the functions E(t) := ∫ Ω (u− 1− lnu) + β 2 ∫ Ω ( w − 1 δ )2 , (5.4) F (t) := ∫ Ω (u− 1)2 + ∫ Ω ( v − 1 δ )2 + ∫ Ω ( w − 1 δ )2 (5.5) satisfy E(t) ≥ 0 for all t > 0, (5.6) EJDE-2022/58 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 13 d dt E(t) ≤ −εF (t) for all t > 0. (5.7) Proof. We define A(t) := ∫ Ω (u− 1− lnu) for all t > 0, B(t) := 1 2 ∫ Ω ( w − 1 δ )2 for all t > 0. Then we can rewritten (5.4) as E(t) = A(t) + β ·B(t) for all t > 0. Next we prove (5.7). By the straightforward calculations we have d dt A(t) = ∫ Ω ut − 1 u ut = ∫ Ω [ ∆u−∇ · (u∇v) + µu(1− u) − 1 u (∆u−∇ · (u∇v) + µu(1− u)) ] = − ∫ Ω ∆u u + ∫ Ω 1 u · ∇ · (u∇v)− µ ∫ Ω (u− 1)2 = − ∫ Ω |∇u|2 u2 + ∫ Ω 1 u ∇u · ∇v − µ ∫ Ω (u− 1)2 ≤ 1 4 ∫ Ω |∇v|2 − µ ∫ Ω (u− 1)2. (5.8) To estimate ∫ Ω |∇v|2, testing the second equation of (1.1) by ( v− 1 δ ) and integrating by parts we have 0 = ∫ Ω ∆v ( v − 1 δ ) + ∫ Ω ( w − 1 δ )( v − 1 δ ) − ∫ Ω ( v − 1 δ )2 = − ∫ Ω |∇v|2 + ∫ Ω ( w − 1 δ )( v − 1 δ ) − ∫ Ω ( v − 1 δ )2 , (5.9) that is ∫ Ω |∇v|2 = ∫ Ω ( w − 1 δ )( v − 1 δ ) − ∫ Ω ( v − 1 δ )2 . (5.10) Combining (5.8) and (5.10), we obtain d dt A(t) ≤ 1 4 ∫ Ω ( w − 1 δ )( v − 1 δ ) − 1 4 ∫ Ω ( v − 1 δ )2 − µ∫ Ω (u− 1)2. (5.11) Rewriting the third equation in system (1.1) as wt = −δw + u = −δ ( w − 1 δ ) + (u− 1). (5.12) Multiplying the equation (5.12) by ( w− 1 δ ) and using Young’s inequality, we derive d dt B(t) = −δ ∫ Ω ( w − 1 δ )2 + ∫ Ω (u− 1) ( w − 1 δ ) . (5.13) Therefore, combining (5.11) and (5.13) we infer that d dt E(t) ≤ −µ ∫ Ω (u− 1)2 − 1 4 ∫ Ω ( v − 1 δ )2 − δβ ∫ Ω ( w − 1 δ )2 14 X. B. YE, L. C. WANG EJDE-2022/58 + 1 4 ∫ Ω ( w − 1 δ )( v − 1 δ ) + β ∫ Ω (u− 1) ( w − 1 δ ) for all t > 0. Now we will prove that there exists ε > 0 such that d dt E(t) ≤ −ε {∫ Ω (u− 1)2 + ∫ Ω ( v − 1 δ )2 + δ ∫ Ω ( w − 1 δ )2} . (5.14) We set a1 := µ, a2 := 1/4, a3 := δβ, a4 := −β, a5 := −1/4, x := u− 1, y := v − 1 δ , and z := w − 1 δ . From (5.3) and µ, δ > 0, we obtain a1 = µ > 0, a2 = 1 4 > 0, a1a2a3 − a1a 2 5 4 − a2a 2 4 4 = µδβ 4 − µ 64 − β2 16 = 16µδβ − 4β2 − µ 64 > 0. Using Lemma 5.2, we obtain there exists ε > 0 such that (5.14) holds. The proof is complete. � In view of Lemma 5.3 we have the asymptotic behavior of u, v and w. Lemma 5.4. Let (u, v, w) be a global bounded classical solution of (1.1) and the initial data (u0, w0) satisfy (1.4). Then we have the following asymptotic behavior ‖u− 1‖L∞(Ω) + ‖v − 1 δ ‖L∞(Ω) + ‖w − 1 δ ‖L∞(Ω) → 0 as t→∞. (5.15) Proof. From u, v, w are bounded in Ω× (0,∞), applying standard parabolic regu- larity theory [19] (see also [3, 21]), there exist θ ∈ (0, 1) and C > 0 such that ‖u‖ C2+θ,1+ θ 2 (Ω×[t,t+1]) + ‖v‖ C2+θ,1+ θ 2 (Ω×[t,t+1]) + ‖w‖ C2+θ,1+ θ 2 (Ω×[t,t+1]) ≤ C (5.16) for all t ≥ 1. Integrating (5.7) over (1,∞) and thanks to E(t) ≥ 0 we have∫ ∞ 1 F (t)dt ≤ 1 ε E(t) <∞. According to (5.16) we infer that F (t) is uniformly continuous in (1,∞). Therefore, from Lemma 5.1 we obtain F (t) = ∫ Ω (u− 1)2 + ∫ Ω ( v − 1 δ )2 + ∫ Ω ( w − 1 δ )2 → 0 as t→∞. (5.17) Using the Gagliardo-Nirenberg interpolation inequality with some CGN > 0 we obtain ‖ϕ‖L∞(Ω) ≤ CGN‖ϕ‖ n n+2 W 1,∞(Ω)‖ϕ‖ 2 n+2 L2(Ω) for all ϕ ∈W 1,∞(Ω). (5.18) Applying (5.16) and collecting (5.17)-(5.18), we have (5.15). � To obtain the rate of convergence for u, v and w, we give the following lemma. Lemma 5.5. Let (u, v, w) be a nonnegative global bounded classical solution of (1.1) and the initial data (u0, w0) fulfill (1.4). Then one can find C > 0 and κ > 0 such that ‖u− 1‖L∞(Ω) + ‖v − 1 δ ‖L∞(Ω) + ‖w − 1 δ ‖L∞(Ω) ≤ Ce−κt for all t > 0. (5.19) EJDE-2022/58 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 15 Proof. Applying the L’Hôpital theorem and letting H(s) := s − ln s for s > 0 we obtain lim s→1 H(s)−H(1) (s− 1)2 = lim s→1 H ′(s) 2(s− 1) = 1 2 . (5.20) Therefore, using (5.20) and Lemma 5.4, there exists t1 > 0 such that 1 4 ∫ Ω (u− 1)2 ≤ A(t) = ∫ Ω (u− 1− lnu) ≤ ∫ Ω (u− 1)2 for all t > t1. (5.21) Hence, by the definitions of E(t) and F (t), there exists C1 > 0 such that E(t) ≤ C1F (t) for all t > t1. (5.22) Plugging (5.22) into (5.7) we obtain d dt E(t) ≤ −εF (t) ≤ − ε C1 E(t) for all t > t1, (5.23) which implies there exist C2 > 0 and l > 0 such that E(t) ≤ C2e −lt for all t > t1. (5.24) Hence, by (5.21) we can find some C3 > 0 such that∫ Ω (u− 1)2 + ∫ Ω ( v − 1 δ )2 + ∫ Ω ( w − 1 δ )2 ≤ C3E(t) ≤ C2C3e −lt (5.25) for all t > t1. 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Jiang, S. Zheng; Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source, J. Math. Anal. Appl. 430 (2015), 585–591. [44] J. Zhao; Convergence rate of a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Math. Anal. Appl. 478 (2019), 625–633. Xiaobing Ye School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China Email address: xiaobingye1130@126.com Liangchen Wang School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China Email address: wanglc@cqupt.edu.cn 1. Introduction 2. Preliminaries 3. A priori estimate for n=2 4. A priori estimate for n=1 and n3 5. Asymptotic behavior of solutions Acknowledgments References