BOUNDEDNESS OF SOLUTIONS TO A FULLY PARABOLIC KELLER-SEGEL SYSTEM WITH NONLINEAR SENSITIVITY

. This paper deals with the global boundedness of solutions to a fully parabolic Keller-Segel system u t = ∆ u − ∇ ( u α ∇ v ), v t = ∆ v − v + u under non-ﬂux boundary conditions in a smooth bounded domain Ω ⊂ R n . The case of α ≥ max { 1 , 2 n } with n ≥ 1 was considered in a previous paper of the authors [Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity, Discrete Contin. Dyn. Syst. B, 21 (2016), 1317–1327]. In the present paper we prove for the other case α ∈ ( 23 , 1) that if (cid:107) u 0 (cid:107) L nα 2 (Ω) and (cid:107)∇ v 0 (cid:107) L nα (Ω) are small enough with n ≥ 3, then the solutions are globally bounded with both u and v decaying to the same constant steady state ¯ u 0 = 1 | Ω | (cid:82) Ω u 0 ( x ) dx exponentially in the L ∞ -norm as t → ∞ . Moreover, the above conclusions still hold for all α ≥ 2 and n ≥ 1, provided (cid:107) u 0 (cid:107) L nα − n (Ω) and (cid:107)∇ v 0 (cid:107) L ∞ (Ω) suﬃciently small.


1.
Introduction. This paper considers the following fully parabolic Keller-Segel system with nonlinear sensitivity x ∈ Ω, t > 0, ∂u ∂n = ∂v ∂n = 0, x ∈ ∂Ω, t > 0, where Ω is a bounded domain in R n with smooth boundary and α ≥ 2 n . Since 1970 the Keller-Segel system proposed by Keller and Segel [9] has been used as a classical chemotaxis model to describe the interaction between the random diffusion of cells and the aggregation of cells due to chemical signals produced by the cells themselves, where u = u(x, t) and v = v(x, t) denote the cell density and the signal concentration respectively.
The classical parabolic-parabolic Keller-Segel system with α = 1 in (1) together with the corresponding parabolic-elliptic form with the second equation replaced by 0 = ∆v − v + u have been well studied with various significant dynamics properties of solutions achieved. Refer to the currently published survey [1] and the references therein. Also see, e.g., [3,4,5,7,8,13,15,16] for results to quasilinear Keller-Segel systems.
Differently, there is a nonlinear sensitivity u α (when α = 1) contained in the fully parabolic Keller-Segel system (1). The more general form is x ∈ Ω, t > 0, (2) where f ∈ C 1+l ([0, ∞)) with f (0) = 0, for which it was shown that if f (s) ≤ cs α for s ≥ 1, 0 < α < 2 n and n ≥ 1, then the solution (u, v) must be globally bounded for arbitrary value of m : n and n ≥ 2, then the system admits unbounded solutions [6]. Moreover, for the case f (u) = u α in (1) with α ≥ max{1, 2 n } and n ≥ 1, the global boundedness of solutions was established in a previous paper of the authors [17], similarly to the case of α = 1 [2,14], by using a new norm to describe the required smallness of (u 0 , v 0 ) instead. Precisely, it was shown that with notations the smallness of u 0 L q * (Ω) and ∇v 0 L p * (Ω) ensures the global boundedness of (u, v) with both u and v converging toū The case of α < 1 in (1) is more difficult, where s α ∈ C 1+l ([0, ∞)), and even does not satisfy the Lipschitz continuity. Rather than the case α ≥ 1, the existence of classical solutions with α < 1 is unclear yet, although that has been obtained for a parabolic-elliptic Keller-Segel system with α < 1 and the second equation replaced by −∆v = u in (1) [11,12]. Recently, a mild solution was established to (1) for α < 1 [10]. The properties of these mild solutions can be found in Section 4.
The goal of this paper is to extend our global boundedness results for (1) obtained in [17] with the following two points.
• The global boundedness of classical solutions under the critical K = nα − n and thus p * = ∞ with α ≥ 2, corresponding to the minimal q * = nα − n. • The global boundedness of mild solutions with 2 n ≤ α < 1 and n ≥ 3 (for which a new norm is necessary to describe the smallness of initial data). Denote by λ 1 > 0 the first nonzero eigenvalue of −∆ in Ω under the homogeneous Neumann boundary condition. The main results of the paper are the following two theorems.
then (1) possesses a global classical solution (u, v) with the decay estimates for t > 0 and some C 1 > 0.
Remark 1. It can be found that the involved norm for α < 1 in Theorem 1.2 is unique, rather than those represented in (3) for the case of α > 1, and there is a gap left for α ∈ [ 2 n , 2 3 ] when n ≥ 3.

2.
Preliminaries. We begin with the known L p -L q estimates for the Neumann heat semigroup on bounded domains as preliminaries, represented in the following two lemmas.
. Next, we will prove Theorems 1.1 and 1.2 in the next two sections respectively. The main techniques are motivated by those in [2,14].
3. Proof of Theorem 1.1. Suppose K = nα − n and thus q * = nα − n, p * = ∞ in (3) with α ≥ 2. The local existence of classical solutions of (1) with α ≥ 1 has been proved in [6, Theorem 3.1]. To establish the global boundedness of solutions, we introduce an important elementary inequality to treat the nonlinearity u α with α ≥ 1. The proof of Theorem 1.1 relies on the following proposition.
then (1) possesses a global classical solution satisfying for t > 0 and some Proof. The framework of the proof is similar to that for [17, Proposition 2] of ours.