BOUNDEDNESS IN A THREE-DIMENSIONAL ATTRACTION-REPULSION CHEMOTAXIS SYSTEM WITH NONLINEAR DIFFUSION AND LOGISTIC SOURCE

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1 Electronic Journal of Differential Equations, Vol. 016 (016, No. 176, pp ISSN: URL: or BOUNDEDNESS IN A THREE-DIMENSIONAL ATTRACTION-REPULSION CHEMOTAXIS SYSTEM WITH NONLINEAR DIFFUSION AND LOGISTIC SOURCE YILONG WANG Abstract. This article concerns the attraction-repulsion chemotaxis system with nonlinear iffusion an logistic source, u t = ((u + 1 m 1 u (χu v + (ξu w + ru µu η, x, t > 0, v t = v + αu βv, x, t > 0, w t = w + γu δw, x, t > 0 uner Neumann bounary conitions in a boune omain R 3 with smooth bounary. We show that if the iffusion is strong enough or the logistic ampening is sufficiently powerful, then the corresponing system possesses a global boune classical solution for any sufficiently regular initial ata. Moreover, it is prove that if r = 0, β > ( an δ > 1 for the latter case, ( 1 then u(, t 0, v(, t 0 an w(, t 0 in L ( as t. 1. Introuction We consier the attraction-repulsion chemotaxis system with nonlinear iffusion an logistic source, u t = ((u + 1 m 1 u (χu v + (ξu w + ru µu η, x, t > 0, v t = v + αu βv, x, t > 0, w t = w + γu δw, x, t > 0, u = v = w = 0, x, t > 0, u(x, 0 = u 0 (x, v(x, 0 = v 0 (x, w(x, 0 = w 0 (x, x, (1.1 in a boune omain R 3 with smooth bounary, where enotes the outwar normal erivative on, m 1, r 0, µ > 0, η > 1, χ > 0, ξ > 0, α > 0, β > 0, γ > 0 an δ > 0 are prescribe parameters. Here u(x, t enotes the cell ensity, v(x, t an w(x, t represent the chemoattractant concentration an the chemorepellent concentration, respectively. In (1.1 we assume that cell kinetics 010 Mathematics Subject Classification. 35K55, 35Q35, 35Q9, 9C17. Key wors an phrases. Chemotaxis; attraction-repulsion; bouneness; nonlinear iffusion; logistic source. c 016 Texas State University. Submitte February 8, 016. Publishe July 6,

2 Y. WANG EJDE-016/176 follows a logistic-type law etermine by parameters r, µ an η. The first crossiffusive term an the secon in the first equation reflect the attractive migration an repulsive movement of cells, respectively. The secon an the thir equations imply that the attractive an repulsive signals are prouce by cells themselves. System (1.1 is a generalize version of the classical Keller-Segel moel [9] an can escribe the aggregation of microglia in Alzheimer s isease ue to the interaction of chemoattractant an chemorepellent [11]. System (1.1 can also aress the quorum effect in the chemotactic process [18]. The mathematical stuies on system (1.1 are much harer than the classical Keller-Segel system ue to lacking a useful Lyapunov functional. To motivate our stuy, we recall some relate works on system (1.1. In the case m = 1, r = 0 an µ = 0, there have been a series of evelopment: Wang et al. [16, 6] prove that system (1.1 possesses a unique global boune solution when n = 1; when n = an the repulsion prevails over the attraction in the sense that ξγ χα > 0, Jin [4] an Tao et al. [15] inepenently prove that system (1.1 amits a unique global boune solution; however, when n = 3 an ξγ χα > 0, the global existence of classical solutions for (1.1 is still open; Jin [4] only prove that system (1.1 possesses a global weak solution; when n =, 3 an the repulsion cancels the attraction (i.e. ξγ = χα, the authors in [13, 5] prove that system (1.1 possesses a unique global boune classical solution an the large time behavior of solutions is consiere for the boune omain an the whole space. The authors in [14] investigate the pattern formation of (1.1 analytically an numerically for n = 1. For the parabolic-elliptic case, the global solvability, critical mass phenomenon, blow-up, an large time behavior were investigate for the boune omain an the whole space (see [1, 1,, 1, 7]. In [6, 7], the global solvability an the uniform bouneness were consiere for the attractionrepulsion chemotaxis system with nonlinear iffusion. In the case m = 1, r > 0 an µ > 0, there are only few results: Li et al. [17, 10] prove that (1.1 with η 1 for n = 1 or η for n = amits a unique global boune solution. For n 3, however, the global existence of classical solutions of (1.1 is still open. In [8], the global solvability, bouneness an large time behavior were only investigate for a quasilinear attraction-repulsion chemotaxis moel with logitic source for parabolic-elliptic type in a boune omain. To the best of our knowlege, there is no rigorous result on the quasilinear attraction-repulsion chemotaxis moel with logistic source for parabolic-parabolic type except [10], where the author only consiere two-imensional case an prove that when m > 1 or η, system (1.1 possesses a global boune classical solution. Thus the goal of this paper is to explore the interactions between the nonlinear iffusion an logistic source on the solutions of system (1.1 for threeimensional case an partially answer the above open problem. Throughout this paper, we assume that the initial ata satisfy u 0 W 1, (, u 0 > 0 on, v 0 W 1, (, v 0 0 on, w 0 W 1, (, w 0 0 on. (1. We now state the main results of this article.

3 EJDE-016/176 BOUNDEDNESS IN A CHEMOTAXIS SYSTEM 3 Theorem 1.1. Let R 3 be a boune omain with smooth bounary an initial ata (u 0, v 0, w 0 satisfy (1.. Suppose that m 1, r 0, µ > 0, η > 1, χ > 0, ξ > 0, α > 0, β > 0, γ > 0 an δ > 0. If one of the following two cases hols: (i η an µ > max { 1 + 4χ + 11α + χ (9 + α + γ, 1 + 4ξ + 11γ + ξ (9 + α + γ } ; (ii η (1, an m > 4/3, then there exists a triplet (u, v, w belonging to C 0( [0, C,1 ( (0, which solves system (1.1 classically. Moreover, u, v an w are boune in (0, in the sense that u(, t L ( + v(, t W 1, ( + w(, t W 1, ( M t (0,, (1.3 where M > 0 is a constant inepenent of t. Remark 1.. Theorem 1.1 answers the open problem in [10, Remark 1.4] an [17, Remark 1.3] for three-imensional case. Remark 1.3. For the case n 4, if m > n, then Theorem 1.1 hols. This can be prove by the same arguments of case (ii in Theorem 1.1. Next, we consier the large time behavior of solutions to (1.1 for a special case r = 0, η, µ > max { 1 + 4χ + 11α + χ (9 + α + γ, 1 + 4ξ + 11γ + ξ (9 + α + γ }, β > 1 ( an δ > 1 ( by using the ieas in [4]. Here r = 0 means that either cells are a prior unable to reprouce themselves, or the consiere time scales are much smaller than those of cell proliferation (see [, 8] for etails. Theorem 1.4. Let R 3 be a boune omain with smooth bounary an initial ata (u 0, v 0, w 0 satisfy (1.. Suppose that m 1, χ > 0, ξ > 0, α > 0, β > 0, γ > 0 an δ > 0. If r = 0, η, µ > max { 1 + 4χ + 11α + χ (9 + α + γ, 1 + 4ξ + 11γ + ξ (9 + α + γ } 1, β > ( an δ > 1 (, then the global solution (u, v, w of system (1.1 satisfies u(, t L ( 0, v(, t L ( 0, w(, t L ( 0 as t. (1.4 This article is organize as follows. In Section, we state the existence of local solutions to (1.1 an some preliminary inequalities. In Section 3, we give some funamental estimates for solutions to (1.1 an prove Theorem 1.1. In Section 4, we consier the large time behavior of solutions to (1.1 for a special case an prove Theorem Preliminaries In this section, we first state the local existence of solutions to the system (1.1 an then present some preliminary inequalities. By irectly aapting the reasoning in [5, Lemma.1], we can erive the following lemma on local existence of classical solutions to (1.1. Lemma.1. Let R 3 be a boune omain with smooth bounary an initial ata (u 0, v 0, w 0 satisfy (1.. Suppose that m 1, r 0, µ > 0, η > 1, χ > 0, ξ > 0, α > 0, β > 0, γ > 0 an δ > 0. Then there exist T max (0, ] an a triplet (u, v, w of nonnegative functions from C 0( [0, Tmax C,1( (0, Tmax

4 4 Y. WANG EJDE-016/176 solving (1.1 classically in (0, T max. These functions satisfy u > 0, v > 0 an w > 0 in (0, T max an, moreover, we have either T max =, or u(, t L ( + v(, t W 1, ( + w(, t W 1, ( as t T max. Let us state the following basic properties associate with u. (.1 Lemma.. There exists some constants M 1 > 0 an C > 0 such that u(x, tx M 1 t (0, T max, (. where t+τ t u η (x, t x s C t (0, T max τ, (.3 τ := min{1, 1 T max}. (.4 Proof. Integrating the first equation of (1.1, we obtain ux + µ u η x = r ux t (0, T max. (.5 t If r = 0, we have t ux = µ uη x for all t (0, T max. The nonnegativity of u implies t ux < 0 for all t (0, T max. Thus, we have ux u 0x for all t (0, T max. If r > 0, then by using Young s inequality we have This implies t ru µu η + µ 1 (r η. ux + r ux µ 1 η (r t (0, Tmax. Thus, { 1 η ux M 1 := max u 0 x, µ (r } r for all t (0, T max. Therefore, (. hols. By a time integration of (.5, we erive that t+τ u(x, t + τx + µ u(x, tx + r M 1 + rτm 1 which implies that (.3 hols. t t+τ t u η (x, s x s u(x, s x s t (0, T max τ, The following auxiliary lemma on a bouneness property in an ODI will be use in our analysis. It is a straightforwar generalization of a particular case τ = 1 which has been prove in [0, Lemma 3.4]. Lemma.3. Let T > 0, τ (0, T, a > 0 an b > 0, an suppose that y : [0, T [0, is absolutely continuous an such that y (t + ay(t h(t for a.e. t (0, T,

5 EJDE-016/176 BOUNDEDNESS IN A CHEMOTAXIS SYSTEM 5 where h L 1 loc ([0, T is nonnegative an satisfies t+τ t h(ss b t [0, T τ. Then b y(t max{y(0 + b, + b} for all t (0, T. aτ As an application of Lemma. an Lemma.3, we can establish the following estimates associate with v an w. Lemma.4. Let η, then there exists a constant C > 0 such that v(x, tx C t (0, T max, (.6 v x C t (0, T max, (.7 w(x, tx C t (0, T max, (.8 w x C t (0, T max. (.9 Proof. We first prove the estimates associate with v. Integrating the secon equation in (1.1 over an then using (., we obtain that vx = β vx + α ux β vx + αm 1 t (0, T max, t which implies (.6 by an ODE comparison. We multiply the secon equation in (1.1 by v an integrate by parts over to erive that 1 v x + v x + β v x t = α u v (.10 1 v x + α u x t (0, T max. Letting y(t := v x an h(t := α u x for t (0, T max, we have y (t + βy(t h(t t (0, T max. Since (.3 hols for all η, we have t+τ h(ss C t 1 for any t (0, T max τ with some constant C 1 > 0 an τ = min { 1, 1 T max}. Thus, accoring to Lemma.3, we establish (.7. Taking a similar proceure for w, we can also establish (.8 an ( Global existence an bouneness of classical solutions In this section, we investigate the global existence an bouneness of classical solutions to system (1.1. In Section 3.1, we establish the global existence for the strong logistic source case (i.e., η an µ > max { 1+4χ +11α +χ (9+α +γ, 1+4ξ +11γ +ξ (9+α +γ }. In Section 3., we establish it for the sufficiently strong iffusion case (i.e., m > 4 3. The key iea is to euce a uniform estimate for u(, t L p (+ v(, t L q (+ w(, t L q ( for sufficiently large p an q.

6 6 Y. WANG EJDE-016/176 A priori estimates for the strong logistic source case. Inspire by Tao an Winkler [4], we first establish an estimate for an appropriate linear combination of the functions (u + 1m+1 x, v 4 x, w 4 x, u v x an u w x. As a beginning, let us erive ifferential inequalities for (u + 1 m+1 x, v 4 x an w 4 x. Lemma 3.1. Let η an the other assumptions in Lemma.1 hol. Then we have (u + 1 m+1 m(m + 1 x + (u + 1 (m 1 u x t χ m(m + 1 u v x + ξ m(m + 1 u w x (3.1 + r(m + 1 (u + 1 m+1 µ(m + 1 x η (u + 1 m+η x + µ(m + 1 (u + 1 m x t (0, T max. Proof. Multiplying the first equation in (1.1 by (u + 1 m an integrating by parts over, we obtain 1 (u + 1 m+1 x m + 1 t = m (u + 1 (m 1 u x + χm u(u + 1 m 1 v ux ξm u(u + 1 m 1 w ux + r u(u + 1 m x µ u η (u + 1 m x m (u + 1 (m 1 u x + χ m u v x + ξ m u w x + r (u + 1 m+1 x µ u η (u + 1 m x for all t (0, T max. Due to the nonnegativity of u, we have (u + 1 η η (u η + 1 an then obtain u η 1 (u + 1 η 1. Inserting it into the above inequality, we η obtain the esire result (3.1. Lemma 3.. Let the assumptions in Lemma 3.1 hol. Then we have v 4 x + 3 v x t v v x + 11α u v x, t (0, T max, w 4 x + 3 w x t w w x + 11γ u w x, t (0, T max. (3. (3.3 Proof. Differentiating the secon equation in (1.1 an then multiplying v, we obtain that ( v t = v D v β v + α u v t (0, T max, (3.4

7 EJDE-016/176 BOUNDEDNESS IN A CHEMOTAXIS SYSTEM 7 where we have use the ientity v = v v + D v. Testing (3.4 against v yiels v 4 x = v v x 4 D v v x t 4β v 4 x + 4α ( u v v x = v v x v x (3.5 4 D v v x 4β v 4 x 4α u v v x 4α u v v x for all t (0, T max. Since v 3 D v, by Young s inequality we have 4α u v v x 4 3α u D v v x 4 D v v x + 3α u v x an (3.6 4α u v v x 1 v x + 8α u v x (3.7 for all t (0, T max. Inserting (3.6 an (3.7 into (3.5, we infer that v 4 x v v x 3 v x t + 11α u v x 4β v 4 x v v x 3 v x + 11α u v x t (0, T max. (3.8 Thus, we obtain (3.. Taking a same proceure for the thir equation in (1.1, we can erive (3.3. To eal with the integrals u v x an u w x on the right-han sie of (3.1-(3.3, we shall establish the following ifferential inequality relate to them. Lemma 3.3. Let the assumptions in Lemma 3.1 hol. Then we have { } u v x + u w x t + (µ 1 4χ u v x + (µ 1 4ξ u w x 1 v x + 1 w x

8 8 Y. WANG EJDE-016/176 + (8 + α + γ (u + 1 (m 1 u x + (r β u v x + + (r δ u w x + for all t (0, T max, where C is a positive constant. u v x u w x + C (3.9 Proof. By employing the ientity v v = 1 v D v, we take a straightforwar computation on t u v x to obtain that u v x t = v { ((u + 1 m 1 u χu v + ξu w + ru µu η} x + u v ( v βv + αux = (u + 1 m 1 u v x + χ u v v x (3.10 ξ u w v x µ u η v x + u( 1 v D v x + (r β u v x + α u v ux for all t (0, T max. We now estimate the integrals on the right of (3.10. By Young s inequality, we have (u + 1 m 1 u v x 1 (3.11 v x + (u + 1 (m 1 u x, 8 χ u v v x 1 v x + χ u v x, (3.1 8 ξ u w v x 1 v x + ξ u w x, ( µ u η v x µ v x µ u v x, (3.14 u( 1 v D v x = u v x u D v x (3.15 = u v x v ux u D v x u v x + 1 v x + u x, 8 α u v ux u v x + α u x (3.16

9 EJDE-016/176 BOUNDEDNESS IN A CHEMOTAXIS SYSTEM 9 for all t (0, T max. Substituting (3.11-(3.16 into (3.10 an using (.7, we obtain that there exists a positive constant C 1 satisfying u v x + (µ 1 χ u v x t 1 v x + (4 + α (u + 1 (m 1 u x ( ξ u w x + (r β u v x + u v x + C 1 for all t (0, T max. By taking a similar proceure, we can erive that t u w x + (µ 1 ξ w x + (4 + γ 1 + χ u w x (u + 1 (m 1 u x u v x + (r δ u w x + u w x + C with a constant C > 0 for all t (0, T max. Aing (3.18 to (3.17 yiels (3.9. Multiplying (3.1 by (9+α +γ m(m+1 obtain the following result. (3.18 an then combining (3., (3.3 an (3.9, we Corollary 3.4. Let the assumptions in Lemma 3.1 hol. Then we have { (9 + α + γ (u + 1 m+1 x + v 4 x + w 4 x t m(m + 1 } + u v x + u w x + v x + w x + [µ 1 4χ 11α χ (9 + α + γ ] u v x + [µ 1 4ξ 11γ ξ (9 + α + γ ] u w x + (u + 1 (m 1 u x + µ(9 + α + γ η (u + 1 m+η x m (r β u v x + (r δ u w x + (9 + α + γ ( r (u + 1 m+1 x + µ (u + 1 m x m + u v x + u w x + v v x + w w x + C for all t (0, T max, where C is a positive constant. (3.19 We now are in a position to establish the estimates on um+1 x, v 4 x an w 4 x. We will show that if µ is taken large enough, then these integrals are uniformly boune for all t (0, T max.

10 10 Y. WANG EJDE-016/176 Lemma 3.5. Suppose that µ > max{1 + 4χ + 11α + χ (9 + α + γ, 1 + 4ξ + 11γ + ξ (9 + α + γ }. Let η an let (u, v, w be a solution to system (1.1. Then we can fin a constant C > 0 such that (u + 1 m+1 x C t (0, T max, (3.0 v 4 x C t (0, T max, (3.1 w 4 x C t (0, T max. (3. Proof. Let y(t := (9+α +γ m(m+1 (u + 1m+1 x + v 4 x + w 4 x + u v x + u w x. Since µ > max{1 + 4χ + 11α + χ (9 + α + γ, 1 + 4ξ + 11γ + ξ (9 + α + γ }, from Corollary 3.4 we have y (t + y(t + v x + w x + (u + 1 (m 1 u x + µ(9 + α + γ η (u + 1 m+η x m (r + 1 β u v x + (r + 1 δ u w x + (9 + α + γ [r(m ] (u + 1 m+1 x (3.3 m(m u v x + u w x + v v x + w w x + µ(9 + α + γ (u + 1 m x m + v 4 x + w 4 x + C 0 t (0, T max, where C 0 is a positive constant. We next use the issipate quantities on the left of (3.3 to estimate all integrals on the right. By the Gagliaro-Nirenberg inequality, we obtain that v 4 x = v L ( (3.4 C 1 v 6 5 L ( v 4 5 L 1 ( + C 1 v L 1 ( for all t (0, T max, with some constant C 1 > 0. From Lemma.4, we know that v C for all t (0, T max with some C > 0. Thus by Young s inequality we can fin a constant C 3 > 0 such that v 4 x 1 8 v L ( + C 3 t (0, T max. (3.5 Similarly, we can also have w 4 x 1 8 w L ( + C 4 t (0, T max (3.6

11 EJDE-016/176 BOUNDEDNESS IN A CHEMOTAXIS SYSTEM 11 with some constant C 4 > 0. By Young s inequality, we obtain (r + 1 β u v (r + 1 β x u x + v 4 x (3.7 4 an hence, upon (3.5 we erive that (r + 1 β u v x + v 4 x (r + 1 β u x + v 4 x 4 (r + 1 β u x v L ( + C 3 µ(9 + α + γ η+1 (u + 1 m+η x + 1 m 4 v L ( + C 5 with some constant C 5 > 0 for all t (0, T max. Similarly, we can obtain (r + 1 δ u w x + w 4 x µ(9 + α + γ η+1 (u + 1 m+η x + 1 m 4 w L ( + C 6 (3.8 (3.9 with some C 6 > 0 for all t (0, T max. Using Young s inequality again, we obtain that there exist positive constants C 7 an C 8 fulfilling (9 + α + γ [r(m ] (u + 1 m+1 x m(m + 1 µ(9 + α + γ (3.30 η+1 (u + 1 m+η x + C 7, m µ(9 + α + γ (u + 1 m x m µ(9 + α + γ (3.31 η+1 (u + 1 m+η x + C 8 m for all t (0, T max. For the bounary integrals in (3.3, by taking similar arguments in [4, (3.0] we obtain that there exist some positive constants C 9 an C 10 fulfilling u v 1 4 u w 1 4 x + v x + 1 x + w x + 1 v v x u x + C 9, (3.3 w w x (3.33 u x + C 10

12 1 Y. WANG EJDE-016/176 for all t (0, T max. Substituting (3.8-(3.33 into (3.3, we obtain that y (t + y(t + v x + w x + (u + 1 (m 1 u x + µ(9 + α + γ η (u + 1 m+η x m 1 v x + 1 w x + u x + µ(9 + α + γ η (u + 1 m+η x + C 11 m for all t (0, T max, where C 11 := C i=5 C i. Hence we have Thus, by Gronwall s inequality, we have y(t = (9 + α + γ u m+1 x + m(m u v x + u w x max{y(0, C 11 } (3.34 y (t + y(t C 11 t (0, T max. (3.35 for all t (0, T max, which implies (3.0, (3.1 an (3.. v 4 x + w 4 x We now use the L m+1 estimate for u + 1 an the L 4 estimate for v an w from Lemma 3.5 to establish higher regularity estimates for (u, v, w. Lemma 3.6. Suppose that µ > max{1 + 4χ + 11α + χ (9 + α + γ, 1 + 4ξ + 11γ + ξ (9 + α + γ }. Let η an let (u, v, w be a solution to system (1.1. Then for all p > 3m 1 an each q > 1 there exists a constant C > 0 such that u(, t L p ( C t (0, T max, (3.36 v(, t L q ( C t (0, T max, (3.37 w(, t Lq ( C t (0, T max. (3.38 Proof. Testing the first equation in (1.1 against (u + 1 p 1 an then integrating by parts over, we erive that 1 (u + 1 p x p t = (p 1 (u + 1 m+p 3 u x + (p 1χ u(u + 1 p v ux (p 1ξ u(u + 1 p w ux + r u(u + 1 p 1 x µ u η (u + 1 p 1 x (p 1 (u + 1 m+p 3 u x + (p 1χ (u + 1 p 1 v u x + (p 1ξ (u + 1 p 1 w u x + r (u + 1 p x

13 EJDE-016/176 BOUNDEDNESS IN A CHEMOTAXIS SYSTEM 13 µ u η (u + 1 p 1 x p 1 (u + 1 m+p 3 u x + (p 1χ (u + 1 p m+1 v x + (p 1ξ (u + 1 p m+1 w x + r (u + 1 p x µ u η (u + 1 p 1 x, t (0, T max, (3.39 where we have use Young s inequality in the last step. Accoring to (3.1 an (3., we note that there exists a constant C 1 > 0 such that v L 4 ( C 1, w L 4 ( C 1 for all t (0, T max. Thus, by using Höler s inequality an the inequality u η (u+1η + 1, we obtain that η (u + 1 p x t p(p 1 (m + p 1 (u + 1 m+p 1 x + p(p 1(χ + ξ C 1 (u + 1 p m+1 L ( + pr (u + 1 p x µp (u + 1 p+ x + µp (u + 1 p 1 x η (3.40 for all t (0, T max. Using that p > 3m 1 implies (m + 1 4(p m + 1 < m + p 1 m + p 1. We note that 4(p m + 1 < 6, m + p 1 because 6(m + p 1 4(p m + 1 = 10(m 1 + p > 0. We may invoke the Gagliaro-Nirenberg inequality to estimate (u + 1 p m+1 L ( = (u + 1 m+p 1 C (u + 1 m+p 1 + C (u + 1 m+p 1 (p m+1 m+p 1 4(p m+1 L m+p 1 ( (p m+1 m+p 1 θ L ( (p m+1 m+p 1 (m+1 L m+p 1 ( (u + 1 m+p 1 for all t (0, T max with some C > 0 an θ (0, 1 etermine by that is, 3(m + p 1 4(p m + 1 = (1 3 3(m + p 1 θ (1 θ; (m + 1 θ = 3(m+p 1 ( 1 m+1 1 3(m+p 1 (m+1 1 (p m+1. (p m+1 m+p 1 (1 θ (m+1 L m+p 1 ( In view of (3.0, we obtain that there exists a constant C 3 > 0 such that (u+1 p m+1 L ( C 3 (u+1 m+p 1 (p m+1 m+p 1 θ L ( +C 3 t (0, T max. (3.41

14 14 Y. WANG EJDE-016/176 It is easy to check that (p m + 1 m + p 1 θ = 3p 9 m + 3 3p + m 4 <, because (3p + m 4 (3p 9 m + 3 = m + 11 (m 1 > 0. Thus, by applying Young s inequality, we have p(p 1(χ + ξ C 1 (u + 1 p m+1 L ( p(p 1 (3.4 (m + p 1 (u + 1 m+p 1 x + C 4 for all t (0, T max with some C 4 > 0. Substituting (3.4 into (3.40 yiels that (u + 1 p x + (u + 1 p x t (pr + 1 (u + 1 p x µp η (u + 1 p+ x ( µp (u + 1 p 1 x + C 4 for all t (0, T max. Using Young s inequality twice again, there exist some positive constants C 5 an C 6 such that (pr + 1 (u + 1 p x µp η+1 (u + 1 p+ x + C 5 t (0, T max, (3.44 µp (u + 1 p 1 x µp (u + 1 p+ x + C 6 t (0, T max. (3.45 η+1 Substituting (3.44 an (3.45 into (3.43, we have (u + 1 p x + (u + 1 p x C 7 t (0, T max t with C 7 := C 4 + C 5 + C 6. We thus conclue that there exists C 8 > 0 such that (u + 1 p x C 8 t (0, T max, which implies (3.36. From the variation-of-constants representation of v, we have v(, t = e t( β v 0 + α t 0 e (t s( β u(, ss t (0, T max, where (e t t 0 is the Neumann heat semigroup in. Using the L p L q estimate for the Neumann heat semigroup, we can fin C 9 > 0 such that v(, t L q ( t C 9 v 0 L q ( + C (t s 1 3 ( 1 p 1 q e β(t s u(, s L p (s 0 (3.46 for all t (0, T max. We note that u(, t LP ( C 10 with C 10 > 0 for all t (0, T max an p > 3m 1. In particularly, if we take p > 3, then we have

15 EJDE-016/176 BOUNDEDNESS IN A CHEMOTAXIS SYSTEM ( 1 p 1 q < 1 for all q > 1. Thus we can fin a constant C 11 > 0 such that v(, t Lq ( t C 9 v 0 Lq ( + C C 10 (t s 1 3 ( 1 p 1 q e β(t s s C 11 t (0, T max. 0 (3.47 Similarly, we have w(, t L q ( C 1 for all t (0, T max with some constant C 1 > 0. In the following subsection, we will eal with the case that the iffusion is strong enough (i.e., m > 4 3. We can also establish the Lp estimate for u an L q estimate for v an w for all p > 1 an q > 1. A priori estimates for the sufficiently strong iffusion case. Lemma 3.7. Suppose that m > 4 3. Let (u, v, w be a solution to system (1.1. Then for all p > 1 an each q > 1 there exists a constant C > 0 such that u(, t L p ( C t (0, T max, (3.48 v(, t L q ( C t (0, T max, (3.49 w(, t L q ( C t (0, T max. (3.50 Proof. Multiplying the first equation in (1.1 by (u + 1 p 1 (p > 1 an integrating it over an using Young s inequality, we obtain that 1 (u + 1 p x p t p 1 (u + 1 m+p 3 u x + (p 1χ (u + 1 p m+1 v x ( (p 1ξ (u + 1 p m+1 w x + r (u + 1 p x µ u η (u + 1 p 1 x for all t (0, T max. Since u η 1 η (u + 1 η 1 an η > 1, by using Young s inequality we obtain r r (u + 1 p x µ (u + 1 p x + µ µ η+1 µ η u η (u + 1 p 1 x (u + 1 p 1 x µ η (u + 1 p+ x (u + 1 p+ x + C 1 + µ (u + 1 p+ x + C (u + 1 p+ x = C 1 + C t (0, T max η+1 (3.5

16 16 Y. WANG EJDE-016/176 with some positive constants C 1 an C. Combining (3.51 with (3.5 yiels that 1 (u + 1 p (p 1 x p t (m + p 1 (u + 1 m+p 1 x + (p 1χ (u + 1 p m+1 v x ( (p 1ξ (u + 1 p m+1 w x + C 3 for all t (0, T max, with C 3 := C 1 + C. In Lemma 3., we obtaine the ientity ( v t = v D v β v + α u v t (0, T max. Testing this against v q yiels 1 v q 4(q 1 x + v q q t q x + D v v q + β v q x q v v x + α v q u vx t (0, T max, where we have use the ientity (q 1 v q 4 v 4(q 1 x = v q q x. (3.54 For the first integral on the right of (3.54, from [3, (3.10] we have q v v x q 1 v q q x + C 4 t (0, T max (3.55 with some constant C 4 > 0. For the secon integral on the right of (3.54, we integrate by parts over an use Young s inequality to erive α v q u v x = α(q 1 u v q 4 v v x α u v q vx q 1 v q 4 (3.56 v x + α (q 1 u v q x + v q v x + 3α u v q x 3 for all t (0, T max. Upon the pointwise inequality v 3 D v, we have v q v x v q D v x t (0, T max. 3 We thus infer from (3.54-(3.56 that 1 v q 3(q 1 x + v q q t q x (q 1 (3.57 α u v q x + C 4

17 EJDE-016/176 BOUNDEDNESS IN A CHEMOTAXIS SYSTEM 17 for all t (0, T max. By taking a similar proceure for w, we erive that 1 w q 3(q 1 x + w q q t q x (q 1 (3.58 γ u w q x + C 5 for all t (0, T max, with some constant C 5 > 0. Thus, collecting (3.53, (3.55 an (3.56 yiels that t{ p (u p x + 1 v q x + 1 } w q x q q (p 1 + (m + p 1 (u + 1 m+p 1 x 3(q 1 + v q 3(q 1 q x + w q q x (3.59 (p 1χ (u + 1 p m+1 v x + (p 1ξ (u + 1 p m+1 w x + (q 1 α (u + 1 v q x + (q 1 γ (u + 1 w q x + C 6 for all t (0, T max with C 6 := C 4 + C 5. Note that (3.59 is similar to [3, (3.14] an our conition m > 4 3 satisfies the conition 1 (m 1 < N (N = 3 in [3]. Thus, the remaining computation shall follow closely with minor moification. We omit it for brevity an easily establish (3.48-(3.50. Proof of Theorem 1.1. Using [3, Lemma A.1] an Lemma 3.6 an Lemma 3.7, we obtain that there exists a positive constant C > 0 such that u(, t L ( C t (0, T max, which together with the extensibility criterion (.1 yiels that T max = +. By well-known arguments from parabolic regularity theory for the secon an thir equations in (1.1, we can fin some constants C > 0 an C 3 > 0 such that v(, t W 1, ( C t > 0, w(, t W 1, ( C 3 t > 0. Thus, we prove that (u, v, w is a global boune classical solution to ( Asymptotic behavior In this section, accoring to the ieas from [4, Section 5], we consier the large time behavior of (u, v, w uner the assumptions r = 0, η an µ > max { 1 + 4χ + 11α + χ (9 + α + γ, 1 + 4ξ + 11γ + ξ (9 + α + γ }. Lemma 4.1. Let r = 0. Suppose that η an µ > max { 1 + 4χ + 11α + χ (9 + α + γ, 1 + 4ξ + 11γ + ξ (9 + α + γ }. Then the solution of (1.1 satisfies u η (x, t x t < 1 u 0 (xx t > 0 (4.1 0 µ

18 18 Y. WANG EJDE-016/176 an there exists a constant C > 0 such that C u(x, tx (t t > 0. (4. Proof. We integrate the first equation in (1.1 in space to obtain u(x, tx = µ u η (x, tx t > 0. (4.3 t Integrating it with respect to t yiels t u(x, tx + µ u η (x, t x t which implies (4.1. Using Höler s inequality, we have ux ( u η x 1 η 1 1 η 0 u 0 (xx t > 0, an then obtain uη x 1 ( ux η. Substituting it into (4.3 yiels u(x, tx µ ( η ux t t > 0. By integrating in time we have ( u(x, tx which implies (4.. 1 ( 1 R + µ( u0(xx t 1 t > 0, Lemma 4.. Let the assumptions in Lemma 4.1 hol. Suppose that β > 1 ( an δ > 1 (. Then there exists C > 0 such that C v(x, tx w(x, tx (t C (t t > 0, (4.4 for all t > 0. (4.5 Proof. Integrating the secon equation in (1.1 an letting y(t := v(x, tx, we see that there exists a constant C 1 > 0 such that y αc 1 (t = βy(t + α u(x, tx βy(t + t > 0. (t We efine Then y(0 = C 1 y (t + βy(t C := max { 1 y(t := v 0 (xx, C (t + 1 v 0(xx = y(0 an αc 1 (t αc1 β 1 t > 0. },

19 EJDE-016/176 BOUNDEDNESS IN A CHEMOTAXIS SYSTEM 19 = = C (η 1(t C (η 1(t (t + 1 C (η 1(t t > 0, + (1 1 t + βc (t + 1 αc 1 (t [ (β 1 η 1 C ( t + 1 ] αc 1 t + 1 (1 1 1 [ 1 + (β (t + 1 η 1 C 1+ 1 ] αc1 1 where we have use the assumption β > ( an the efinition of C. By comparison, we thus establish (4.4. By taking a similar proceure for w, we can also obtain (4.5. Lemma 4.3. Let η an µ > max { 1 + χ + 11α + χ (9 + α + γ, 1 + ξ + 11γ + ξ (9 + α + γ }. Then there exists θ (0, 1 an a constant C > 0 such that u C θ, C t > 1. (4.6 θ ( [t,t+1] Proof. We write the first equation in (1.1 in the form u t a(x, t, u = b(x, t x, t > 0, where a(x, t, u := (u+1 m 1 u h(x, t, h(x, t := χu v ξu w an b(x, t := ru(x, t µu η (x, t for x an t > 0. Here we note that a(x, t, u u = (u + 1 m 1 u h u 1 u 1 h, a(x, t, u C 1 u + h in (0, with C 1 := (M + 1 m 1, where we have use (1.3. Accoring to Lemma 3.6, we obtain that h an b belong to L ((0, ; L q ( for any q (1,. By parabolic Höler regularity [19, Theorem 1.3], this implies (4.6. Proof of Theorem 1.4. Let us assume that the first claim in (1.4 oes not hol. Then we can fin a sequence (t j j N (1, an a constant C 1 > 0 such that t j as j an u(, t j L ( C 1 j N. (4.7 In view of Lemma 4.3 an the Arzelà-Ascoli theorem, we see that (u(, t j j N is relatively compact in C 0 (. We can extract a subsequence (still enote by (u(, t j j N such that u(, t j u in L ( as j with a certain nonnegative u C 0 (. However, from Lemma 4.1 we can obtain that u(, t 0 in L 1 ( as t. Therefore, we have u 0, which contraicts (4.7. Thus, we prove u(, t L ( 0 as t. Upon Lemma 3.6 an Lemma 4., we can take a similar arguments to prove v(, t L ( 0 an w(, t L ( 0 as t.

20 0 Y. WANG EJDE-016/176 Acknowlegements. The author is very grateful to the referees for their etaile comments an valuable suggestions, which greatly improve the manuscript. This work was partially supporte by the Natural Science Project of Sichuan Province Department of Eucation (No.16ZB0075. References [1] E. Espejo, T. Suzuki; Global existence an blow-up for a system escribing the aggregation of microglia, Appl. Math. Lett. 35 (014, [] E. Espejo, T. Suzuki; Reaction terms avoiing aggregation in slow fluis, Nonlinear Anal. Real Worl Appl. 1 (015, [3] S. Ishia, K. Seki, T. Yokota; Bouneness in quasilinear Keller-Segel systems of parabolicparabolic type on non-convex boune omains, J. Differential Equations 56 (014, [4] H. Y. Jin; Bouneness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl. 4 (015, [5] H. Y. Jin an Z. Liu; Large time behavior of the full attraction-repulsion Keller-Segel system in whole space, Appl. Math. Lett. 47 (015, [6] H. Y. Jin, Z. A. Wang; Asymptotic ynamics of the one-imensional attraction-repulsion Keller-Segel moel, Math. Methos Appl. Sci. 38 (015, [7] H. Y. Jin, Z. A. Wang; Bouneness, blowup an critical mass phenomenon in competing chemotaxis, J. Differential Equations 60 (016, [8] A. Kiselev an L. Ryzhik; Biomixing by chemotaxis an enhancement of biological reactions, Commun. Partial Differential Equations 37 (01, [9] E. F. Keller, L. A. Segel; Initiation of slime mol aggregation viewe as an instability, J. Theor. Biol. 6 (1970, [10] X. Li; Bouneness in a two-imensional attraction-repulsion system with nonlinear iffusion, Math. Methos Appl. Sci. 39 (016, [11] M. Luca, A. Chavez-Ross, L. Eelstein-Keshet, A. Mogilner; Chemotactic signalling, microglia, an Alzheimers isease senile plague: Is there a connection? Bull. Math. Biol. 65 (003, [1] Y. Li, K. Lin, C. Mu; Asymptotic behavior for small mass in an attraction-repulsion chemotaxis system, Electron. J. Differential Equations 146 (015, [13] K. Lin, C. L. Mu, L. C. Wang; Large time behavior for an attraction-repulsion chemotaxis system, J. Math. Anal. Appl. 46 (015, [14] P. Liu, J. Shi, Z. A. Wang; Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Cont. Dyn. Systems-B 18 (013, [15] D. Liu, Y. Tao; Global bouneness in a fully parabolic attraction-repulsion chemotaxis moel, Math. Methos Appl. Sci. 38 (015, [16] J. Liu, Z. A. Wang; Classical solutions an steay states of an attraction-repulsion chemotaxis in one imension, J. Biol. Dynam. 6 (01, [17] X. Li, Z. Xiang; On an attraction-repulsion chemotaxis system with a logistic source, IMA J. Appl. Math. 81 (016, [18] K. J. Painter, T. Hillen; Volume-filling an quorum-sensing in moels for chemosensitive movement, Can. Appl. Math. Quart. 10 (00, [19] M. M. Porzio, V. Vespri; Höler estimates for local solutions of some oubly nonlinear egenerate parabolic equations, J. Differential Equations 103 (1993, [0] C. Stinner, C. Surulescu, M. Winkler; Global weak solutions in a PDE-ODE system moeling multiscale cancer cell invasion, SIAM J. Math. Anal. (46 014, [1] R. K. Shi, W. K. Wang; Well-poseness for a moel erive from an attraction-repulsion chemotaxis system, J. Math. Anal. Appl. 43 (015, [] Y. Tao, Z. A. Wang; Competing effects of attraction vs. repulsion in chemotaxis, Math. Moels Methos Appl. Sci. 3 (013, [3] Y. Tao, M. Winkler; Bouneness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations 5 (01, [4] Y. Tao, M. Winkler; Bouneness an ecay enforce by quaratic egraation in a threeimensional chemotaxis-flui system, Z. Angew. Math. Phys. 66 (015,

21 EJDE-016/176 BOUNDEDNESS IN A CHEMOTAXIS SYSTEM 1 [5] Y. Tao, M. Winkler; A chemotaxis-haptotaxis moel: the roles of nonlinear iffusion an logistic source, SIAM J. Math. Anal. 43 (011, [6] Y. Wang; Global existence an bouneness in a quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type, Boun. Value Probl. 016 (016, 1. [7] Y. Wang, Z. Xiang; Bouneness in a quasilinear D parabolic-parabolic attraction-repulsion chemotaxis system, Discrete Cont. Dyn. Systems-B 1 (016, [8] Y. Wang; A quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type with logistic source, J. Math. Anal. Appl. 441 (016, Yilong Wang School of Sciences, Southwest Petroleum University, Chengu , China aress: wangelongelone@163.com