An attraction-repulsion chemotaxis system with nonlinear productions

J. Math. Anal. Appl. 484 (2020) 123703

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

An attraction-repulsion chemotaxis system with nonlinear productions ✩ Liang Hong a,c , Miaoqing Tian b,c , Sining Zheng c,∗ a

School of Mathematics, Liaoning University, Shenyang 110036, PR China College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, PR China c School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, PR China b

a r t i c l e

i n f o

Article history: Received 22 August 2019 Available online 22 November 2019 Submitted by Y. Yamada Keywords: Attraction-repulsion Nonlinear production Chemotaxis Logistic source Boundedness

a b s t r a c t This paper studies the semilinear attraction-repulsion chemotaxis system with nonlinear productions and logistic source: ut = Δu −χ∇ ·(u∇v) +ξ∇ ·(u∇w) +f (u), 0 = Δv+αuk −βv, 0 = Δw+γul −δw, in bounded domain Ω ⊂ Rn , n ≥ 1, subject to the non-flux boundary conditions, where the nonlinear productions for the attraction and repulsion chemicals are described via αuk and γul respectively, and the logistic source f ∈ C 2 [0, ∞) satisfying f (u) ≤ u(a − bus ) with s > 0, f (0) ≥ 0. It is proved that if one of the random diffusion, logistic source and repulsion mechanisms dominates the attraction with max{l, s, n2 } > k, the solutions would be globally bounded. Furthermore, under the three balance situations, namely, k = s > l, k = l > s or k = s = l, the boundedness of solutions depends on the sizes of the coefficients involved. This extends the global boundedness criteria established by Zhang and Li (2016) [20] for the attraction-repulsion chemotaxis system with linear productions and logistic source. © 2019 Elsevier Inc. All rights reserved.

1. Introduction In this paper, we study a Keller-Segel system of attraction-repulsion chemotaxis model with nonlinear productions and logistic source of the form ✩ Supported by the National Natural Science Foundation of China (11171048), the Science Foundation of Liaoning Education Department (LYB201601) and the Key Research Plan of Henan Province Colleges (20B110021). * Corresponding author. E-mail addresses: [email protected] (L. Hong), [email protected] (M. Tian), [email protected] (S. Zheng).

https://doi.org/10.1016/j.jmaa.2019.123703 0022-247X/© 2019 Elsevier Inc. All rights reserved.

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L. Hong et al. / J. Math. Anal. Appl. 484 (2020) 123703

⎧ ⎪ ut = Δu − χ∇ · (u∇v) + ξ∇ · (u∇w) + f (u), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 = Δv + αuk − βv, ⎪ ⎪ ⎪ ⎨ 0 = Δw + γul − δw, ⎪ ⎪ ⎪ ∂u ∂v ∂w ⎪ ⎪ ⎪ = = = 0, ⎪ ⎪ ∂ν ∂ν ∂ν ⎪ ⎪ ⎩ u(x, 0) = u (x), 0

(x, t) ∈ Ω × (0, T ), (x, t) ∈ Ω × (0, T ), (x, t) ∈ Ω × (0, T ),

(1.1)

(x, t) ∈ ∂Ω × (0, T ), x ∈ Ω,

where Ω ⊂ Rn (n ≥ 1) is a bounded domain with smooth boundary, ν denotes the outward normal vector, the logistic source f ∈ C 2 [0, ∞) satisfying f (u) ≤ u(a − bus ) with f (0) ≥ 0, and the parameters α, β, γ, δ, χ, ξ, a, b, s, k, l > 0. In the model (1.1), u represents the cell density, v and w describe the concentrations of two kinds of chemical signals produced by the cells, possessing attractive and repulsive effects for u with the chemotactic sensitivity χ, ξ > 0 respectively. It is mentioned that, instead of the linear production u, the nonlinear production uα was used to model the aggregation patterns formed by some bacterial chemotxis (refer to [10, Chapter 5] and [3,11–13]). Correspondingly, here the productions of signals v and w in the model are both nonlinear with the forms of αuk and γul . It will be observed that this would substantially affect the behavior of solutions. The Keller-Segel chemotaxis systems, proposed in 1970, have gained a lot of interesting results in the past years [8,4–7,18,19]. Among them Tello and Winkler [15] proved for the chemotaxis system 

ut = Δu − χ∇ · (u∇v) + f (u) in Ω × (0, T ), 0 = Δv + αuk − βv

in Ω × (0, T )

(1.2)

with f (u) ≤ u(a − bus ), k = s = 1 that the solutions are globally bounded whenever n−2 n χ < b. While for the model (1.2) with more general case with k, s > 0, Wang and Xiang [17] obtained that the chemotaxis model has global solutions if either s > k, or s = k with

kn − 2 χ < b. kn

(1.3)

For the attraction-repulsion system (1.1) with linear productions, i.e., ⎧ ut = Δu − χ∇ · (u∇v) + ξ∇ · (u∇w) + f (u) in Ω × (0, T ), ⎪ ⎪ ⎨ 0 = Δv + αu − βv in Ω × (0, T ), ⎪ ⎪ ⎩ 0 = Δw + γu − δw in Ω × (0, T )

(1.4)

with f (u) ≤ u(a − bu) Zhang and Li [20] proved that the problem possesses a globally bounded classical ¯ if one of the following holds: solution for any nonnegative u0 (x) ∈ C(Ω), (a1 ) αχ − γξ ≤ b; (b1 ) n ≤ 2; (c1 )

n−2 (αχ − γξ) < b with n ≥ 3. n

(1.5)

In the present paper, we consider the case of nonlinear productions with s, k, l > 0 to establish the global boundedness of solutions for the attraction-repulsion model (1.1). The main result is the following theorem. ¯ Theorem 1. Let nonnegative initial data u0 (x) ∈ C(Ω). (i) If k < max{l, s, n2 }, then Eq. (1.1) admits a globally bounded classical solution. (ii) Asuume k = max{l, s} ≥ n2 . Then Eq. (1.1) admits a globally bounded classical solution if one of the following assumptions holds:

L. Hong et al. / J. Math. Anal. Appl. 484 (2020) 123703

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(a) k = l = s, (kn−2) kn (αχ − γξ) < b; (b) k = l > s, αχ − γξ < 0; (c) k = s > l, (kn−2) kn αχ < b. Remark 1. The behavior of solutions to (1.1) is determined by the interaction among the four mechanisms of random diffusion, attraction, repulsion and logistic source. Besides the attraction, all the other three mechanisms benefit the global boundedness of solutions. Theorem 1 shows in what way the nonlinear exponents k, l, s > 0 (describing the productions for components v and w as well as the logistic source for u respectively) influence the evolution of solutions. More precisely, if the attraction is dominated by one of the random diffusion, logistic source and repulsion mechanisms with max{l, s, n2 } > k, then the solutions would be globally bounded. Under the three balance situations with k = s > l, k = l > s or k = s = l, the boundedness of solutions would be determined by the sizes of the coefficients involved. Remark 2. Now compare our result with that for the system (1.4) possessing linear productions. Obviously, Theorem 1 covers the criteria (1.5) for (1.4) obtained in [20, Theorem 1.1] as its special case by letting k = s = l = 1 in the model (1.1). It is mentioned that, in fact, the item (a1 ) itself is included by the other items (b1 ) and (c1 ) already. Also, the global boundedness condition (1.3) in [17] for the chemotaxis system (1.2) with nonlinear production αuk can be realized by taking ξ = l = 0 in Theorem 1. Remark 3. It can be observed as well that the boundedness criteria obtained for the attraction-repulsion chemotaxis system (1.1) with nonlinear productions are parallel to those for the corresponding attractionrepulsion chemotaxis model with some nonlinear diffusion and linear productions [16]. 2. Preliminaries In this section we deal with local solutions as preliminaries. The local existence of solutions to (1.1) can be addressed by the standard theory of parabolic equations in a suitable framework of fixed point theory [2,9,15]. We give a lemma on the local existence of solutions without proof. Lemma 2.1. Suppose that Ω ⊂ Rn (n ≥ 1) is a bounded domain with smooth boundary. Then for any ¯ there exist nonnegative functions u, v, w ∈ C 0 (Ω×[0, ¯ ¯ nonnegative u0 (x) ∈ C(Ω), Tmax )) ∩C 2,1 (Ω×(0, Tmax )) with Tmax ∈ (0, ∞] classically solving (1.1). Moreover, if Tmax < ∞, then lim u(·, t) L∞ (Ω) = ∞.

t→Tmax

Next, we introduce a crucial lemma to establish the global existence of solutions, i.e., an extended version of [16, Lemma 2.2] with general exponent l > 0. Lemma 2.2. Let (u, v, w) be a solution to (1.1) ensured by Lemma 2.1. Then for any l, η > 0, θ > 1, there is c0 = c0 (η, θ, l) > 0 such that 

 w ≤η

ulθ + c0 , t ∈ (0, Tmax ).

θ

Ω

Ω

Proof. Integrate the first equation in (1.1) to get d dt





Ω

 u(a − bus ) ≤ a

u= Ω

u− Ω

b  |Ω|s



1+s u

Ω

, t ∈ (0, Tmax )

(2.1)

L. Hong et al. / J. Math. Anal. Appl. 484 (2020) 123703

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by Jensen’s inequality, and hence  u ≤ max



Ω

a 1  u0 , ( ) s |Ω| := M, t ∈ (0, Tmax ). b

(2.2)

Ω

Moreover, integrating the third equation of (1.1) yields w L1 =

γ l u L1 . δ

(2.3)

Multiplying the third equation of (1.1) by wθ−1 , and integrating on Ω, we have 4(θ − 1) θ2



θ 2



|∇w | + δ 2

Ω

 θ

l

w =γ Ω

uw

θ−1

θ−1 ≤ δ θ

Ω



γθ w + θ−1 θδ

 ulθ , t ∈ (0, Tmax )

θ

Ω

(2.4)

Ω

by Young’s inequality, and thus γ w Lθ (Ω) ≤ ul Lθ (Ω) , t ∈ (0, Tmax ), δ   θ 4(θ − 1) γθ 2 2 |∇w | ≤ θ−1 ulθ , t ∈ (0, Tmax ). θ δ Ω

(2.5) (2.6)

Ω

¯ θ) > 0 such that By Ehrling’s lemma, for any η > 0, θ > 1, there is C¯ = C(η, 2 ¯ φ 2L2 (Ω) ≤ η φ 2W 1,2 (Ω) + C φ 2

L θ (Ω)

, φ ∈ W 1,2 (Ω).

θ

Take φ = w 2 . We know from (2.3), (2.5) and (2.6) that 

 wθ ≤ η

Ω

ulθ + C1 ul θL1

(2.7)

Ω

with C1 = C1 (η, θ) > 0. For l ∈ (0, 1], we have by the Hölder inequality with (2.2) that ul θL1 < C2

(2.8)

with C2 = C2 (η, θ, l) > 0. For l > 1, by interpolation inequality, Young’s inequality and (2.2), we have ul θL1

≤

l θ(1−τ ) ul θτ Lθ u L1/l

 ≤η

ulθ + C3 ,

(2.9)

Ω

where τ =

l−1 l− θ1

∈ (0, 1) and C3 = C3 (η, θ, l) > 0. Combining (2.7)–(2.9) yields (2.1). 

3. Proof of Theorem 1 In this section, we deal with the global boundedness of solutions to prove Theorem 1. Proof of Theorem 1. We at first prove that for any p > 1 there is c = c(p) > 0 such that  up ≤ c, Ω

t ∈ (0, Tmax ).

(3.1)

L. Hong et al. / J. Math. Anal. Appl. 484 (2020) 123703

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In the sequel, for simplicity, denote by c and cε positive constants c = c(p) and c = c(p, ε) respectively, independent of t, which may change from line to line. Multiply the first equation of (1.1) by up−1 , integrate on Ω, and then use the rest two equations to get 1 d p dt



 up ≤ −(p − 1) Ω



Ω

 Ω

4(p − 1) ≤− p2



4(p − 1) p2

Ω

up Δv + Ω



αχ(p − 1) p

p

|∇u 2 |2 +

Ω

ξ(p − 1) p



γξ(p − 1) p

Ω



 up − b

up Δw + a Ω

 up+k −



up − b

+a

Ω



χ(p − 1) p

p

|∇u 2 |2 −

Ω



up−1 ∇u · ∇w

up+s

Ω

≤−

up−1 ∇u · ∇v − (p − 1)ξ Ω



up − b

+a



up−2 |∇u|2 + (p − 1)χ

Ω

 up+l +

up+s Ω

δξ(p − 1) p

Ω



up w Ω

up+s , t ∈ (0, Tmax ).

(3.2)

Ω

By Young’s inequality and (2.1), we have δξ(p − 1) p



ε 2

up w ≤ Ω



 up+l + cε

Ω

w



p+l l

≤ε

Ω

up+l + cε , t ∈ (0, Tmax )

(3.3)

Ω

with any ε > 0. Substitute (3.3) into (3.2) to get 1 d p dt

 up ≤ −



4(p − 1) p2

Ω

p

|∇u 2 |2 + Ω

 Ω

 up+k −

 γξ(p − 1) p

−ε

Ω



up − b

+a

αχ(p − 1) p

up+s + cε , t ∈ (0, Tmax ). Ω

Case (i) k < max{l, s, n2 }. Let k < s. We know by Young’s inequality that αχ(p − 1) p

 up+k ≤

b 2

Ω

Taking ε :=

γξ(p−1) p

 up+s + c, t ∈ (0, Tmax ). Ω

in (3.4), we have 1 d p dt



 up ≤ a

Ω

up − Ω

b 2

 up+s + c, t ∈ (0, Tmax ). Ω

This proves (3.1). Let k < l. By Young’s inequality, we know αχ(p − 1) p

 up+k ≤ Ω

γξ(p − 1) 2p

 up+l + c, t ∈ (0, Tmax ). Ω



up+l

Ω

(3.4)

L. Hong et al. / J. Math. Anal. Appl. 484 (2020) 123703

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Substitute into (3.4), 1 d p dt





   γξ(p − 1) u ≤− up+l + a up − b up+s + cε , t ∈ (0, Tmax ) −ε 2p p

Ω

Taking ε :=

γξ(p−1) 2p

Ω

Ω

Ω

> 0, we have 1 d p dt







up ≤ a Ω

up − b Ω

up+s + c, t ∈ (0, Tmax ), Ω

and hence arrive at (3.1). Let k < n2 . Without loss of generality, suppose k ≥ max{l, s} > 0. With ε := γξ(p−1) > 0 in (3.4), we 2p obtain by Young’s inequality that     p 1 d 4(p − 1) αχ(p − 1) p+k 2 |2 + up ≤ − |∇u u + a up + c p dt p2 p Ω

Ω

≤−

4(p − 1) p2



Ω



p

|∇u 2 |2 + c Ω

Ω

up+k + c, t ∈ (0, Tmax ).

(3.5)

Ω

Now using the Gagliardo-Nirenberg inequality with (2.2), we have 

 p  2(p+k) p up+k = u 2  2(p+k) p

L

Ω

2(p+k)   2(p+k)    2(p+k) p λ u p2  2p (1−λ) + c u p2  2p ≤ c ∇u 2 L2 p

Lp

Lp

2(p+k)  p λ ≤ c ∇u 2 L2 p + c, t ∈ (0, Tmax )

with λ :=

p p 2 − 2(p+k) p 1 1 + − 2 n 2

∈ (0, 1), where 

2(p+k) λ p

< 2 due to k <

2 n.

This yields

 p 2 up+k ≤ η ∇u 2 L2 + cη , t ∈ (0, Tmax )

(3.6)

Ω

for any η > 0 by Young’s inequality. Now combining (3.5) with (3.6), and letting η small enough, we get   1 d p u ≤ − up+k + c, t ∈ (0, Tmax ). p dt Ω

Ω

This concludes (3.1). Case (ii) k = max{l, s} ≥

2 n.

For k = l, we know by (3.4) that 1 d p dt



4(p − 1) u ≤− p2



Ω



p 2

Ω

Ω

2

(αχ − γξ)(p − 1) +ε p

uk+s + cε , t ∈ (0, Tmax ). Ω

 up+k Ω



up − b

+a



|∇u | +

p

(3.7)

L. Hong et al. / J. Math. Anal. Appl. 484 (2020) 123703

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(a) Let k = l = s. Then the inequality (3.7) becomes 1 d p dt



4(p − 1) u ≤− p2



p 2

Ω



|∇u | +

p

2

 (αχ − γξ)(p − 1) up+k +ε−b p

Ω



Ω

up + cε , t ∈ (0, Tmax ).

+a

(3.8)

Ω

If αχ − γξ ≤ 0, we have by taking ε := 1 d p dt



 (γξ−αχ)(p−1)

+

2p

b 2



> 0 that

  (γξ − αχ)(p − 1) b + up ≤ − up+k + a up + c, t ∈ (0, Tmax ). 2p 2

Ω

Ω

Ω

This yields (3.1).  αχ−γξ If αχ − γξ > 0, take p ∈ (1, (αχ−γξ−b) ) to ensure ε := 12 b − + 1 d p dt

 up ≤ −

(αχ−γξ)(p−1)  p

> 0 in (3.8). We have



  1 (αχ − γξ)(p − 1) up+k + a up + c, t ∈ (0, Tmax ). b− 2 p

Ω

Ω

(3.9)

Ω

αχ−γξ This concludes (3.1) with p ∈ (1, (αχ−γξ−b) ). So, it suffices to deal with the case of αχ − γξ − b > 0. Since  nk+ αχ−γξ  (kn−2) kn (αχ − γξ) < b, we can take p0 ∈ 2 , αχ−γξ−b . By the Gagliardo-Nirenberg inequality, we know



 p  2(p+k) p up+k = u 2  2(p+k) p

L

Ω

2(p+k)     2(p+k)  2(p+k) p σ u p2  2pp (1−σ) + c u p2  2pp ≤ c ∇u 2 L2 p 0 0 p

L

 p ≤ c ∇u 2 

2(p+k) σ p L2

with σ :=

p 2p0 p

p − 2(p+k)

2p0

1 +n − 12

L

p

+ c, t ∈ (0, Tmax )

∈ (0, 1) when p > p0 . Due to p0 >

nk 2 ,

we have

2(p+k) σ p

< 2. We have by Young’s

inequality with any η > 0 that 

 p+k 2 up+k ≤ η ∇u 2 L2 + cη , t ∈ (0, Tmax ).

Ω

Taking ε := b in (3.8), we can choose η small enough to get 1 d p dt

 u ≤ −c Ω

This yields (3.1). (b) Let k = l > s. Since αχ − γξ < 0, taking ε =: 1 d p dt

 up ≤ − Ω

 p

 u

p+k

Ω

(γξ−αχ)(p−1) 2p

up + c, t ∈ (0, Tmax ).

+a Ω

in (3.7), we have

(γξ − αχ)(p − 1) 2p



 up + c, t ∈ (0, Tmax ).

up+k + a Ω

Ω

L. Hong et al. / J. Math. Anal. Appl. 484 (2020) 123703

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This concludes (3.1). (c) Let k = s > l. Take ε := γξ(p−1) in (3.4) to get p 1 d p dt

 Ω

   αχ(p − 1) up ≤ − b − up+k + a up + c, t ∈ (0, Tmax ), p Ω

Ω

the same as (3.9) with γξ = 0. By repeating the arguments for the case of k = l = s with αχ − γξ > 0, we can prove (3.1) as well. Let p > max{kn, ln, 1}. By the elliptic Lp -estimate to the two elliptic equations in (1.1), we get v(·, t) W 2,p/k , w(·, t) W 2,p/l < c,

t ∈ (0, Tmax ),

and hence v(·, t) C 1 (Ω) ¯ , w(·, t) C 1 (Ω) ¯ < c,

t ∈ (0, Tmax )

by the Sobolev imbedding theorem. Now the Moser iteration technique [1,14] ensures u L∞ (Ω) ≤ C for all t ∈ (0, Tmax ). This concludes by Lemma 2.1 that Tmax = ∞.  References [1] N.D. Alikakos, Lp bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations 4 (1979) 827–868. [2] X. Cao, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl. 412 (2014) 181–188. [3] E. Galakhov, O. Salieva, J.I. Tello, On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations 261 (2016) 4631–4647. [4] T. Hillen, K.J. Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009) 183–217. [5] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Dtsch. Math.-Ver. 105 (2003) 103–165. [6] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. II, Jahresber. Dtsch. Math.-Ver. 106 (2004) 51–69. [7] D. Horstmann, M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations 215 (2005) 52–107. [8] W. Jäger, S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992) 819–824. [9] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasi-Linear Equations of Parabolic Type, AMS, Providence, 1968. [10] J.D. Murray, Mathematical Biology, I. An Introduction, third edition, Interdisciplinary Applied Mathematics, vol. 17, Springer-Verlag, New York, 2002, xxiv+551 pp. [11] E. Nakaguchi, M. Efendiev, On a new dimension estimate of the global attractor for chemotaxis-growth systems, Osaka J. Math. 45 (2008) 273–281. [12] E. Nakaguchi, K. Osaki, Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation, Nonlinear Anal. 74 (2011) 286–297. [13] E. Nakaguchi, K. Osaki, Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete Contin. Dyn. Syst. Ser. B 18 (2013) 2627–2646. [14] Y. Tao, M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations 252 (2012) 692–715. [15] J. Tello, M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations 32 (2007) 849–877. [16] M. Tian, X. He, S. Zheng, Global boundedness in quasilinear attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal. Real World Appl. 30 (2016) 1–15. [17] Z. Wang, T. Xiang, A class of chemotaxis systems with growth source and nonlinear secretion, arXiv:1510.07204, 2015. [18] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations 248 (2010) 2889–2905. [19] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl. 100 (2013) 748–767. [20] Q. Zhang, Y. Li, An attraction-repulsion chemotaxis system with logistic source, Z. Angew. Math. Mech. 96 (2016) 570–584.