BOUNDEDNESS AND LARGE TIME BEHAVIOR OF AN ATTRACTION-REPULSION CHEMOTAXIS MODEL WITH LOGISTIC SOURCE

In this paper, we study an attraction-repulsion Keller-Segel chemotaxis model with logistic source  ut = ∆u− χ∇ · (u∇v) + ξ∇ · (u∇w) + f(u), x ∈ Ω, t > 0, vt = ∆v + αu− βv, x ∈ Ω, t > 0, wt = ∆w + γu− δw, x ∈ Ω, t > 0, (∗) in a smooth bounded domain Ω ⊂ Rn(n ≥ 1), with homogeneous Neumann boundary conditions and nonnegative initial data (u0, v0, w0) satisfying suitable regularity, where χ ≥ 0, ξ ≥ 0, α, β, γ, δ > 0 and f is a smooth growth source satisfying f(0) ≥ 0 and f(u) ≤ a− bu, u ≥ 0, with some a ≥ 0, b > 0, θ ≥ 1. When χα = ξγ (i.e. repulsion cancels attraction), the boundedness of classical solution of system (∗) is established if the dampening parameter θ and the space dimension n satisfy  θ > max{1, 3− 6 n }, when 1 ≤ n ≤ 5, θ ≥ 2, when 6 ≤ n ≤ 9, θ > 1 + 2(n−4) n+2 , when n ≥ 10. Furthermore, when f(u) = μu(1−u) and repulsion cancels attraction, by constructing appropriate Lyapunov functional, we show that if μ > χα(β−δ) 8δβ2 , the solution (u, v, w) exponentially stabilizes to the constant stationary solution (1, α β , γ δ ) in the case of 1 ≤ n ≤ 9. Our results implies that when repulsion cancels attraction the logistic source play an important role on the solution behavior of the attraction-repulsion chemotaxis system.

When χα = ξγ (i.e.repulsion cancels attraction), the boundedness of classical solution of system ( * ) is established if the dampening parameter θ and the space dimension n satisfy n+2 , when n ≥ 10.
Furthermore, when f (u) = µu(1 − u) and repulsion cancels attraction, by constructing appropriate Lyapunov functional, we show that if µ > 1. Introduction and main results.We consider the initial-boundary value problem of the attraction-repulsion Keller-Segel model with logistic source 856 SHIJIE SHI, ZHENGRONG LIU AND HAI-YANG JIN x ∈ Ω, t > 0, w t = ∆w + γu − δw, x ∈ Ω, t > 0, ∂u ∂ν = ∂v ∂ν = ∂w ∂ν = 0, x ∈ ∂Ω, t > 0, u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), w(x, 0) = w 0 (x), x ∈ Ω, (1) in a bounded domain Ω ⊂ R n (n ≥ 1) with smooth boundary, where u denotes the cell density, v represents the concentration of chemoattractant and w accounts for the concentration of chemorepellent.The positive parameters χ and ξ are called the chemotactic coefficients, and α, β, γ, δ are chemical production and degradation rates.Here ≥ 0 is a nonnegative scaling constant.The logistic source f (u) describes the cell proliferation and death.The system (1) with two chemical signals was proposed by Luca et al. in [29] to examine whether the combined chemicals (chemoattractant and chemorepellent) may interact to produce aggregation of microglia, which also was proposed in [34] to describe the quorum effect in the chemotactic process.
The attraction-repulsion Keller-Segel model ( 1) can be viewed as a generalization of the following Keller-Segel chemotaxis model whose solution behavior has been extensively studied in the past four decades in various perspectives (see the survey articles [3,12] and references therein).When f (u) = 0, the system (2) was called minimal Keller-Segel chemotaxis model.A striking feature of the minimal model ( 2) is the blow-up of solutions in two or higher dimensions [7,11,14,31,46,48], which limits the application of the model to explain the aggregation phenomena observed in experiment.To prevent blowup of solutions, differential mechanisms have been proposed (see [8,9,10] and references therein).The chemotaxis model (2) with non-trivial logistic source has been studied [20,33,44,45], and the result showed that this mechanism can enforce the boundedness of solutions so that blow-up is inhibited.More precisely, when = 1 and f (u) = µu(1 − u) with µ > 0, the system (2) has a uniform-in-time bounded classical solution in two dimensional bounded domain [33] and a weak solution in three dimensional convex domain [20].In higher dimensions (n ≥ 3), the existence of the classical solution with uniform-in-time bound was also established if the logistic source f (u) ≤ a − bu 2 for some a ≥ 0 and b > 0 with = 0 in [41] and = 1 in [45], respectively.Moreover, for more general logistic Winkler [44] also constructed some global 'very weak' solutions to the system (2), which however whether or not there exist global classical solutions was left as an interesting and challenging open problem.We should point out that from the point of mathematical intuition, the logistic damping has a balance effect on the formation of possible singularity.However, the blow-up is possible in a slightly modified version of system (2) with logistic source in [47].
Although the attraction-repulsion Keller-Segel model ( 1) is a direct generalization of the Keller-Segel system (2), the mathematical analysis on the boundedness and blow-up of solutions confront great challenges due to the complicated interactions between three species u, v and w, and the difficulty of constructing a Lyapunov functional.For the attraction-repulsion chemotaxis system (1) without growth source (i.e.f (u) = 0), the global existence of classical solutions, non-trivial stationary state, asymptotic behavior and pattern formation of the system (1) with Neumann boundary conditions were studied [16,27,28] with = 1 in one dimension.By introducing a novel transformation s = ξw−χv, Tao and Wang [40] studied the global solvability, boundedness, blow-up, existence of steady states in a bounded domain with homogeneous Neumann boundary conditions in higher dimensions (n ≥ 2) and first found that the solution behavior of (1) essentially depends on the sign of parameter Θ := χα − ξγ, which interprets the competing effect between attraction and repulsion as follows (see also [18]): The recent progress on the solution behavior of the system (1) without logistic source can be found in [5,15,18,21,23,24,25,26] for bounded domain and in [17,36] for whole space.
However, to our knowledge, there are few results on the attraction-repulsion chemotaxis model (1) with non-trivial logistic source.When = 0, based on the L p energy estimates and Moser iteration, Zhang and Li [50] established the existence of global classical solutions of the system (1) with f (u) = µu(1 − u) if one of the following conditions holds: (1)χα − ξγ ≤ µ; (2)n ≤ 2; (3) n−2 n (χα − ξγ) < µ and n ≥ 3.Moreover, they also showed that the global classical solution will converge to the unique constant steady (1, α β , γ δ ) if µ > 2χα.The global existence of classical solution also was studied by Li and Xiang [22] for the logistic source f (u) ≤ a − bu θ with some a ≥ 0 and b > 0. Specially, Li and Xiang [22] proved that the classical solution will exist for all n ≥ 2 in the case θ > 2.Moreover, when χα = ξγ (i.e.repulsion cancels attraction), they proved the classical solution with uniform-in-time bound exists if θ > 1 2 ( √ n 2 + 4n − n + 2).However if = 1, they only established the boundedness of solutions when n = 1 with θ ≥ 1 or n = 2 with θ ≥ 2 [22].According to the above results [22,50], the blow-up of solution is prevented when = 0 and the power parameter θ is large in higher dimensions.Whileas for the full parabolic attraction-repulsion chemotaxis model (1) (i.e.= 1 ) with logistic source, the global existence of classical solutions was only obtained when the space dimension n ≤ 2, which was left as an open problem for the higher dimensions (n ≥ 3).Moreover, as far as we know, there is not any result on the large time behavior of solutions for the full attraction-repulsion chemotaxis model (1) with logistic source.
The main purpose of this paper is to investigate the effect of the logistic source on the solution behavior of the following full attraction-repulsion chemotaxis model where the kinetic term f satisfies f (0) ≥ 0 and To study the dampening effect of the logistic source, we focus our study on the case of χα = ξγ.We find the lower bound of the power parameter θ depending on n to guarantee the existence of global bounded solutions.Moreover, by constructing Lyapunov functional, we also study the large time behavior of the solution for the system (3) with logistic source f (u) = µu(1 − u).Our first main results are stated as follows.
We have several remarks concerning the boundedness results in Theorem 1.1.

Remarks.
• In Theorem 1.1, we give the lower bound of θ (which may be not optimal) to prevent the blow-up of solutions.For the prototype logistic source f (u) = µu(1 − u) (i.e.θ = 2), when 1 ≤ n ≤ 9 we can obtain the existence of the classical solution with uniform-in-time bound directly from Theorem 1.1 by noting θ = 2 > max{1, 3 − 6 n } for 1 ≤ n ≤ 5. • For the cubic growth source f (u) = u(u − c)(d − u) with c, d > 0 as originally introduced by Mimura and Tsujikawa in [27], it satisfies the condition (4) with θ = 3 and some a, b > 0. For any n, one can easily check that 1 + 2(n−4) n+2 < 3, hence the system (3) with cubic growth source has a unique global classical solution satisfying (5) for all biologically meaningful parameters.
is a bounded domain with smooth boundary, if ξγ = χα and f (u) = µu(1 − u), then for any µ > 0, system (3) has a non-negative solution (u, v, w) then the classical solution (u, v, w) of system (3) satisfies where c and λ are positive constants independent of t.
Remark 1.When β = δ, from (6) we know that the solution will converge to the unique non-trivial constant state (1, α β , γ δ ) for any µ > 0. We conjecture that the same asymptotic stability results hold for β = δ, which however is left as an open problem due to the technical reasons.
Outline of main approaches: Inspired by the ideas in [3,49], we first establish the boundedness criterion for the solution of the system (3).More precisely, by combining the semigroup theory, Gagliardo-Nirenberg inequality, the L p energy estimate and Moser-Alikakos iteration, we show that when repulsion cancels attraction(i.e.ξγ = χα), the uniform boundedness of L r -norm of u(•, t) for some r > n 4 can rule out the blow-up of solutions for the system (3) (see Lemma 2.3).With the boundedness criterion established in Lemma 2.3 in hand, we use the coupled energy estimate as in [38] together with the method of heat Neumann semigroup to study the boundedness of the solution to the system (3) in higher dimensions.The relations of dampening parameter θ and the space dimension n are found to ensure the boundedness of solution for system (3).Specially, our results show that when 1 ≤ n ≤ 9 and repulsion cancels attraction the global classical solution with uniform-in-time bound exist for the prototype logistic source f (u) = µu(1 − u) with µ > 0, which is substantially different from the classical chemotaxis model with logistic source.Moreover, based on the ideas in [2,39], we show that under , the functional F(t) defined as for all t > 0, act as a Lyapunov functional for the system (3) with ξγ = χα and appropriate choices of the positive constant γ 1 and γ 2 , which will be used to study the large time behavior of solutions.The remainder of the paper is organized as follows.In section 2, we establish the boundedness criterion for the solution of the system (3) in the case of repulsion cancels attraction.With the aid of the boundedness criterion in Lemma 2.3, we show the existence of globally boundedness classical solutions to the system (3) for arbitrary dimension in section 3.In section 4, The global dynamic of solutions to the system (3) with f (u) = µu(1 − u) will be studied.

Local existence and preliminaries.
In what follows, without confusion, we shall abbreviate Ω f dx as Ω f for simplicity.Moreover, we shall use c i or C i (i = 1, 2, 3, • • • ) to denote generic constants which may vary in the context.The existence of local solutions of the problem (3) can be proved by the fixed point theorem and the maximum principle along the same line shown in [22,40,42].
Furthermore, the L 1 -norm of u is uniformly bounded, i.e. there exists a constant Proof.The proof of local-in-time existence of classical solutions to the system (3) is quite standard, see [22,40,42] for details.Since f (0) ≥ 0, using the maximum principle we can derive u, v, w are nonnegative, as shown in [22,40].Integrating the first equation of the system (3) and using (4), one can derive that The L 1 -norm of u is uniformly bounded by using the standard Grönwall's inequality.
The following Gagliardo-Nirenberg inequality will be frequently used later.

Boundedness criterion.
Inspiring by the works in [3,49], we will show the boundedness criterion of solutions for the system (3) as follows.
Lemma 2.3 (Criterion for boundedness).Suppose the conditions in Lemma 2.1 hold.Let (u, v, w) be the solution of system (3) defined on its maximal existence time interval [0, T max ).If ξγ = χα and there exists a constant M > 0 such that for all p > n 4 u(•, t) L p ≤ M, for all t ∈ (0, T max ), then one can obtain a constant C > 0 independent of t such that Next, we will prove Lemma 2.3.Before that we first present some basic estimates of solutions.Letting s := ξw − χv and noting ξγ = χα, then the system (3) can be transformed into (8) Then for the transformed system (8), we have the following results: Lemma 2.4.Let (u, s, v) be a solution of (8) defined on its maximal existence interval [0, T max ).Suppose p ≥ 1 and for i = 1, 2 If there exists a constant M > 0 such that for some T ∈ (0, T max ), it holds that u(•, t) L p ≤ M for all t ∈ (0, T ), (10) then for all t ∈ (0, T ), one has Proof.Suppose that there exists a constant K > 0 such that Then using the Hölder's inequality and ( 12), for all r ∈ [1, p] one has Hence, we may assume that q i > p(i = 1, 2) in the proof of this lemma for convenience.
Using the variation of constants representation of v, then from the third equation of system (8) we have which together with (10) gives ) )dτ ) )dσ, (13) and ) )dτ ) )dσ, (14) where the smoothing properties of (e τ ∆ ) τ ≥0 have been used (see [6,Lemma 3.3] or [46,Lemma 1.3]).Thanks to the conditions (9) of q 1 and q 2 , we know that c 5 : ) )dσ < ∞ and c 6 : ) )dσ < ∞.Hence from ( 13) and ( 14), we can derive that and thereby prove (11).Hence the proof of this lemma is completed.Lemma 2.5.Let q 2 ≥ 1 and If for all M > 0 there exists a constant C s (q 2 , q 3 , M ) > 0 such that for some T ∈ (0, T max ), we have Proof.Without loss of generality, we assume that q 3 > q 2 for simplification.Using the variation of constants representation of s, from the second equation of the system (8), we can derive that Then using again the smoothing properties of (e τ ∆ ) τ ≥0 and noting ( 16), one has dτ. ( Since q 3 satisfies (15), we have Then (17) follows from (18).Hence the proof of this Lemma 2.5 is completed.
Next, we will give the proof of Lemma 2.3.
Proof of Lemma 2.3.If p > n 3 , from (9) we can choose q 2 > n such that v L q 2 is uniformly bounded.Then from Lemma 2.5, one has which together with the well-known Moser-Alikakos iteration technique (cf.[1,40]) gives the L ∞ -bound of u.Since u(•, t) L 1 (i.e.p = 1) is uniformly bounded (see Lemma 2.1), when n ≤ 2, we have (19) and hence the L ∞ -bound of u.
Next, we will consider the case n 4 < p ≤ n 3 with n ≥ 3, which gives n Hence from ( 17) and ( 15), we can obtain Multiplying the first equation of the system (8) by u k−1 (k > n 3 ≥ 1) and integrating by parts over Ω, then using Young's inequality, we end up with 1 k Using Hölder's inequality, (20) and Gagliardo-Nirenberg inequality (see Lemma 2.2), one has where (22) together with Young's inequality, we can derive that Using Gagliardo-Nirenberg inequality again, we can find Substituting ( 23) and ( 24) into (21) and choosing ε 1 and ε 2 small enough, we have which together with Grönwall's inequality yields u(•, t) L k ≤ c 13 , for all k > n 3 .
Then using Lemma 2.4 again, we can derive that (19) holds for all n ≥ 3, which together with the well-known Moser iteration gives Moreover, through a straightforward reasoning involving standard parabolic regularity theory ( [19]), we have which combines with (25) gives (7).Then the proof of Lemma 2.3 is completed.
3. Proof of Theorem 1.1.In this section, we are devoted to proving Theorem 1.1 based on the boundedness criterion of solutions for system (3) (see Lemma 2.3).From Lemma 2.1, one has u L 1 ≤ c 1 for all θ ≥ 1.Then using the boundedness criterion established in Lemma 2.3, we know that the system (3) has a global classical solution with uniform-in-time bound for n ≤ 3. Hence to completed the proof of Theorem 1.1, we only need to consider the case n ≥ 4.
3.1.Parameter conditions.Before proving our main results in Theorem 1.1, we first introduce some notations that will be used later. and , for i = 1, 2, as well as , for i = 1, 2, Next, we will show some results on the parameters which will be used in the proof of the boundedness of global solutions based on some ideas in [4,51].
Next, we show that the regularity can be improved if θ ≥ 2. Multiplying the third equation of ( 8) by 2v and −2∆v respectively, and then integrating them with respect to x, we end up with and The combination of ( 45) and (46) gives From the first equation of the system (3), we have Combining ( 47) and ( 48) and using the facts u L 1 ≤ c 4 and θ ≥ 2, one has which together with Grönwall's inequality gives Hence using the Sobolev inequality, from (49) we have for all From ( 50), using the similar argument as (18) one has ∇s L r ≤ C with r ∈ [1, nq n−q ), which together with ( 49) and ( 50) gives (44) with q ∈ [1, 2n n−2 ), r = 2 and r ∈ [1, nq n−q ).Then the proof of this lemma is completed.
With the results obtained in Lemma 3.1 and Lemma 3.2 in hand, we will establish the following key lemma in the proof of Theorem 1.1.Lemma 3.3.Assume that the conditions in Lemma 3.2 hold.If (30) holds, then for sufficiently large k ∈ ( n 4 , ∞), there exists a constant c 1 > 0 independent of t such that the solution of the system (8) satisfies Proof.Using a similar argument as in proof of ( 21), one has Differentiating the third equation of ( 8) and then multiplying it with ∇v, applying the identity We multiply (53) with 2κ 1 |∇v| 2(κ1−1) (κ 1 > 1) and integrate it to get Noting , then the first term of (54) can be rewritten as Using the trace inequality [37, Remark 52.9] that for any ε > 0: and the inequality ∂|∇v| 2 ∂ν ≤ 2m|∇v| 2 on ∂Ω for some constants m > 0 (cf.[30, Lemma 4.2] and [13]), we can estimate the first term on the right hand of (55) as follows Substituting ( 55), ( 56) into (54), we have Since κ 1 > 1 and ∇v L 1 ≤ C, then using the Gagliardo-Nirenberg inequality we can find θ = Moreover, we can estimate the first item on the right hand side of (57) as follow Combining ( 58), ( 59) with (57), one has Similarly, we differentiate the second equation of ( 8) and multiply it with ∇s to have 1 2 Multiplying the above identity with 2κ 2 |∇s| 2(κ2−1) and integrating it with respect to x, we end up with Using Hölder's inequality and Young's inequality, we have Similarly, using Hölder's inequality and Young's inequality together with Gagliardo-Nirenberg inequality, we can find a λ > q 2 satisfying 1 λ + 1 λ = 1 and θ = where the last identity holds by noting that θ(n+q) Combining Lemma 3.1 and Lemma 3.2, using Gagliardo-Nirenberg inequality and Young's inequality, we have for i = 1, 2 and for i = 3, 4 Substituting ( 65) and (66) into (64), one has Using Young's inequality, we have Substituting ( 68) into (67), we have which together with Grönwall's inequality gives (51).

Proof of Theorem 1.1.
Proof of Theorem 1.1.Since the L 1 -norm of u is uniformly bounded for all θ ≥ 1 (see Lemma 2.1) ,then we can apply Lemma 2.3 with p = 1 to find a constant c 1 > 0 independent of t such that u(•, t) L ∞ + (v, w)(•, t) W 1,∞ ≤ c 1 for all t ∈ (0, T max ) when n ≤ 3.This along with Lemma 2.1 proves Theorem 1.1 in the case of n ≤ 3. When n ≥ 4, Theorem 1.1 is a direct result from the combination of Lemma 2.3, Lemma 3.3 and Lemma 2.1.

4.
Proof of Theorem 1.2.In this section, we are devoting to prove Theorem 1.2.When f (u) = µu(1 − u), noting ξγ = χα and using the transformation s = ξw − χv, the system (3) becomes (69) The existence of global classical solution can be obtained directly from Theorem 1.1 by noting θ = 2 in this case.Hence in the following subsections, we will focus on studying the large time behavior of solutions to complete the proof of Theorem 1.2.4.1.Construction of an energy functional.Next, based on the idea in [39], we will construct a Lyapunov functional by studying the time evolution of each of the integrals therein.
for all t > 0.
Proof.Motivated by the ideas from [39], we multiply the first equation of the system (69) by 1 Using Young's inequality, one has which together with (71) gives (70).
Lemma 4.2.Suppose (u, s, v) is a global classical solution of the system (69).Then it has Proof.Testing the second equation of (69) by s − χα(β−δ) δβ , we have 1 2 which yields (72), where we have used Lemma 4.3.Assume that (u, s, v) is a global classical solution of the system (69).
Then we have Proof.We multiply the third equation of the system (69) by v − α β , and then use Young's inequality to derive that which yields (73).
Lemma 4.4.Let (u, s, v) be the global classical solution of the system (69).Suppose then there exist positive constants γ 1 , γ 2 such that for all t > 0, the function , then we can find a positive constant γ 2 satisfying 2βµ The combination of Lemma 4.1,4.2 and 4.3 gives ) (76) Proof.From (82), we can get a t 0 > 0 such that for all t > t 0 u − 1 L ∞ < 1 2 , which gives (87) immediately.From Lemma 4.5, we can get two constants c 1 , c 2 > 0 such that Hence, using (87) and (89) and choosing , and suppose (u, s, v) is the global classical solution of the system (69).Then there exists a constant c > 0 such that for all t > 0 u − 1 where γ 3 is given in Lemma 4.9.
Next, we will use the interpolation inequality to derive the uniform exponential stabilization property., and let (u, s, v) be the global classical solution of the system (69).Then there exist two constants c > 0 and γ 4 > 0 such that for all t > 0 u(•, t) and Proof.Based on (79), one can readily get a constant c 3 > 0 (e.g.see [39,Lemma 3.14]) such that u(•, t) W 1,∞ ≤ c 3 , for all t > 1.
Proof of Theorem 1.2.The existence of global classical solution is a direct result of Theorem 1.1 by noting θ = 2 if f (u) = µu(1 − u).From Lemma 4.11, we only need to prove the convergence rate of w to complete the proof of Theorem 1.2.In fact, since s = ξw − χv and χα = ξγ, then using (94), we have and hence which together with the convergence rate of u, v in Lemma 4.11 finish the proof of Theorem 1.2.

Lemma 4 . 1 .
Let (u, s, v) be the global classical solution of the system (69).Then we have d dt Ω