ASYMPTOTIC DECAY FOR THE CLASSICAL SOLUTION OF THE CHEMOTAXIS SYSTEM WITH FRACTIONAL LAPLACIAN IN HIGH DIMENSIONS

. In this paper, we study the generalized chemotaxis system with fractional Laplacian. The existence and the uniqueness of global classical so- lution are proved under the assumption that the initial data are small enough. During the proof, with the help of the ﬁxed point theorem, the asymptotic decay behaviors of u and ∇ v are also shown.


1.
Introduction. This paper is devoted to the study of a parabolic-parabolic Keller-Segel system, involving the fractional Laplacian    ∂ t u + (−∆) s u + ∇ · (u∇v) = 0, x ∈ R n , t > 0, x ∈ R n , t > 0, u(0, x) = u 0 (x), v(0, x) = v 0 (x), x ∈ R n . (1.1) Here, u represents the density of the cells, v stands for the concentration of the chemical. As usual, this model is developed to describe the biological phenomenon chemotaxis with anomalous diffusion. The action of the integro-differential operator (−∆) s , s ∈ (0, 1), on a smooth bounded function v: R n → R is defined by (−∆) s v(x) = γ n P.V. R n v(x) − v(y) |x − y| n+2s dy with γ n := Γ( n+2s 2 ) π (n+2s)/2 , (1.2) and the notation P.V. means that the integral is taken in the Cauchy principle value sense (see for example [18]). When s = 1, system (1.1) is the classical model   It was proposed by Keller and Segel [13] in 1970, which has played an increasing important role in the past four decades. Corrias and Perthame [9,10] proved that small initial data give rise to the global weak solutions. In the integral sense, the weak solution vanish as the heat equation for large time and exhibit a regularizing effect of hyper-contractivity type was shown as well. Moreover, the review [1] by 4004 XI WANG, ZUHAN LIU AND LING ZHOU Bellomo, Bellouquid, Tao and Winkler showed the qualitative analysis of a variety of chemotaxis models, such as the existence of weak solutions, blow-up and asymptotic behavior.
When s ∈ (0, 1), system (1.1) is the fractional chemotaxis model, which was first studied by Escudero in [11]. More precisely, the author focused on the Keller-Segel model as follows    ∂ t u + (−∆) s u + χ∇ · [u∇v] = 0, x ∈ R n , t > 0, −∆v + v = u, x ∈ R n , t > 0, u(0, x) = u 0 (x), x ∈ R n , (1.4) and proved that the model has blowing-up solutions for large initial conditions in dimensions n 2. In dimension n = 1, the author also obtained the global existence with the initial data u 0 ∈ L 1 (R) ∩ H 1 (R). This system has been widely studied by Biler et al. [2,4,5,6,7] and Li et al. [14,15,16], as well. For example, in [2], the conditions for local and global in time existence of positive weak solutions in dimensions n = 2, 3 were obtained. In [6], they deduced the existence of local in time mild solutions and global mild solutions under the small initial data, based on applications of the linear analytic semigroup theory to quasi-linear evolutions equations. In [7], the blow-up of solutions, with regard to the Keller-Segel model with either classical or fractional diffusion in two dimensions, in terms of suitable Morrey spaces norm is derived. Moreover, Li, Rodrigo and Zhang [16] obtained the local existence and uniqueness of solutions, and also attained mass conservation and non-negative solutions. Compared with the former works, we study parabolic-parabolic Keller-Segel system and assume that the models both move by anomalous diffusion. Moreover, the purpose of this paper is to prove the global existence and the uniqueness of the classical solution to system (1.1), as well as to attain the asymptotic decay behavior. Specifically, with the aid of the fixed point theorem, under the small initial data, the existence, the uniqueness are proved and the asymptotic decay behavior of the solution is also obtained. More precisely, we clarify the asymptotic decay behaviors of the W m−n−3,∞ -norm of u and ∇v (see Theorem 3.1).
At last, we present the outline of the paper. In Section 2, we prove some preliminary results. Section 3 deals with the proof of the main results in the paper.

2.
Preliminary. In this section, we will give some results which will be used in the proof of the existence of the solution to the problem (1.1) in the next section. As in [3,8,10], the solution of (1.1) can be written as Here, K t (x) is the heat kernel, which is defined by and Here, C is a positive constant.
We perform the following L 1 -norm of K t (x) combining with Lemma 2.1, which will be used later in the article.
Lemma 2.2. For 0 < γ < 1, we have the following inequality for K t (x) that where C is a constant, not depending on t. (2.10) Using Fourier transformation, the solution of (2.10) can be written as Let N 0 be an integer. Then the following estimates hold and Proof. Using (2.4) and (2.5) with p = ∞, we have (2.14) We obtain from (2.11) and (2.14) that and

XI WANG, ZUHAN LIU AND LING ZHOU
Multiplying the first equation of (2.10) by u, integrating on [0, t] with respect to τ , we find (2.18) The same argument as in the proof of (2.17) gives By Sobolev Embedding Theorem, we have where C is a positive constant. Combining (2.15) with (2.20) we obtain where C is a constant, not depending on t. Again, applying (2.21), (2.16) for w, the conclusions (2.12), (2.13) follow.
Lemma 2.4. Let N 0 be an integer. Assume u = K t (x) * φ(x) is a solution of (2.10), then the following estimate holds where 1 < q p < ∞ and C is a positive constant, not depending on t. Proof.
is a solution of (2.10), then the following estimates hold where C is a positive constant, not depending on t.
Proof. By (2.11) we have and Using (2.4) and (2.6) with p = ∞, we have By Young's inequality and (2.34), we have With the same arguments as in the proof of (2.27), we get Replacing u by ∇u in (2.20) and (2.16), we get and Combining (2.35) with (2.37) we get Combining (2.36) with (2.38) we get
Proof. For multiindex γ, from (2.11), we have by Young's inequality, we deduce Lemma 2.7. Consider the Cauchy problem Given a positive number T > 0. Assume then problem (2.45) has a unique weak solution satisfying u ∈ L 2 (0, T ; H 2s (R n )) and ∂ t u ∈ L 2 (0, T ; L 2 (R n )). (2.47) Moreover, the following estimate holds where the constant C is independent of T .
Proof. Since H s (R n ) is separable, we may take {w k } ∞ k=1 to be a basis of H s (R n ). Given a positive integer m, we will look for a function u m (t) of the form where we hope to select the coefficients g ik (t) (0 t T , k = 1, 2, · · ·, m), so that Using (2.49), (2.50) and (2.51) can be rewritten as According to standard existence theory for ordinary differential equations, there exists a unique solution And then u m defined by (2.49) solves (2.53) for a.e. 0 t T and Next we will give some estimates of u m (t).
(2.70) In order to show that u is a solution of problem (2.45), upon passing to weak limits in L 2 (0, T ; H s (R n )), we set m = m l in (2.50), by (2.69) and (2.70), to find (2.71) Note that w j ∈ H s (R n ), (2.71) can be rewritten as where < ·, · > denotes the pairing of H −s (R n ) and H s (R n ).
Since Hence, (2.47) follows. We need to prove that u satisfies initial data. In fact, from (2.67)-(2.68) we obtain that u m l (0) u(0, x), weakly in L 2 (R n ) as l → ∞. (2.77) Note that (2.52), we get Multiplying (2.45) by u and ∂ t u, respectively, and then integrating from 0 to t with respect to t, we find for 0 t T that where m ∈ N , s ∈ (0, 1), then problem (2.45) possesses a unique weak solution u = u(t, x) satisfying u ∈ L 2 (0, T ; H m+2s (R n )) (2.82) and ∂ t u ∈ L 2 (0, T ; H m (R n )), where the constant C is independent of T .

XI WANG, ZUHAN LIU AND LING ZHOU
3. Existence and asymptotic decay. In this section, we prove the main existence result of the system (1.1) applying the fixed point theorem, meanwhile, obtain the asymptotic decay behaviors of the W m−n−3,∞ norm of u and ∇v.
Theorem 3.1. Assume Assume n 2 and s ∈ ( 2 3 , 1). For every integer m n + 4, there exists a constant E ∈ (0, 1), such that if initial data satisfy then the system (1.1) admits a unique global classical solution (u, v). Moreover, the temporal decay estimates hold for every positive α := m − n − 3 and Proof. We divide the proof into five steps.
where E will be chosen later. Set the distance of any u 1 , u 2 ∈ X m,E as Then, (X m,E , d m,E ) is a nonempty complete metric space. Given the initial data u 0 and v 0 , for any fixedū ∈ X m,E , combining with (2.2), we setv Step 2. Essential inequalities.
As an application, we obtain some estimates onv defined in (3.7), which will be fundamental throughout the sequel.
We have proved that T is a mapping from X m,E to X m,E .
The map T : X m,E → X m,E is a strict contraction.
Combining the above five steps, applying fixed point theorem, we prove that the problem (1.1) has a unique global classical solution. Moreover,(3.24) and (3.13) imply (3.2)-(3.3). This complete the proof of Theorem 3.1.