LARGE TIME BEHAVIOR IN THE LOGISTIC KELLER-SEGEL MODEL VIA MAXIMAL SOBOLEV REGULARITY

. The fully parabolic Keller-Segel system with logistic source (cid:26) u t , ( ,T , τv ( x,t (0 ,T ) , ( (cid:63) ) is considered in a bounded domain Ω ⊂ R N ( N ≥ 1) under Neumann boundary conditions, where κ ∈ R , µ > 0, χ > 0 and τ > 0. It is shown that if the ratio χµ is suﬃciently small, then any global classical solution ( u,v ) converges to the spatially homogenous steady state ( κ + µ , κ + µ ) in the large time limit. Here we use an approach based on maximal Sobolev regularity and thus remove the restrictions τ = 1 and the convexity of Ω required in [17].

Going beyond the boundedness results, the study of global dynamic is a natural continuation [17], we refer to [10,12,2] for Keller-Segel model including multiple species. We note that (1) can be seen as a subsystem in a multiple species model. In the case τ = 0, the results from [10,12,2] can be summarized as: if the quotient χ µ is suitably small, (1) admits a global classical solution and it converges to ( κ µ , κ µ ). Considering the fully parabolic system, that is τ > 0, [17] proves the same conlusion under the restrictions that τ = 1 and Ω is convex, which are quite critical in the proof. Under these assumptions, the combination y(x, t) = u + χ 2 |∇v| 2 satisfies a scalar parabolic inequality with some C > 0 for all t > 0 [13]. The comparison principle immediately yields that With this information, one can finally show convergence combined on the basis of estimates for the Neuman semigroup. However, if τ = 1, the first step already fails; we can not find any combination like y(x, t) satisfying a single parabolic inequality on its own. In a recent paper [1], the authors develop a functional approach to prove convergence for global bounded solutions if χ 2 µ is small. This approach also works for τ = 1.
It is our purpose in the present paper to investigate how the size of the quotient χ µ affects the global dynamics for any choice of τ > 0. We find a replacement of (4): with sufficiently large p and for some C > 0, which is sufficient for the conlclusion in [17]. Our main result reads as follows: Then there exists θ 0 > 0 with the property that if then for all initial data (u 0 , v 0 ) fulfilling (2), the system (1) possesses a global classical solution Moreover, (u, v) satisfies 2. Preliminaries. Before going into details, we first introduce the following local existence result for (1). The proof can be found in many previous work [15].
, is a bounded domain with smooth boundary, µ > 0 and χ > 0, and u 0 ∈ C 0 (Ω) and v 0 ∈ W 1,q (Ω) (with some q > N ) both are nonnegative. Then there exist T max ∈ (0, ∞] and a pair nonnegative The main ingredient in this paper heavily relies on the maximal Sobolev regularity with time weighted function, as obtained in [4,18].
We consider the following evolution equation For each v 0 ∈ W 1,r (Ω) and any u ∈ L r ((0, T ); L r (Ω)), there exists a unique solution t0 Ω e r τ s u r dxds 3. Large time behavior of L p norm. As already mentioned in the introduction, our first and the most important goal is to identify the large time behavior of u(·, t) L p (Ω) . The proof is very similar to that of Lemma 3.1 in [18]. We have Proof. First we see that for any a, b > 0, Young's inequality provides k p > 0 such that Let C r+1 denote the constant from Lemma 2.2 for r ∈ (1, ∞). Now we can find θ 1 > 0 small enough such that

XINRU CAO
We multiply the first equation in (1) by u p−1 to obtain that for all t ∈ (0, T max ). Now Young's inequality (14) yields that We see that (25-27) imply for all t ∈ (0, ∞). Let s 0 ∈ (0, ∞), using the Gronwall's inequality to the above inequality we obtain that for all t ∈ (t 0 , T max ). An application of Lemma 2.2 implies for all t ∈ (t 0 , T max ). In view of the condition (15), we see that the term −pµ( for all t ∈ (t 0 , T max ). This implies (12). Suppose that T max = ∞. Letting t to ∞, we obtain that lim sup t→∞ Ω u p (·, t) ≤ C µ p with some C > 0. Taking the p-th root on both sides, we finish the proof.

4.
Large time behavior of L ∞ norm. Using the variation of constants formula to the second equation in (1) and the L p -L q estimate for the Neumann semigroup, we readily have the following: for some C 1 > 0. Then for all there exists C 2 > 0 such that Proof. Assume that p ∈ ( N 2 , N ) without loss of generality. First we find r < N p N −p , c 1 > 0 and t 0 > 0 such that for all t > t 0 . Let s 0 ∈ (t 0 , ∞). Using the constants formula for the first equation in (1), we have for all t ∈ (s 0 , s 0 + 2). We begin with t s0 for all t ∈ (s 0 , s 0 + 2). By the L p -L q estimate for the Neumann heat semigroup, there exists a constant k 1 > 0 fulfilling for all t ∈ (s 0 , s 0 + 2). Let q satisfy 1 q ∈ ( 1 r , 1 N ), we can find r > q such that 1 q = 1 r + 1 r , and a = 1− 1 r ∈ (0, 1). Using the Hölder inequality and the interpolation inequality, the second term can be estimated as . Now we collect the above estimates (24-27) to see that M (t). Because 1 2 − N 2q + N 2p (1 − a) > 0, we take the supremum on both sides of the above inequality to obtian that Since a < 1, this implies that with some C 2 > 0. It also holds that for all t ∈ (s 0 + 1, s 0 + 2). According to the choice of s 0 , we conclude the assertion.
Applying the constants variantion formula to the first equation in (28) and employing the same argument used in [17,Lemma 4.2], we show that with some c 2 > 0. We again follow the idea of [17,Lemma 4.3] to find that lim sup with some c 3 > 0. By the embedding theorem (30), there is c 4 > 0 fulfilling The proof is complete.

5.
Refined estimate for u. In this section, we show that after suitably large time, u lies in a neighborhood of κ µ whose radius is measured by θ. We can prove it by using maximum principle and the pointwise bound of ∆v. for some C 1 > 0. Then there exists C 2 > 0 such that Proof. By the assumption, we can find t 0 > 0 and c 1 > 0 fulfilling We use (41) and the first equation in (1) to estimate that for all x ∈ Ω and t > t 0 , where we use θ = χ µ . Let z := z(t) be the solution to It is easy to see that z(t 0 ) > 0 by the strong maximum principle. The comparison principle implies Thus we can derive that This leads to Similarly ( see also in [17] ), using the lower bound of ∆v in the first equation in (1) and constructing subsolution by the corresponding ODE show that lim sup Combining (45) and (46), we establish (40).
6. Decay of (U, V ). In the last section, we prove that U is in a neighborhood of 0 after suitably large time. This enables us to show that U in fact decays in the large time limit if θ is sufficiently small. At the same time, the decay of V is also obtained. Letting λ 1 be the first non-zero eigenvalue of −∆ associated with Neumann boundary conditions, we have the following: Lemma 6.1. Suppose that κ = 0. Let 0 < ζ < min{ 1 τ , λ 1 }, θ := χ µ . If for some C 1 > 0, it holds that Then there exist θ 2 > 0, C > 0 such that if θ < θ 2 , for each solution of (1) with initial data fulfilling (2), we have (U, V ) defined as (28) satisfies for all t ≥ 0.
Proof. In view of the condition, we infer from Lemma 3.1 that u(·, t) L p (Ω) (p > N 2 ) is bounded, this implies the global boundedness of (u, v) by standard iteration [1,Lemma 2.6]. For such global bounded solutions, [5, Theorem 1.1] implies the assertion.