GLOBAL WEAK SOLUTION AND BOUNDEDNESS IN A THREE-DIMENSIONAL COMPETING CHEMOTAXIS

. We consider an initial-boundary value problem for a parabolic-parabolic-elliptic attraction-repulsion chemotaxis model > 0 in a domain Ω ⊂ R 3 with positive parameters χ,ξ,α,β,γ and δ . It is ﬁrstly proved that if the repulsion dominates in the sense that ξγ > χα , then for any choice of suﬃciently smooth initial data ( u 0 ,v 0 ) the corresponding initial-boundary value problem is shown to possess a globally deﬁned weak solution. To the best of our knowledge, this situation provides the ﬁrst result on global existence of the above system in the three-dimensional setting when ξγ > χα , and extends the results in Lin et al. (2016) [19] and Jin and Xiang (2017) [14] to more general case. Secondly, if the initial data is appropriately small or the repulsion is enough strong in the sense that ξγ is suitable large as related to χα , then the classical solutions to the above system are

It is firstly proved that if the repulsion dominates in the sense that ξγ > χα , then for any choice of sufficiently smooth initial data (u 0 , v 0 ) the corresponding initial-boundary value problem is shown to possess a globally defined weak solution. To the best of our knowledge, this situation provides the first result on global existence of the above system in the three-dimensional setting when ξγ > χα, and extends the results in   [19] and Jin and Xiang (2017) [14] to more general case.
Secondly, if the initial data is appropriately small or the repulsion is enough strong in the sense that ξγ is suitable large as related to χα, then the classical solutions to the above system are uniformly-in-time bounded.

HUA ZHONG, CHUNLAI MU AND KE LIN
x ∈ Ω, t > 0, 0 = ∆w − δw + γu, 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), x ∈ Ω, (1) in an arbitrary bounded smooth domain Ω ⊂ R 3 with sufficiently smooth initial data u 0 (x) and v 0 (x). The above chemotaxis system describes the process of chemotactic interaction between one species (the density denoted by u(x, t)) and two different chemical signals ( the concentration represented by v(x, t) and w(x, t), respectively). More precisely, the positivity of the attractive chemotactic coefficient χ implies the species move towards to the chemical signal v, but gets away from the repulsive signal w because of the positivity of the repulsive chemotactic coefficient ξ. Here the positive parameters α, β, γ and δ represent the production and degradation rates of the chemicals, respectively.
Some experimental examples for (1) can be found in [24] which models the aggregation of microglia in Alzheimer's disease or in [31] to address the quorum sensing effect in the chemotactic movement. This model can be regard as the combination between the following one-species and one-single attractive Keller-Segel chemotaxis model (see [15]) x ∈ Ω, t > 0, (2) and the repulsive system (see [4]) u t = ∆u + ξ∇ · (u∇w), x ∈ Ω, t > 0, τ w t = ∆w − δw + γu, x ∈ Ω, t > 0, where τ = 0, 1. From a mathematic point of view and the actual environment, for the attractive chemotaxis system (2), the solution may blow up in finite time in higher dimensions n ≥ 2 if the initial data is suitable large [28,9,25,38,40,3], although it remains bounded if n = 1 [30] or n = 2 with small initial data [3,8,29]. However, all the solutions of (3) would globally exist whenever n ≥ 2 [4,26,27]. Compared with results in the classical attractive or repulsive chemotaxis system, the solution behavior of (1) could be essentially dominated by the competition of attraction and repulsion: namely, when the repulsion dominates (i.e., ξγ > χα) or the repulsion cancels attraction (i.e., ξγ = χα), the solution remains globally bounded in higher dimensions; In the case attraction dominates (i.e., ξγ < χα), unbounded solutions may exist. With more complicate consideration and for convenience, a general situation for (1) can be written as with τ 1 , τ 2 ≥ 0, and with nonlinear diffusion D satisfying D(s) ≥ D 0 s θ for all s > 0 with some D 0 > 0, and θ ∈ R.
For the simple case τ 1 = τ 2 = 0 and D ≡ 1, quite a comprehensive understanding has been achieved. Accordingly, global smooth and bounded solutions could be shown to exist if ξγ ≥ χα and n ≥ 2 [33,17,6], and the large time behavior of solution for any initial data was studied in [21], while suitable large data admits solutions blowing up in finite time if ξγ < χα and n = 2 [6,16,41].
The respective knowledge is much less complete in fully parabolic situations in the sense that τ 1 = τ 2 = 1 in higher dimensions. Even in the comparatively simple case D ≡ 1 and ξγ > χα seems complex enough so as to allow for global boundedness results only in the two-dimensional setting or global existence of weak solutions in three-dimensional case up to now [11]. The balanced case ξγ = χα is considered in [20] and the global existence of classical solutions for any n ≤ 3 has been shown by semigroup techniques (see also [7,Theorem 1.1]). However, the blow-up phenomenon for ξγ < χα is not clear, although the global classical solutions with uniform-in-time bound was established if u 0 is suitable small if n = 2 [18]. For more details about the global existence of classical solution, asymptotic behavior and pattern formation in one-demensional space, we refer the readers to [12,23,22].
While for the case τ 1 = 1, τ 2 = 0, relying on the Lyapunov functional, the authors in [13] firstly show that the problem (1.4) with D ≡ 1 is globally solvable in the two-dimensional setting if either ξγ ≥ χα, or ξγ < χα and Ω u 0 < 4π χα−ξγ , whereas under the largeness condition that Ω u 0 > 4π χα−ξγ and ξγ < χα, there exists solution which blows up in the finite time. With a nonlinear diffusion D(u) = D 0 u θ , the number θ = 1 − n 2 has been uniquely detected to be the critical blow-up exponent in [19]: namely, if θ > 0, the corresponding initial-boundary problem possesses a nonnegative globally bounded solution, whereas θ ≤ 0, blow up may occur in a ball Ω ⊂ R n (n ≥ 3). However, taking into account that the effect of the repulsion plays an important role in boundedness, the range of θ has been extended in [14] to θ > 1 − 4 n+2 if the repulsion dominates or the repulsion cancels attraction in the sense that either ξγ > χα or ξγ = χα, β ≥ δ.
However, these results are not available for (1) even if n = 3, since θ in (5) always remains positive for any n ≥ 2. The first object in our paper is to develop an approach to make sure that the system (1) is globally solvable in the weak sense when ξγ > χα. As for the initial data, throughout the text we may assume that (u 0 , v 0 ) ∈ C 0 (Ω) are nonnegative, u 0 = 0 and v 0 = 0.
Then there exists at least one pair (u, v, w) of nonnegative functions which forms a global weak solution (in the sense of Definition 2.1 below) of (1).
A nature question connected to regularity is whether or not there exists globally bounded solution for (1). In the following theorem, we could see that whenever the initial data u 0 is suitable small or the repulsion is enough strong, and hence each of solutions becomes eventually smooth and bounded in the three-dimensional setting.
The rest of this paper is organized as follows. In Section 2, we will introduce the definition of a weak solution and give an approximate problem, and some basic properties. In Section 3, some priori estimates are given to prove the existence of global weak solution. Section 4 is devoted to showing the global boundedness of solution under some smallness assumption on the initial data.

2.
Preliminaries. The concept of (global) weak solution for (1) we shall purse in this sequel will be given in the follows.
In order to obtain the existence of global weak solution, we then need to consider the approximate system with some κ > 5 and ∈ (0, 1). The exponent κ > 5 which ensures the global existence of (15) is not optimal and we pick κ > 5 here for our convenience.
In contrast to (1), a fundamental difference between both systems consists in the circumstance that the term − u κ in (15) apparently destroys the construction of the Lyapunov functional, which plays an important role in establishing the behavior of solutions for (1) (see [13]). Therefore, motivated by the ideas in [13, Lemma 4.1], we explore similar appropriately energy structures (see Lemma 3.1) associated with (15) and achieve a priori estimates for solutions (u , v , w ), ∈ (0, 1), which would facilitate a global existence of weak solutions. In deriving the assertion about boundedness, some estimates for the functional z(t) := Ω u 2 (t) + Ω |∇v(t)| 4 would be given if the initial data u 0 is suitable small. Then, with these estimates at hand, we can further achieve eventual boundedness and regularity of (u, v, w).
According to the well-known arguments for parabolic-parabolic or parabolicelliptic logistic-type chemotaxis model (see [39,37,33]), a local existence theory of classical solution will be obtained in the following sense.

Lemma 2.2.
Let Ω ⊂ R n (n ≥ 2) be a bounded domain, and let χ, ξ, α, β, γ and with homogeneous Neumann boundary conditions, for any initial data u 0 and v 0 fulfilling (6) and which solve (16) classically in Ω × (0, T max ), and which are such that Now, we let (u , v , w ) denote the local solution to (15) with the maximal time give by T max, . In view of above criterion for extensibility of local solutions, we can show that T max, = ∞. Lemma 2.3. Let χ, ξ, α, β, γ, δ > 0 and κ > 5. For each ∈ (0, 1) the problem (15) has a (unique) nonnegative global classical solution in the sense that T max, = ∞.
Proof. Given T ∈ (0, T max, ), we first integrate the first equation in (15) to see that is valid for all t ∈ (0, T ). Making use of estimates for the heat semigroup in the variations-of-constants formula for v , we invoke Hölder's inequality to obtain c 3 , c 4 and c 5 > 0 such that To estimate ||∇w (·, t)|| L ∞ (Ω) with t ∈ (0, T ), we need to multiply the first equation by u p−1 with p > 3 and use Young's inequality to find that with c 6 , c 7 and c 8 > 0, where we have used (19) and the following inequality according to [42,Lemma 3.1]. Now integration over time yields c 9 > 0 such that Ω u p ≤ c 9 for all t ∈ (0, T ) with any p > 3 and thus, in conjunction with Agmon-Douglis-Nirenberg estimates (see [1,2]) on linear elliptic equation, proves that with some c 10 > 0. Then one may apply standard a priori estimates techniques to infer the existence of c 11 > 0 such that ||u (·, t)|| L ∞ (Ω) ≤c 11 for all t ∈ (0, T ) and ||v (·, t)|| W 1,∞ (Ω) ≤c 11 as well as ||w (·, t)|| W 1,∞ (Ω) ≤ c 11 for all t ∈ (0, T ).
As a consequence of these and Lemma 2.2, we actually have T max, = ∞.
Finally, let us briefly collect some elementary properties of the solution (u , v , w ) to (15).
Proof. We need to integrate the first equation in (15) over Ω and drop nonnegative term.

3.1.
A priori estimates. In this section we proceed to provide some estimates for the solution (u , v , w ) based on the analysis of the coupled functional under the assumption ξγ > χα. Here we note that since u ≥ 0, it is reasonable for us to use the term Ω (u + 1) ln(u + 1) instead of Ω u ln u in (29).
Proof. First, observing the first equation in (15) can be rewritten as we test this by ln(u + 1) and integrate by parts to see that Here, in the second integral on the right side we again integrate by parts and make use of v t = ∆v − βv + αu to obtain Similarly, the third integral can be estimated as where we recall (25) to find As for the term χ Ω ∆v ln(u + 1), we use Young's inequality to gain and then use (23) with η = ξγ−χα 4χ 2 and with some c 2 = c 2 (χ, ξ, α, β, γ) > 0 Moreover, we take similar procedure to deal with with some c 3 = c 3 (χ, ξ, α, γ, δ) > 0, thanks to (26). Finally, we may choose c 4 = c 4 (χ, ξ, α, γ) > 0 such that where collecting above inequalities imply (30).
Upon integration in time, (30) first provides the following statements on regularity of solution (u , v , w ) under the assumption ξγ > χα.
The following estimates for u are immediate.
The next lemma gives estimates on the derivatives of the second solution component v .
3.2. Passing to the limit. To prepare our subsequent compactness properties of (u , v , w ) by means of the Aubin-Lions lemma, we use Lemmas 3.3-3.5 to obtain the following regularity property with respect to the time variable.
Based on above lemmas and by extracting suitable subsequences in a standard way, we could see the solution of (1) is indeed globally solvable.

Preservation of smallness.
We next address the question how the local solution of (1) obtained from Lemma 2.2 becomes smooth and bounded. A cornerstone for all our subsequence analysis is obtained by constructing ordinary differential equations for the functional z(t) := Ω u 2 (·, t) + Ω |∇v(·, t)| 4 with any t ∈ (0, T max ), thus a smallness condition relating the physically relevant total mass may ensure the local solution in fact becomes globally bounded. The main idea in our proof comes from [36].
Then it follows from Lemma 4.2, applied to t 0 = 0 and arbitrary k > 2, yields c 2 > 0 such that whereupon (91) finally can be derived from (96) by a Moser-type iteration in conjunction with standard parabolic and elliptic regularity arguments.
Proof of Theorem 1.2. The conclusion in Theorem 1.2 follows from Lemmas 2.2 and 4.3.