Comptes Rendus Mathématique

. This paper deals with the chemotaxis system with nonlinear signal secretion (cid:40) u t =∇· ( D ( u ) ∇ u − S ( u ) ∇ v ), x ∈ Ω , t > 0, v t = ∆ v − v + g ( u ), x ∈ Ω , t > 0, under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ (cid:82) n ( n ≥ 2). The di ﬀ usion function D ( s ) ∈ C 2 ([0, ∞ )) and the chemotactic sensitivity function S ( s ) ∈ C 2 ([0, ∞ )) are given by D ( s ) ≥ C d (1 + s ) − α and0 < S ( s ) ≤ C s s (1 + s ) β − 1 forall s ≥ 0with C d , C s > 0and α , β ∈ (cid:82) .Thenonlinearsignalsecretion function g ( s ) ∈ C 1 ([0, ∞ )) is supposed to satisfy g ( s ) ≤ C g s γ for all s ≥ 0 with C g , γ > 0. Global boundedness of solution is established under the speciﬁc conditions: 0 < γ ≤ 1 and α + β < min (cid:189) 1 + 1 n ,1 + 2 n − γ (cid:190) . The purpose of this work is to remove the upper bound of the di ﬀ usion condition assumed in [9], and we also give the necessary constraint α + β < 1 + 1 n , which is ignored in [9, Theorem 1.1].

under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n (n ≥ 2).The diffusion function D(s) ∈ C 2 ([0, ∞)) and the chemotactic sensitivity function S(s) ∈ C 2 ([0, ∞)) are given by D(s) ≥ C d (1+s) −α and 0 < S(s) ≤ C s s(1+s) β−1 for all s ≥ 0 with C d ,C s > 0 and α, β ∈ R. The nonlinear signal secretion function g (s) ∈ C 1 ([0, ∞)) is supposed to satisfy g (s) ≤ C g s γ for all s ≥ 0 with C g , γ > 0. Global boundedness of solution is established under the specific conditions: The purpose of this work is to remove the upper bound of the diffusion condition assumed in [9], and we also give the necessary constraint α + β < 1 + 1 n , which is ignored in [9,Theorem 1.1].

Introduction
In the present work, we consider the following system, which describes the fully parabolic chemotaxis system with nonlinear diffusion, sensitivity and signal secretion with homogeneous Neumann boundary conditions, where Ω ⊂ R n (n ≥ 2) is a bounded domain, and ∂/∂ν is the derivative of the normal with respect to ∂Ω.In system (1), u = u(x, t ) and v = v(x, t ) represent the density of population and the concentration of chemicals, respectively.In this article, the diffusion function D ∈ C 2 ([0, ∞)) and the chemotactic sensitivity function S ∈ C 2 ([0, ∞)) with S(0) = 0 are given by The well-known chemotaxis model for the chemotactic movement of one specie [4] proposed by Keller and Segel, which describes the aggregation phenomenon of the Dictyostelium discoideum, there are many results about this system [1,3,9,12,13,15,16].For instance, in case g (u) = u, the asymptotics of S(u) D(u) u 2 n is critical to distinguish the blow-up and global boundedness: under the condition S(u) D(u) ≤ cu 2 n − for all u > 1 with > 0, Tao and Winkler [12] obtained the global boundedness of solution; while if S(u) D(u) ≤ cu 2 n + for all u > 1 [16], the solution of (1) blow-up either in infinite time or finite time.We note that in [9], global boundedness of solution is established under the conditions that α + β + γ < 1 + 2 n and d 0 (1 + u) α ≤ D(u) ≤ d 1 (1 + u) α 1 with d 0 , d 1 > 0 and α, α 1 ∈ R. The purpose of this work is to remove the upper bound of the diffusion condition and give the necessary constraint α + β < 1 + 1 n that is ignored in [9,Theorem 1.1].The main result of this article is described below.

Theorem 1.
Let Ω ⊂ R n (n ≥ 2) be a smooth bounded domain.The nonnegative initial data (u 0 , v 0 ) ∈ C 0 (Ω) ×C 1 (Ω).Assume that (2) and (3) hold.If 0 < γ ≤ 1 and then system (1) possesses a unique global bounded classical solution (u, v) in the sense that there exists some constant C > 0 satisfying Remark 2. Compared with the previous study in [9, Theorem 1.1], we give the necessary constraint α + β < 1 + 1 n that is ignored in it, and we also remove the restriction on the upper bound of the diffusion function D(s).
Assume that (2) and (3) hold, then there exists t ∈ (0, T max ) such that system (1) has a unique non-negative solution and satisfies where T max denotes the maximal existence time.
In order to obtain the global boundedness of solution to system (1), we first establish a series of prior estimates; then we treat the dissipative terms on the right hand side of the inequality by using the Gagliardo-Nirenberg inequality; last, we get our final results by controlling the parameter range in the inequality.The ideas come from [9,[12][13][14].
Assume that (2) and (3) hold, then the first term of the solution to system Furthermore, assume that 0 Proof.Integrating the first equation of (1) over Ω, (4) can be easily obtained.From the Neumann semigroup estimates method in [5, Lemma 1], ( 5) can be obtained.
Before we give the result of main part, we first select the appropriate parameters.
Proof of Theorem 1.In view of [12, Lemmas 3.3 and A.1], we obtain the desired results.

Funding.
This work is supported by the Chongqing Research and Innovation Project of Graduate Students (No. CYS20271) and Chongqing Basic Science and Advanced Technology Research Program (No. cstc2017jcyjXB0037).