CHEMOTAXIS MODEL WITH NONLOCAL NONLINEAR REACTION IN THE WHOLE SPACE

. This paper deals with a parabolic-elliptic chemotaxis system with nonlocal type of source in the whole space. It’s proved that the initial value problem possesses a unique global solution which is uniformly bounded. Here we identify the exponents regimes of nonlinear reaction and aggregation in such a way that their scaling and the diﬀusion term coincide (see Introduction). Comparing to the classical KS model (without the source term), it’s shown that how energy estimates give natural conditions on the nonlinearities implying the absence of blow-up for the solution without any restriction on the initial data.


Introduction.
In this work, we analyze qualitative properties of non-negative solutions for the chemotaxis system in dimension n ≥ 3 with linear diffusion given by u t = ∆u − ∇ · (u σ ∇v) + u α 1 − R n u β dx , x ∈ R n , t > 0, u(x, 0) = u 0 (x) ≥ 0, x ∈ R n .
Here v(x, t) expresses the chemical substance concentration and it is given by the fundamental solution v(x, t) = K * u(x, t) = c n R n u(y, t) |x − y| n−2 dy, where c n = 1 n(n − 2)b n , b n = π n/2 Γ(n/2 + 1) , b n is the volume of n-dimensional unit ball. This system without the reaction has been proposed as a model for chemotaxis driven cell movement [13]. Here σ ≥ 1 in chemotaxis is to model the nonlinear aggregation, u α 1 − R n u β dx with α > 1, β > 1 is the reaction term representing nonlinear growth under nonlocal resource consumption of the bacteria [5,14]. Initial data will be assumed Under the initial assumptions, we consider the case σ ≥ 1 and prove that in the following cases, (1) σ + 1 ≤ α and α < 1 + 2β/n, (2) σ + 1 > α and (σ + 1)(n + 2) < 2β + 2α + n, the solution of (1) is unique and global without any restriction on the initial data.
In the follows, we define the exponent p arising from Sobolev inequality [16] and the notation Q T p := 2n n − 2 , Q T := R n × (0, T ) for all T > 0.
Here C 1 is a positive constant only depending on u 0 L q (R n ) but not on t.
Actually, problem (1) contains three terms, the diffusion term ∆u, the nonlocal aggregation term ∇·(u σ ∇v) and the nonlinear growth term u α (1− R n u β dx) (where −u α R n u β dx can be viewed as the death contributing to the global existence), then the competition arises between the diffusion, the death and the aggregation, the growth. Indeed, (1) can be recast as If σ + 1 = α, then the particular nonlinear reaction exponent α = 2β/n + 1 gives the balance of diffusion and aggregation, reaction. In fact, plugging u λ (x, t) = λ n β u(λx, λ 2 t) into (1), it's easy to verify that u λ (x, t) is also a solution of (1) and the scaling preserves the L β norm in space, the diffusion term λ n/β+2 ∆u(λx, λ 2 t) has the same scaling as the aggregation term λ (σ+1)n/β ∇ · (u∇(K * u))(λx, λ 2 t) and the reaction term λ nα/β u α (1 − R n u β dx)(λx, λ 2 t) if and only if α = 2β/n + 1. From observing the rescaled equation we can see that when n(α − 1)/β < 2, for low density (small λ), the aggregation dominates the diffusion thus prevents spreading. While for high density (large λ), the diffusion dominates the aggregation and thus blow-up is precluded. Hence, in this case, the solution will exist globally (Theorem 1.2). On the other hand, if n(α − 1)/β > 2, then the diffusion dominates for low density and the density had infinite-time spreading, the aggregation manipulates for high density and the density has finite time blow-up. Therefore, our conjecture is that there exists finite time blow-up for α − 1 > 2β/n. Moreover, for α − 1 = 2β/n, similar to [3], we guess that there is a critical value for the initial data sharply separating global existence and finite time blow-up.
Moreover, noticing that (8) includes u t , ∆u, u σ+1 , u α , ∇ · (u σ ∇v). Therefore, we detect the following equations are similar to (8): and In order to compare (9), (10) with (8), we take σ + 1 = α (for simplicity) to find the effects of the nonlocal reaction u α 1 − R n u β dx . Concerning (8) and (9), in this paper, we prove that for α < 1 + 2β/n, (1) admits a unique and global solution. While for the Fujita type equation (9), it's known that for α < 1 + 2/n, there is no global solution [6] (For comparison, λ n β u(λx, λ 2 t) in (1) is just the mass invariant scaling). As to (10) and (8), the most remarkable difference is that the mass conservation holds for (10) but not for (8), using this property it's been proved that the solution of (10) exists globally with small initial data [2,4,12,19,20], while (8) has a unique and global solution without any restriction on the initial data. Thus we conclude that the reaction term can prevent blow-up.
In brief, comparing to the above models, the absence of mass conservation (model (10)) and the comparison principle (model (9), (11)) are two obstacles in our model (1) [6,19,20,23]. Besides, the nonlocal reaction makes the key energy estimates more difficult and many tools in the bounded domain can't work in the whole space [7,17,21,23]. In our results, we will use analytical methods in the energy estimates and derive the conditions on α, β, σ for global existence (Theorem 1.2).
The main work is devoted to the global unique solution of model (refstar00) for α > 1, β > 1, σ ≥ 1, with that aim Section 3 considers the local existence and uniqueness of the solution. In Section 4, the a priori estimates are performed and show that −u α R n u β dx plays a crucial role on the global existence, thus complete the proof of Theorem 1.2. Here we split the arguments into several parts strongly depending on the exponents α, β and σ, consequently the uniformly boundedness is obtained by virtue of the Moser iterative method. Section 5 discusses some open questions of Eq. (1).

2.
Preliminaries. We firstly state some lemmas which will be used in the proof of local existence and Theorem 2.
Consider the Cauchy problem then the solution z(x, t) can be expressed from semigroup theory [18] as follows: given by [4,20] z is the Green function associated to the heat equation.
The following lemma is an immediate consequence from Sobolev inequality [16] which will play an important role in the proof of global existence of solutions for equation (1).
Here C(n) is a constant depending on n, C 0 is an arbitrarily positive constant and 3. Local existence and uniqueness. This part concerns local existence of the strong solution of (1). The result is standard, more detailed arguments can be found in [4,20,21].
Remark 1. Here β is chosen to be β > n/2 in order to prove the local existence and the a priori estimates Proposition 2. By Sobolev embedding theorem, Proof of Proposition 1. The proof can be divided into 2 steps.
Step 1 investigates a semilinear parabolic equation and shows the local existence of the strong solution of Eqn. (1).
Step 2 gives the uniqueness of the strong solution.
Step 1 (Local existence). In this step, we show the local existence of the strong solution, the proof is refined in spirit of [4,20]. Here, we denote X T by for some C 1 , C 2 are constants only depending on n, α, β, σ and T > 0 to be determined later in Remark 2. We also define Firstly, we consider where f ∈ X T , then by the weak Young inequality [16] ∇V In addition, by the Maximum principle [15] one has Now we introduce the following problem for 0 < t < T Let u 0 satisfies (4). Assume f ∈ X T , then 1 − R n f β dx is bounded by a constant C u 0 L 1 (R n )∩L ∞ (R n ) which only depending on the initial data, then by virtue of [15,Theorem 9.1] and [6,8,20], equation (22) corresponding to the initial data u 0 has a strong solution u f ∈ W u and can be expressed by where G(·, t) is the Green function as in Lemma 2.1. Next we define a mapping Φ by We show the solution u f is nonnegative as follows. For simplicity, in the following we use u instead of u f . Firstly, multiplying (22) by |u| k−2 u (k > 1) and using Young's inequality we have Plugging it into (25) one has Taking k → ∞ we derive Then using Gronwall's inequality it's obtained that Further integrating (22) over R n arrives at The nonnegativity of u can be obtained by multiplying (22) with u − := − min(u, 0) that 1 2 which directly assures that for all 0 ≤ t < T Furthermore, Φ is a contraction map in L ∞ (0, T ; L n−1 (R n )). In fact, we consider the complete metric space ( from (22) one has The multiplication (35) by |w| n−3 w and using Hölder's inequality give that for β > n/2 Here u and f satisfy αu α−1 = u α 1 −u α 2 and βf β−1 = f β 1 −f β 2 by mean value theorem. Thus (35) follows that

SHEN BIAN, LI CHEN AND EVANGELOS A. LATOS
Applying Gronwall's inequality yields Hence there exists Using Banach's fixed point theorem, we have that Φ has a fixed point Φ(f ) = u f = f ∈ X T * . Iterating the method we prove the existence of the strong solution u of equation Step 2 (Uniqueness). The uniqueness of the strong solution can be shown in the follows. Assume (u 1 , v 1 ) and (u 2 , v 2 ) solve (39) in Q T with initial data (4), then Multiplying (40) with u 1 − u 2 we obtain Here u comes from mean value theorem by By Hölder's inequality and weak Young's inequality [16] we have where L n n−1 w is the weak norm defined in [16, formula (6), pp.107] and Taking (41), (43) and (44) together one has This yields u 1 = u 2 in Q T which implies the uniqueness of solutions. Thus we complete the proof of local existence and uniqueness of the strong solution.
Remark 2. In Proposition 1, the bounded time T in X T can be preestimated as follows Multiplying (46) by rf r−1 (r > 1) obtains that Thus Letting r → ∞ one has Hence from ODE inequality we have that there is a maximum existence time T = T ( u 0 L ∞ (R n ) ) such that f is bounded from above in [0, T ).

Proof of Theorem 1.2.
In this section, we derive the a priori estimates of the strong solution and complete the proof of Theorem 1.2.
where C is a positive constant depending on u 0 L k (R n ) and T max . Especially, when σ + 1 = α, we have that for β ≤ k ≤ ∞ where C only depends on u 0 L k (R n ) not on T max .
Proof of Proposition 2. For the rigorous proof, we should multiply (1) by ku k−1 ψ l , where ψ l is a standard cut-off function. By the limiting process we can justify the following formal calculation. Throughout the proof, we suppose σ + 1 = η.
Step 1 (A priori estimates). Firstly multiplying (1) with ku k−1 (k ≥ 1) one has In order to control the right hand side of (55) by using the two nonnegative terms in the left hand side of (55), we apply Hence taking (56) and (57) together we will conduct further estimates of (55) for and max where we have used the fact that 1 ≤ r is equivalent to k 2 ≤ k . Combining (56) and (57) we infer from (55) that We further assume k > β and use the following interpolation inequalities such that and where To use Young's inequality, we need the following three conditions that Firstly for (63), thanks to the arbitrariness of k , we take
Plugging the above formula into (79) we have which follows the uniformly boundedness in time On the other hand, letting in Lemma 2.2 one has that for n ≥ 1 where k 1 = kr 2 < k. Furthermore, for β < k 1 < k + α − 1 we can take Combining (85) and (87) together yields Substituting (88) into (79) one has It can obtained that for any β Furthermore, for the L ∞ norm, according to (55), we can conduct similar procedures as Step 4 in [1] in terms of R n u k+α−1 dx and R n u k+η−1 dx together on the right hand side of (55) and get Step 3 (L β estimates). For σ + 1 = α, taking k = β in (55) one has It follows that for all 0 < t < T max Step 4 (L k estimates for 1 ≤ k < β + α − 1). By virtue of (79), we have that for any 0 < t < T max and β − (α − 1) < k < β + α − 1, Furthermore, integrating (1) over R n and using (91) and (94) get Thus for any 0 < t < T max u(·, t) L 1 (R n ) ≤ e C u0 L ∞ (R n ) , u0 L β (R n ) ,Tmax Tmax u 0 L 1 (R n ) .
This completes the a priori estimates.
Proposition 2 together with the blow-up criterion (16) allow us to state without further arguments Proof of Theorem 1.2. With the aid of the blow-up criterion (16) and the uniformly boundedness of the solution in Proposition 2, there exists a positive constant C u 0 L β+α−1 (R n ) , u 0 L ∞ (R n ) such that u(·, t) L ∞ (R n ) ≤ C for all 0 < t < ∞.
By Proposition 1 we obtain the desired result. The proof of Theorem 1.2 is completed.

5.
Conclusions. This paper concerns Eq. (1) in terms of different reaction and aggregation exponents. If α ≥ σ + 1, then the growth in reaction dominates and letting α < 1 + 2β/n in the death u α R n u β dx can prevent blow-up. While for α < σ + 1, the aggregation dominates and let the aggregation exponent σ + 1 < (2α + 2β + n)/(n + 2) thus the solution will exist globally. Moreover, if σ + 1 = α, n(α − 1)/β = 2, then u λ (x, t) = λ n β u(λx, λ 2 t) is also a solution of (1) and the scaling preserves the L β norm in space. When n β (α − 1) < 2, for low density (small λ), the aggregation dominates the diffusion thus prevents spreading. While for high density (large λ), the diffusion dominates the aggregation and thus blow-up is precluded. Hence, in this case, the solution will exist globally (Theorem 1.2) and we believe that the solution converges to the stationary solution as time goes to infinity. On the contrary, both global existence and finite time blow-up may occur for n β (α − 1) > 2, so our conjecture is that there exists finite time blow-up for α − 1 > 2β/n. As to the case α − 1 = 2β/n, similar to [3], whether there is a critical value for the initial data sharply separating global existence and finite time blow-up is also unknown. Our result is to be considered as the first step to a more general theory of chemotaxis system with nonlocal nonlinear reaction. This will be a fertile area to explore.