Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions

We consider the following attraction-repulsion Keller-Segel system: \begin{document}$\begin{equation*}\begin{cases}u_t=\nabla· (D(u) \nabla u)-χ\nabla·( u\nabla v)+ξ\nabla·( u\nabla w), x'>with homogeneous Neumann boundary conditions in a bounded domain $Ω\subset \mathbb{R}^n(n>2)$ with smooth boundary. Here all the parameters \begin{document}$χ, ξ, α, β, γ$\end{document} and \begin{document} $δ$\end{document} are positive. The smooth diffusion \begin{document}$D(u)$\end{document} satisfies \begin{document}$D(u)≥ d u^θ, u>0$\end{document} for some \begin{document}$d>0, θ∈\mathbb{R}$\end{document} . It is recently known from [ 25 ] that boundedness of solutions is ensured whenever \begin{document}$θ>1-\frac{2}{n}$\end{document} . Here, it is shown, if repulsion dominates or cancels attraction in the sense either \begin{document}$\{ξγ> χα\}$\end{document} or \begin{document}$\{ξγ=χα, β≥ δ\}$\end{document} , the corresponding initial-boundary value problem possesses a unique global classical solution which is uniformly-in-time bounded for large initial data provided \begin{document}$θ>1-\frac{4}{n+2}$\end{document} . In this way, the range of \begin{document}$θ>1-\frac{2}{n}$\end{document} of boundedness is enlarged and thus the repulsion effect on boundedness is exhibited.

(1.2) Intuitively, large values of θ seem to enhance boundedness of solutions. Neglect of the repulsion effect, the model (1.1) reduces to the widely known (attractive) Keller-Segel chemotaxis model with nonlinear diffusion: x ∈ Ω, t > 0, (1.3) whose solution behavior has been extensively studied in the past four decades in various perspectives (see the survey articles [12,4] and references therein). More often than not, when D(u) is a constant, the system (1.3) is known as the classical Keller-Segel model. A striking feature of the classical model (1.3) is the blowup of solutions in two or higher dimensions [31,41,42]. While, when D(u) is a nonlinear function, there is a critical exponent θ * = 1 − 2 n which distinguishes between occurrence and impossibility of blow-up. More precisely, the solution of system (1.3) will globally exist [14,15,36,39] if θ > θ * and blow up in finite time [5,7,8,13,30,32,16] if θ < θ * with n ≥ 3; when θ = θ * , based on the availability of Lyapunov functional, it has been proved there exists radially symmetric initial data such that the solution of (1.3) will blow up in finite time [17,22].
Another important subsystem of (1.1) is the following repulsive Keller-Segel model: u t = ∇ · (D(u)∇u) + ξ∇ · (u∇w), x ∈ Ω, t > 0, Based on a Lyapunov functional different from that of the attractive Keller-Segel model (1.3), the global existence of classical solutions in two dimensions and weak solutions in three and four dimensions were established in [6] with D(u) = 1. Further results on a repulsive Keller-Segel model with nonlinear chemosensitivity can be found in [37].
The attraction-repulsion model (1.1) is a mixed combination of the attractive KS model (1.3) and the repulsive one (1.4). Hence, the mathematical analysis on the boundedness and blow-up of solutions offers great challenges due to the complex interactions between the three components u, v and w, and the difficulty of constructing a Lyapunov functional. To motivate our study, we summarize the known results on boundedness versus blow-up for (1.1) in the literature as follows: • τ 1 = τ 2 = τ and D(u) = 1: In one dimensional space, the global existence of classical solutions, non-trivial stationary state, asymptotic behavior and pattern formation of the system (1.1) have been studied in [20,28,27] with τ = 1. In higher dimensions (n ≥ 2), Tao and Wang [38] studied the global solvability, boundedness, blow-up, existence of steady states by introducing the transformation s = ξw − χv. It has been proved that when repulsion dominates or cancels attraction in the sense of ξγ ≥ χα, the global classical solution will exists for both τ = 0 [38] and τ = 1 [18,26]. Whereas, if attraction dominates in the sense of ξγ < χα, then the solution of system (1.1) with τ = 0 will blow up in finite time for large initial mass and exist globally with small initial mass in two dimensional spaces [9,23,43]. The large time behavior of solution with small initial data was established in [24]. Part of the above-mentioned results have been recently carried over to the whole space in [19,35]. However, whether or not the boundedness and blow-up of solution hold for higher dimensions was left as an open problem. • τ 1 = 1, τ 2 = 0: When D(u) = 1, Jin and Wang [21] firstly detected a Lyapunov functional, and then they established the global existence of uniformly-in-time bounded classical solutions in two dimensional bounded domain with large initial data if the repulsion dominates or cancels attraction (i.e., ξγ ≥ αχ). If the attraction dominates (i.e. ξγ < αχ), a critical mass blow-up phenomenon was found. With a nonlinear diffusion D(u), the similar results in higher dimensions were available in [25]; more precisely, therein the authors showed that, for the prototypical choice D(u) = du θ and θ > 1 − 2 n , the corresponding initial-boundary problem possesses a nonnegative globally bounded solution.
On the other hand, if θ < 1 − 2 n and ξγ < χα, there exist some symmetric initial data such that the corresponding solution blow-up in finite time in the case of Ω = B R (0) ⊂ R n (n ≥ 3). For the borderline case θ = 1 − 2 n with n = 3, there exist radially symmetric solutions which may blow up in finite time in Ω = B R (0) with R > 0.

HAI-YANG JIN AND TIAN XIANG
Employing the same Lyapunov functional as in [25] in a close way, when the repulsion dominates or cancels attraction in the sense we establish the refined L 1+θ -boundedness of u compared to the L 1 -boundedness of u as used in [25] as the starting point toward our further bootstrap type argument and then the remaining procedure leading to its L ∞ -boundedness is standard and developed since [39]. Namely, we first use the gained L 1+θ -bound for u and the vequation to improve the regularity of ∇v and then, with an elliptic Agmon-Douglis-Nirenberg L p -estimate applied to the w-equation, the restriction θ > 1− 4 n+2 enables us to derive a Grownwall type inequality for the coupled quantity Ω u p + Ω |∇v| 2q for all large p, q > 2 and, finally, we utilize the well-known Moser-Alikakos type iteration technique (cf. [3] or [39, Lemma A.1]) to conclude the L ∞ -boundedness of u and then W 1,∞ -boundedness of v and w. The mathematical formulation of our main result is stated in the next theorem. Case 2: 1 − 4 n+2 < θ ≤ 1 − 2 n and either {ξγ > χα} or {ξγ = χα, β ≥ δ}. Accordingly, our boundedness especially enlarges the range of θ in [25, Theorem 1.1] provided that the repulsion dominates or cancels attraction in the sense of (1.6). Hence, it exhibits the repulsion effect on boundedness in attraction-repulsion models.
The authors in [25, Remark 1.5] remarked that the set of initial data (u 0 , v 0 , w 0 ) enforcing finite-time blow-up in [25,Theorem 1.2] can be proved to be dense with respect to an appropriate topology. While, under (1.6), such blow-up set of initial data is empty, as ensured by Theorem 1.1.

2.
Local existence and preliminaries. The local existence theorem of (1.1) can be proved by the fixed point theorem and maximum principle along the same line as shown in [38]. We omit the details for convenience.

REPULSION EFFECTS IN AN ATTRACTION-REPULSION CHEMOTAXIS MODEL 3075
By the blowup criterion (2.1) of Lemma 2.1, it suffices to derive u(·, t) L ∞ < ∞ for all t > 0 to obtain the global-in-time solutions. We first notice that L 1 -norm of the solution triple of (1.5) is bounded by integrating equations in (1.5) over Ω.
The following version of Gagliardo-Nirenberg inequality will be used in several places in our upcoming discussions.

Lemma 2.3 (Gagliardo-Nirenberg inequality [10]).
Let Ω be a bounded domain in R n with smooth boundary. Let l and k be any integers satisfying 0 ≤ l < k, and let Then, for any f ∈ W k,q (Ω) ∩ L r (Ω), there exist a constant c depending only on Ω, q, k, r and n such that: with the following exception: if 1 < q < ∞ and k − l − n q is a nonnegative integer, then (2.5) holds only for a satisfying l k ≤ a < 1. We should mention that the original Gagliardo-Nirenberg inequality (e.g. see [33]) is stated only for r ≥ 1, but this condition can be easily relaxed to r ∈ (0, p) by means of the Hölder's inequality (cf. [40,Lemma 3.2] ).
The existence of Lyapunov function for system (1.5) was firstly found in [21] with D(u) = 1, which is further developed in [25] for nonlinear diffusion. We here once again will employ this functional in a close way to obtain the key L 1+θ -boundedness of u.
Lemma 2.4. Let (u, v, w) be the classical solution of (1.5) obtained in Lemma 2.1.
Proof. Multiplying the first equation of (1.5) by G (u) − χv + ξw and integrating the result with respect to x over Ω, we obtain 3076 HAI-YANG JIN AND TIAN XIANG (2.8) A use of integration by part yields It follows from the second equation of (1.5) that u = 1 Similarly, a substitution of the third equation of (1.1) shows which gives rise to The combination of (2.9), (2.10) and (2.12) leads to which together with (2.8) implies (2.7). The proof of Lemma 2.4 is completed.
Employing the energy identity obtained in Lemma 2.4, we next derive L 1+θboundedness of the u-component provided the repulsion dominates or cancels attraction as specified in (1.6), which serves a key starting point towards our further analysis.

REPULSION EFFECTS IN AN ATTRACTION-REPULSION CHEMOTAXIS MODEL 3077
Substituting (2.14) into (2.7), we obtain (2.15) Integrating (2.6) with respect to t and using (2.15), we have (2.16) In the first case of (1.6), i.e., ξγ > χα, using the Gagliardo-Nirenberg inequality (2.3) and the boundedness of v(·, t) L 1 in (2.3), we have (2.17) While, in the second case of (1.6), i.e., ξγ = χα and β ≥ δ, one trivially has Once we gain the refined L 1+θ -boundedness of u as in (2.13), the remaining procedure leading to its L ∞ -boundedness is standard and developed since [39]. To make our presentation self-contained, we would like to offer all the necessary details.
where the following well-known smoothing L p -L q type estimates of (e t∆ ) t≥0 have been used (see [11, formula (16)] or [41, Lemma 1.3]): for 1 ≤ q ≤ p ≤ ∞, one can find c 4 , c 5 > 0 such that as well as Here, λ 1 is the first positive eigenvalue of −∆ under homogeneous boundary condition.
Thanks to the restriction of s in (2.20), we know that and thereby proves (2.21). The proof of this lemma is thus accomplised.

Lemma 2.7.
Let Ω be a bounded domain in R n (n ≥ 2) with smooth boundary. For any ε > 0, p > 1, there exists a constant C > 0 such that for each u ∈ L 1 (Ω), the solution of the elliptic Neumann problem Proof. First, we apply the Agmon-Douglis-Nirenberg L p -estimates (p > 1) [1,2] to the linear elliptic problem (2.23) to find a constant c 1 > 0 such that Then we deduce from the boundedness of w L 1 in (2.4) due to the fact that u ∈ L 1 (Ω), the elliptic estimate (2.24), Gagliardo-Nirenberg inequality stated in Lemma 2.3 and Young's inequality with that where we used the fact p > 1 to guarantee The desired estimate is nothing but (2.25).

REPULSION EFFECTS IN AN ATTRACTION-REPULSION CHEMOTAXIS MODEL 3079
Remark 3. The larger the exponent θ is, the easier the boundedeness of u is. Indeed, if θ + 1 > n, then Lemma 2.6 tells us that ∇v(·, t) L ∞ is bounded. Then (3.1) of Lemma 3.1 quickly shows that u(·, t) L p is bounded for any p > 2. Thus, we directly jump to the proof of Theorem 1.1 on P. 13 to obtain that u(·, t) L ∞ is bounded. Accordingly, we shall proceed henceforth with θ ≤ n − 1.
The condition θ > 1 − 4 n+2 (along with θ ≤ n − 1 as noted in Remark 3) indeed enables us to select certain parameters in an appropriate way such that we have the license to apply Gagliardo-Nirenberg inequality in Lemma 3.2.
3. Proof of Theorem 1.1. In this section, we show the proof of Theorem 1.1; the idea for the proof starts with the following coupled energy estimates.
Let Ω ⊂ R n (n ≥ 2) be a bounded domain with smooth boundary. Suppose that D(u) satisfies (1.2). Assume that p > 2 and q > 2. Then there exist two constants C 1 > 0 and C 2 > 0 such that for all t ∈ (0, T max ), the solution of (1.5) satisfies (3.1) Proof. We multiply the first equation of (1.5) by pu p−1 , and integrate the equation with respect to x over Ω to obtain Now, we use Young's inequality and Lemma 2.7 to estimate the last term in (3.2) as Even with the lower regularity of ∇v L s (s ∈ [1, n n−1 )) compared to that of Lemma 2.6 and the standard technique to control the resulting boundary integral [14], the following estimate associated with the v-equation in (1.5) is quite known, cf. [25,Lemma 3.4]: Then a combination of (3.4) and (3.5) gives rise to (3.1).

REPULSION EFFECTS IN AN ATTRACTION-REPULSION CHEMOTAXIS MODEL 3081
In the sequel, we shall utilize the parameters chosen in Lemma 2.8 to bound the two coupling integrals on the right-hand side in terms of the dissipation terms on its left-hand side. To start off, the Hölder inequality gives that and where λ ∈ (1, n n−2 ) and µ > max{1, n 2 } such that 1 λ The facts λ > 1 and p > 1 + 2θ as in (2.26) On the other hand, from the first inequality of (2.27), we have These two inequalities enable us to apply the Gagliardo-Nirenberg inequality to bound By noting the L 1+θ -boundedness of u by (2.13) and setting due to λ ∈ (1, n n−2 ), we then get from (3.12) that (3.14) Fixing s ∈ [1, n(θ+1) (n−1−θ) + ), using the second inequality in (2.27), (3.11) and the Gagliaro-Nirenberg inequality again, we find a k 2 = q s − q 2λ q s + 1 n − 1 2 ∈ (0, 1) such that This along with the boundedness of ∇v L s as shown in (2.20) of Lemma 2.6 entails implied by (2.29). Similarly, in light of (2.28) and the boundedness of u ∈ (0, 1). Then the Gagliardo-Nirenberg inequality together with the boundedness of ∇v L s implies ∈ (0, 1). (3.20) A collection of (3.9), (3.10), (3.14), (3.15), (3.17) and (3.19) shows that For a > 0 and b > 0 with a + b < 1, a couple of simple uses of Young's inequality with epsilon, for any ε > 0 and X, Y ≥ 0, there exists a constant C ε > 0 such that Solving the Grownwall's inequality (3.26) gives the existence of a constant c 17 such that which yields the desired estimates (3.6) and (3.7). This completes the proof.