GLOBAL EXISTENCE AND LARGE TIME BEHAVIOR OF A 2D KELLER-SEGEL SYSTEM IN LOGARITHMIC LEBESGUE SPACES

. This paper is devoted to the global analysis for the two-dimensional parabolic-parabolic Keller-Segel system in the whole space. By well balanced arguments of the L 1 and L ∞ spaces, we ﬁrst prove global well-posedness of the system in L 1 × L ∞ which partially answers the question posted by Kozono et al in [19]. For the case µ 0 > 0, we make full use of the linear parts of the system to get the improved long time decay property. Moreover, by using the new formulation involving all linear parts, introducing the logarithmic-weight in time to modify the other endpoint space L ∞ × L ∞ , and carefully decompos-ing time into several pieces, we are able to establish the global well-posedness and large time behavior of the system in L ∞ ln × L ∞ .

1. Introduction. We study the Cauchy problem of the two-dimensional (2D) Keller-Segel system where µ 0 ≥ 0, u = u(t, x) and v = v(t, x) are density of amoebae and concentration of chemical attractant respectively, (u 0 , v 0 ) is the given initial value. Chemotaxis is a biological phenomenon describing the change of motion of a cell population density in response to an external chemical stimulus spreads in the environment where the cells reside. Chemotaxis plays essential roles in various biological processes such as embryonic development, wound healing, and disease progression. For the u(t, x), v(t, x) → λ 2 u(λ 2 t, λx), v(λ 2 t, λx) for any λ > 0.
Our first goal is to answer Kozono, Sugiyama and Wachi's question proposed in [19]: whether there exists a solution to (1) even locally in time for (u 0 , v 0 ) ∈ L 1 (R 2 ) × L ∞ (R 2 )? We give an affirmative answer to the existence of small solution in Theorem 1.2 by introducing t u(t, ·) ∞ in the L p -framework, see e.g. [13].
When µ 0 > 0, the chemical concentration decays exponentially for large time and the coupled linear operators yields better a-priori estimates, which indicate that L ∞ (R 2 ) × L ∞ (R 2 ) is another critical Lebesgue space for (1). Indeed, by making full use of the linear part of (1) and by defining where λ ≥ 0 and S 0 (t) = exp{t∆}, we rewrite (1) into integral equations via Duhamel's principle as It is known that a solution to system (4) is called a mild solution of system (1). It is worth pointing out that system (4) captures all the linear information of the original system (1) and it does not treat u as an external force, which is one of our new points in the current paper.
It seems to be an interesting problem that whether global existence is true for system (1) Our second goal is to prove global well-posedness of (1) in a subspace L ∞ ln × L ∞ , a natural substitution of L ∞ × L ∞ . The results are summarized in Theorem 1.3. Now we recall some results concerning the parabolic-elliptic/parabolic-hyperbolic Keller-Segel systems. For the parabolic-elliptic Keller-Segel model where u and v are concentration of cell and chemical, µ, χ, f, g, ν > 0 are constants, Childress and Percus conjectured in [4] that, in a radial symmetric two dimensional domain Ω, there exists a critical number c * such that if Ω u 0 (x)dx < c * , then the radial symmetric solution exists globally in time, and if Ω u 0 (x)dx > c * , then blowup happens. For different versions of the Keller-Segel models, the conjecture has been essentially proved. For a complete review of this topic, we refer the readers to [11] and the references therein, also see e.g. Diaz-Nagai-Rakotoson [7], Blanchet-Dolbeault-Perthame [1]. A parabolic-hyperbolic system which was derived from the Keller-Segel model where u is the concentration of cell, v = ∇c c and c is the concentration of chemical, D > 0, was studied in [31,24] for one dimensional case, was extended to multidimensional cases in [8,23,22], and was studied in [21,28] with a comprehensive qualitative and numerical analysis. We refer the readers to references [5,6,9,15,16,25,27,29,30,33] for more discussions in this direction.
Throughout this paper, let C α,β,··· and c α,β,··· be positive constants which depend on α, β, · · · and which may vary from line to line. We denote for short. For any 1 ≤ q ≤ ∞, we denote L q (R 2 ) by L q . Here and hereafter, we focus on the case n = 2 and denote · L q (R 2 ) by · q .
where 1 ≤ q < ∞. Therefore, any bounded 1 L q function with 1 ≤ q < ∞ belongs to L ∞ ln , for instance, w 1 = ε 2 q ϕ(εx) for small > 0 and ϕ ∈ L 1 . In fact, by standard argument of the characterization theory for Besov space, it is easy to know that sup t≥e ln t S 0 (t)w ∞ is determined only by the low frequency information of w. As a consequence, L ∞ ln can be thought as a set of bounded functions with some L q -perturbations, see (9). Moreover, the Fourier transformation of the perturbations are supported in a bounded domain. For instance, where s is a small positive constant, is a large positive integer, k is a small negative number, andφ(ξ) is compactly supported near the origin with φ(ξ)dξ = 1.
Remark 2. From (10)- (11) we observe that solutions to system (1) with µ 0 > 0 decay faster than that of µ 0 = 0. Moreover, in the case µ 0 > 0, one can also prove the existence of small global mild solution to system (1) of (u 0 , v 0 ) ∈ L q × L ∞ for any 1 < q < ∞, as well as its long time decay estimates. However, it seems quite difficult to establish the desired a priori bilinear estimates with To complete the investigation in Lebesgue space framework, it remains to analyze the other end-point case, i.e., L ∞ × L ∞ . However, as mentioned above, this case seems quite difficult. As a consequence, we modify the L ∞ slightly and introduce the logarithmic bounded space L ∞ ln instead. Luckily, this logarithmic space works and yields the following result.
being the set of weakly-star continuous functions on [0, ∞) valued in Banach space L ∞ ln , and as t → +∞.
Plan of the paper: In Section 2, we give preliminary. We prove Theorem 1.2 in Section 3. In Section 4, we give several important Definitions and Lemmas on the analysis in the logarithmic space L ∞ ln × L ∞ which lead to the proof of Theorem 1.3.
Moreover, for any γ > 0, there exists a positive constant C γ depending only on γ such that .
Proof. We prove (13) first. In order to give the detailed proof, we divide the proof into two cases: 0 < t ≤ 9 and t > 9.
Hence we get where .
The last lemma of this section is a slightly different version of the well-known Picard contraction principle, see [20], Theorem 13.2, p.124. Lemma 2.2. Let µ 0 > 0. Let X × Y, · X + · Y be an abstract Banach product space, and B 0 : X × Y → X and B 1 : X × Y → Y be two bilinear operators. If for any (u, v) ∈ X × Y , there exists a positive constant c such that if then for any (u 0 , v 0 ) satisfying the following system has a solution (u, v) in X × Y . In particular, the solution is the only one such that Proof. For completeness, we sketch the proof. As usual, we first define the map J as follows == "Right hand side of (24)".
Then from (22)-(23), we have Applying (23) to (27), we obtain that J (u, v) maps a bounded ball centered at origin of radius 3 2 A in X × Y into itself.
Next, for any (u 1 , v 1 ), (u 2 , v 2 ) satisfy (25), we derive that In fact, From the definition of J, (26), (24), we have that As a direct consequence of contraction mapping theorem, the existence and uniqueness of solution follow. It remains to prove the inequality on the left of (25). In fact, it follows immediately from Hence we complete the proof.
3.1. Analysis in L 1 × L ∞ . In this subsection, we prove global well-posedness of (1) with (u 0 , v 0 ) ∈ L 1 × L ∞ in the Kato L p -framework. At first, we define the working space as follows: The next lemma is about the estimates for the linear parts of (4).
Lemma 3.1. Assume that µ 0 ≥ 0. For any (u 0 , v 0 ) ∈ L 1 × L ∞ , there exists a positive constant c 0 independent of µ 0 such that for any t > 0 there holds In particular, if µ 0 > 0, then we have and Proof. Noticing that the two-dimensional heat kernel is 1 4πt exp{− |x| 2 4t }, then for any t > 0, 1 ≤ p ≤ ∞, As a direct consequence of (33), 1−exp{−µ0t} µ0t ≤ 1 and the Young's inequality, it follows The remained parts follow in the similar way. Hence we finish the proof.
It remains to establish the key bilinear estimates in X × Y . We denote Lemma 3.2. Assume that µ 0 ≥ 0. There exists a constant c > 0 independent of µ 0 such that Moreover, if µ 0 > 0, then we have Proof. In order to estimate B 0 (u, w) X , we apply (33) to (34) to get Therefore, we complete the proof. 4. Analysis in the logarithmic space L ∞ ln × L ∞ . In this section, we apply the logarithmic time weighted norm to the local well-posedness obtained in Section 2 to get global well-posedness. Define ln(e + t)w(t) ∞ + sup t>0 √ t ln(e + t)∇w ∞ < ∞ .
Lemma 4.1. Assume that u 0 ∈ L ∞ ln and v 0 ∈ L ∞ . Then there exists a positive constant c such that and Proof. From