GLOBAL EXISTENCE AND OPTIMAL DECAY RATE OF SOLUTIONS TO HYPERBOLIC CHEMOTAXIS SYSTEM IN BESOV SPACES

. In this paper, we study the qualitative behavior of hyperbolic system arising from chemotaxis models. Firstly, by establishing a new product estimates in multi-dimensional Besov space ˙ B d 2 2 ,r ( R d )(1 ≤ r ≤ ∞ ), we establish the global small solutions in multi-dimensional Besov space ˙ B d 2 − 1 2 ,r ( R d ) by the method of energy estimates. Then we study the asymptotic behavior and ob- tain the optimal decay rate of the global solutions if the initial data are small in B d 2 − 1 2 , 1 ( R d ) ∩ ˙ B 01 , ∞ ( R d ).

1. Introduction. In this paper, we consider the Cauchy problem of a hyperbolic system arising from chemotaxis models    ∂ t a − div(aq) = D∆a, x ∈ R d , t ≥ 0, ∂ t q − ∇a − ∆q = ∇(|q| 2 ), x ∈ R d , t ≥ 0, (a, q)| t=0 = (a 0 , q 0 ), Chemotaxis is a biological phenomenon describing the oriented movements of cells and micro-organisms population density in response to an external chemical stimulus that spreads in their environment. The mathematical prototype of chemotaxis models, dating to Keller and Segel [8,9] was proposed to describe the aggregation of cellular slime molds Dictyostelium discoideum. The extensively studied Keller-Segel chemotaxis model takes the following formulation ∂ t u = div(D(u, c)∇u − χ(u, c)∇φ(c)), ∂ t c = (u, c)∆c + g(u, c), where x = (x 1 , x 2 , · · ·, x d ) ∈ R d , u = u(x, t) and c = c(x, t) > 0 are called the cell density and the chemical concentration, respectively. D > 0 and > 0 denote the diffusion rate of the cells(bacteria) and chemical substance, respectively. The function χ(u, c) represents the sensitivity with respect to chemotaxis and χ > 0(< 0) corresponds to attractive(repulsive) chemotaxis. The potential function φ(c), also called chemotactic sensitivity function, describes the signal detection mechanism, and g(u, c) characterizes the chemical growth and degradation. With different choices of φ(c) and g(u, c), various results have been established, see [7,13,11,2] and the references therein. If we assume D and are positive constants and choose χ(u, c) = χ 0 u, φ(c) = ln c and g(u, c) = uc − µc, then the Keller-Segel system (2) takes the following form We consider the repulsive (χ 0 < 0) case for (3), then system (1) is derived from (3) under Hopf-Cole transformation where c = e µt c, together with scalings Now, let us briefly review some results related to our problem. When the diffusion of chemical substance is so small that it is negligible, i.e. = 0, (1) becomes to the hyperbolic-parabolic chemotaxis model and there are many results concerning the qualitative behavior of solutions. Guo et al. [5], Zhang et al. [22] investigated the large-amplitude classical solutions on R without any smallness assumption on the initial data. The long-time behavior of one-dimensional small/large-amplitude classical solutions were established in [22] and [12], respectively. Related results on finite interval in one-dimensional case, we refer the readers to [4,23,17] and the references therein. For multi-dimensional case, Li et al. [11] investigated the local/global existence in H s (R d ) with s > 1 + d 2 . Moreover, blow up criterion and quantitative decay of perturbations of classical solutions were also shown. Hao [6] obtained the global small-amplitude strong solution for initial data close to a constant equilibrium state in critical hybrid Besov spaces B Fan and Zhao [3] extended the local well-posedness of large-amplitude classical solutions in [11] to H s (R d ) with s > d 2 and gave several blow up criteria of local strong solutions.
Compared with the wide study on (1) with = 0, the results on the chemically diffusible mode (i.e. (1) with > 0) are relatively less. Recently, the authors in [14,15,16,10] considered the traveling wave solution of (1) and its nonlinear stability were established on R. The global well-posedness, long-time behavior of one-dimensional classical solutions for large data on finite intervals with Neumann-Dirichlet boundary conditions and Dirichlet-Dirichlet boundary conditions were established in [20,17], respectively. Wang et al. [21] established the global existence, asymptotic decay rates and diffusion convergence rate of small solutions in H k (R d ) with d = 2, 3 and some integer k ≥ 2. Martinez et al. [18] obtained the global asymptotic stability of constant ground states, and showed the explicit decay rate of solutions under very mild conditions on initial data in H 2 (R). The purpose of this paper is to establish the asymptotic behavior (global existence and time decay rate) of solutions of (1) in Besov spaces which is a larger important space containing Sobolev space and L p space. Using the Besov space technique, by establishing a new product estimates in multi-dimensional Besov spaceḂ Our first result of this paper is stated as follows.
2,r ). Moreover, there exists a constant C 0 > 0, and we have Our second result concerns the asymptotic behavior of solutions. The classical The classical Littlewood-Paley decomposition also yields that the low frequency is a natural function space for system (1). Oktia [19] established the perturbations decay for global solutions to the compressible Navier-Stokes equations in critical Besov spaces, if the initial perturbations of density and velocity are small in B respectively. Inspired by Okita's work, our second purpose in this paper is to study the asymptotic behavior for the obtained solutions. Precisely, we have the following decay estimates.
There exist two positive constants σ and C 0 such that if Remark 1. We note that the decay rate of solutions obtained in Theorem 1.2 is optimal in the sense that it attains the decay rate of solutions to the linearized system (see Section 3 and Lemma 4.1). Indeed, by B The rest of this paper is structured as follows. In Section 2 we present some notions and basic tools. In Section 3 we establish a priori estimates for the linearized equations of system (1) which will be crucial in the proof of Theorem 1.1. In Section 4 we prove Theorem 1.1 and Theorem 1.2 in detail.
2. Littlewood-Paley theory and Besov spaces. Let S(R d ) be the Schwartz class of rapidly decreasing functions.
The inverse of F is denoted by F −1 , which is defined by Now we provide a characterization of Besov space based on the Littlewood-Paley decomposition [1].
We start with the dyadic partition of unity. Choose two nonnegative radial functions χ, ϕ ∈ S(R d ), supported in the ball B = {ξ ∈ R d , |ξ| ≤ 4 3 } and in the ring For every f ∈ S (R d ), the homogeneous (or nonhomogeneous) dyadic blocks∆ j (or ∆ j ) and homogeneous (or nonhomogeneous) low-frequency cut-off operatorṠ j (or S j ) are defined as follows and Then we have the formal Littlewood-Paley decomposition in the nonhomogeneous case Unfortunately, for the homogeneous case, the Littlewood-Paley decomposition is invalid. We need a new space to modify it, namely, Thus we have the formal Littlewood-Paley decomposition in the homogeneous case With suitable choice of χ and ϕ, one can easily verify that the Littlewood-Paley decomposition satisfies the property of almost orthogonality: Next we recall Bony's decomposition from [1]: The operators∆ j and ∆ j help us recall the definition of the homogenous Besov spaces and the inhomogenous Besov spaces (see [1]).
The nonhomogeneous Besov space B s p,r is the set of tempered distribution f satisfying ||f || B s p,r = (2 js ||∆ j f || L p ) j r < ∞. Now we present some useful lemmas which will play an important role in the proof of our main results.

Lemma 2.2. ([1]
) Let 1 ≤ p ≤ q ≤ ∞ and B be a ball and C a ring of R d . Assume that f ∈ L p , then for any α ∈ N d , there exists a constant C independent of f , j such that Proof. Using the Bony paraproduct decomposition (9) and the property of quasiorthogonality (8), we havė which means that We shall estimate the above three terms separately. Using the Young inequality and the Hölder inequality, we get I 2 can be estimated in a similar way, and we have For the term I 3 , the Hölder inequality together with the Young inequality yield Taking (12)- (14) into (11) and using the Besov embeddingḂ (ii) Let a, b > 0 and f ∈ L 1 (0, ∞). Then 3. Reformulation and a priori estimate. Due to the physical significance of and D as diffusive parameters, without loss of generality, we assume = D = 1 in what follows. Let p = a −ā, then system (1) can be rewritten as We first consider the following linearized equations of system (15) with a general function (f, g): Let by using operator A, problem (16) is written as We introduce a semigroup associated with A. We set here E d denotes the identity matrix of d dimension.
We have the following lemma. .
Proof. Applying the operator∆ j to Equations (16) 1 and (16) 2 , respectively, yields that here and in the sequel, we always denote φ j =∆ j φ. Taking L 2 inner product of Equations (18) 1 and (18) 2 with p j andāq j , respectively, then integrating by parts, we get Notice the fact that (ādivq j , a j ) = −(∇a j ,āq j ) and use Lemma 2.2, then we get from (19) that Multiplying both sides of (20) by 2 j( d 2 −1) and taking l r norm over j ∈ Z, we infer that Hence, we complete the proof of lemma 3.
Similarly, we have Then, according to Lemma 3.1, we conclude that , which together with condition (4) imply the result in Theorem 1.1 by using a standard continuity argument provided σ is suitable small. Let U (t) = (p(t), q(t)) T be a solution of (16) with (f, g) = (div(pq), ∇(|q| 2 )), then U (t) can be written as the following integral form: where U 0 = (p 0 , q 0 ) T and F (U (t)) = (f, g) T .
To simplify the notation, we set where M (t) = M (t) + M h (t). Then, we have since ∆ j coincides with∆ j for j ≥ 0.
To obtain the decay estimate of Theorem 1.2, we state the estimate of the semigroup operator E(t) associated with A.
Lemma 4.1. Let U 0 = (p 0 , q 0 ) T . Then the operator E(t) satisfies the following estimates Proof. According to the action of semigroup of the heat equation over spectrally supported funtions [1], it can be checked that Due to the fact that [1]: for any σ > 0 there exists a constant C σ such that sup t≥0 k∈Z Firstly using the homogeneous Littlewood-Paley decomposition (7) and then the property of almost orthogonality (8), we arrive at We also find that This implies the first estimate of our result.