Boundedness criteria for the quasilinear attraction-repulsion chemotaxis system with nonlinear signal production and logistic source

: This paper deals with the following quasilinear attraction-repulsion chemotaxis system


Introduction
Chemotaxis is one of the basic physiological reactions of organisms, which refers to the directional movement of biological cells or organisms along the concentration gradient of stimulants under the stimulation of some chemicals in the environment. The chemotaxis phenomenon has been described in the pioneering work proposed by Keller and Segel [1], which is given by x ∈ Ω, t > 0, where Ω ⊂ R n (n ≥ 1) is a bounded domain with smooth boundary, χ > 0, τ ∈ {0, 1}, u(x, t) and v(x, t) denote the density of cells and the concentration of the chemical signal, respectively. It is well known that chemotaxis research has many important applications in biology and medicine, such as in bacterial colonies [2], tumor invasion processes [3,4] and embryonic development [5], so that it has been one of the hottest research focuses in applied mathematics nowadays. In the past few decades, a large number of valuable theoretical results have been obtained by scholars [6][7][8]. Among them, one of the main issues related to (1.1) is to study whether there is a globally in-time bounded solution or when blow-up occurs. For τ = 1, it has been shown that the system (1.1) has globally bounded classical solution when n = 1 [9] or n = 2 and Ω u 0 dx < 4π χ [10,11], whereas the system (1.1) has finite time blow-up solution in the case of n = 2 and Ω u 0 dx > 4π χ [12,13] or in the case of n ≥ 3 [14,15]. When the chemical substance diffuses much faster than the diffusion of cells [16], model (1.1) can be reduced to the simplified parabolic-elliptic model, namely, the second equation in system (1.1) is replaced by |Ω| Ω u 0 (x)dx. Compared with the fully parabolic version of system (1.1), the similar results on global boundedness and blow-up of solutions can be found in [17][18][19][20], which still depend on the dimensions of space. As described in system (1.1), the term of signal production is a linear function of the cell density in the classical Keller-Segel model. Nevertheless, the mechanism of signal production might be very complex, particularly, it could be in a nonlinear form. When the second equation in system (1.1) is replaced by v t = ∆v − v + g(u) with g(u) ∈ C 1 ([0, +∞)) and 0 ≤ g(u) ≤ Ku α for some constants K, α > 0, Liu and Tao [21] proved that if 0 < α < 2 n , the system (1.1) possesses a globally bounded classical solution. When the second equation in system (1.1) degenerates into an elliptic equation, u is replaced by g(u) and v is replaced by µ(t) := 1 |Ω| Ω g(u(·, t)), g(u) = u κ with κ > 0, Winkler [22] derived a blow-up critical exponent κ = 2 n , which asserted that the radially symmetric solution blows up in finite time if the parameter κ satisfies κ > 2 n . Conversely, when κ < 2 n they proved that there existed suitable initial data u 0 such that the system has globally bounded classical solutions. In many biological processes, the proliferation and death of cells should be considered, from which one can derived the related chemotaxis-growth model x ∈ Ω, t > 0, (1.2) Here it is worth mentioning that logistic source term f (u) should somewhat decrease the possibility of blow-up. When τ = 0, Tello and Winkler [23] considered the system (1. 2) with f (u) ≤ u(a − bu) and g(u) = u for a, b > 0 and they proved that the system has globally bounded classical solution whenever n−2 n χ < b. For the more general case f (u) ≤ u(a − bu s ) and g(u) = u k with k, s > 0, Wang and Xiang [24] showed that the system (1.2) has globally bounded classical solutions if either s > k or s = k with kn−2 kn χ < b. For f (u) = au − bu s and g(u) = u with s > 1, a ≥ 0, b > 0, Winkler [25] proved global existence of very weak solutions of (1.2) under the assumption that s > 2 − 1 n , moreover, boundedness properties of the constructed solutions were studied. When τ = 1, g(u) = u and f (u) is controlled by −c 0 (u+u s ) and a−bu s , respectively, for all u ≥ 0 with some s > 1, b, c 0 > 0 and a ≥ 0, by an appropriate definition of very weak solution, Viglialoro [26] constructed the such global solutions under the assumptions that n ≥ 2 and s > 1 − 2 n , and in [27], a relaxation of these hypotheses could be achieved so as to ensure solvability even for any s > 2n+4 n+4 when n ≥ 2. Furthermore, when f (u) = au − bu s and g(u) = u, Winkler [28] proved that if s ≥ 2 − 2 n , under an appropriate smallness assumption on χ any such solution at least asymptotically exhibits relaxation by approaching the nontrivial spatially homogeneous steady state ( a b ) in the large time limit. Continuing the research initiated in [26], Viglialoro and Woolley [29] studied the boundedness and regularity of these solutions after some time.
From a physical point of view, the equation modeling the migration of cells should rather be regarded as nonlinear diffusion [37], especially the slow diffusion with finite propagation property, which reads where the positive functions D(u) and S (u) are used to describe the strength of diffusion and chemoattractant, respectively. When τ = 1, n ≥ 2, Ω ⊂ R n is a ball and f (u) = 0, Winkler [38] proved that if D(u) S (u) grows faster than u 2 n as u → ∞ and some further technical conditions are fulfilled, then there exist solutions that blow up in either finite or infinite time, which implies that the result is optimal. Inter alia, there still exist many results on global boundedness and blow-up in (1.3), please see [39][40][41][42]. For τ = 0, when f (u) = au − bu κ for all u ≥ 0 with a ≥ 0, b > 0 and κ > 1, and the second equation |Ω| Ω u(x, t)dx. In [43], for D(u) ≥ D 0 u −m and S (u) = χ for all u > 0 with some m ∈ R, χ, D 0 > 0, it was shown that the system (1.3) possesses a unique globally bounded classical solution for any initial data u 0 ∈ C 0 (Ω) and n ≥ 2 if κ > max{m + 3 − 4 n+2 , 2}; and for D(u) = D 0 u −m and S (u) = χ with 4 n − 1 < m ≤ 0, the system (1.3) blows up in finite time in a ball if κ ∈ 1, (3−m)n−2 2n−2 and n ≥ 5. For more boundedness results and blow-up of solutions to system (1.3) with or without logistic source, we refer the interested readers to [44][45][46][47][48].
As stated in [49], studies have shown that the reaction of one species to multiple stimuli is given by the motion of microglia in Alzheimer disease tethered to a glass side in a conflict situation involving β-amyloid (an attractant) and tumor necrosis factor α (a repellent). To model such biological processes, the following attraction-repulsion chemotaxis system was proposed in [50] where u, v and w represent the density of cell, chemical concentration of attractant and chemical concentration of repellent, respectively, the parameters χ, ξ > 0. To better understand system (1.4), let us mention some previous contributions in this direction. We first introduce some global boundedness of classical solutions related to system (1.4). For the case of f (u) ≤ u(a − bu), when g 1 (u) = αu and g 2 (u) = γu with a, b, α, γ > 0, Zhang et al. [51] obtained that for any nonnegative u 0 (x) ∈ C 0 (Ω), if one of the following conditions holds then the system (1.4) has a globally bounded classical solution. For the case of f (u) ≤ u(a − bu s ), when g 1 (u) = αu k and g 2 (u) = γu l with a, b, α, γ, k, l, s > 0, Hong et al. [52] showed that if k < max{l, s, 2 n }, then the system (1.4) admits a globally bounded solution. Furthermore, when k = max{l, s} ≥ 2 n , the system (1.4) also admits a globally bounded solution if one of the following assumptions holds: More recently, on the basis of [52], in high dimension (n ≥ 2), Zhou et al. [53] have further studied the boundedness of globally classical solution for the critical cases: Apart from that, there are some interesting findings about blow-up behavior of solutions for system (1.4). When f (u) = 0, g 1 (u) = αu and g 2 (u) = γu with α, γ > 0, Li et al. [54] proved that the nonradial solutions to system (1.4) would blow-up in finite time if either αχ − ξγ > 0, β δ and Ω u 0 dx > 8π αχ−ξγ or αχδ − ξγβ > 0, δ < β and Ω u 0 dx > 8π αχδ−ξγβ in the case n = 2. Yu et al. [55] improved the above finite-time blowup result under the condition of αχ − ξγ > 0 and Ω u 0 dx > 8π αχ−ξγ . More recently, when f (u) = 0, the second and third equations are replaced by 0 = ∆v − m 1 (t) + g 1 (u), m 1 (t) = 1 |Ω| Ω g 1 (u) and 0 = ∆w − m 2 (t) + g 2 (u), m 2 (t) = 1 |Ω| Ω g 1 (u), respectively, for g 1 (u) ≥ k 1 u γ 1 and g 2 (u) ≤ k 2 u γ 2 for all u ≥ 0 with k 1 , k 2 , γ 1 , γ 2 > 0, Liu et al. [56] proved that if γ 1 > max{γ 2 , 2 n }, the radial solutions to system (1.4) would blow-up in finite time, and if γ 1 < 2 n , the classical solution would be globally bounded. Later on, Wang et al. [57] extended such blow-up results to a quasilinear system with logistic source. Similar to classical Keller-Segel system, when considering the effect of nonlinear diffusion and logistic source, Chiyo et al. [58] studied the following parabolic-elliptic-elliptic system with m, p, q ∈ R, where they classify boundedness and blow-up into the cases p < q and p > q without any condition for the sign of χα − ξγ and the case p = q with χα − ξγ > 0 or χα − ξγ < 0. In contrast to the systems mentioned above, we find that there are very few results on the existence of globally bounded classical solutions to the attraction-repulsion system with nonlinear diffusion and logistic source as well as nonlinear signal production at the same time. On the basis of work [52], the purpose of the present paper is to continue to detect the effect among nonlinear diffusion and logistic source as well as nonlinear signal production on the boundedness of the solution to the following attraction-repulsion system where Ω ⊂ R n (n ≥ 1) is a bounded domain with smooth boundary ∂Ω, ν denotes the outward unit normal vector on ∂Ω. Here, u, v and w represent the density of cell, chemical concentration of attractant and chemical concentration of repellent, respectively, and the parameters satisfy m, θ, l ∈ R, χ, ξ, a, b, α, β, γ, δ, γ 1 , γ 2 > 0, κ > 1. We state our main results to system (1.6) as follows.
Remark 1.5. Theorem 1.1 and Theorem 1.3 also leave an interesting problem, i.e. it is still unknown whether the boundedness criteria obtained in Theorem 1.1 and Theorem 1.3 are optimal for system (1.6). We will also further study the finite-time blow-up criteria of the solution for system (1.6) in the future research.
We carry out this paper as follows. In Section 2, we state a result on the existence local solutions and give some useful lemmas. In Section 3, we construct the L p −estimates for component u and then use the Moser iteration to prove Theorem 1.1 and Theorem 1.3.

Preliminaries
In this section, we first state the existence of local solutions to system (1.6). The proof relies on Schauder fixed theorem. We omit it for brevity and refer the readers to [59,60] for more details.

Global existence and boundedness
In this section, we construct the L p −estimate for component u under different conditions to prove Theorem 1.1 and Theorem 1.3.
Proof. Multiplying the first equation of system (1.6) by (u + 1) p−1 and integrating by parts over Ω, we derive for all t ∈ (0, T max ).