Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production

Abstract

This paper deals with the chemotaxis system with nonlinear signal secretion { ut =∇· (D(u)∇u −S(u)∇v), x ∈Ω, t > 0, vt =∆v − v + g (u), x ∈Ω, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ Rn (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) ≥ Cd (1+s)−α and 0 < S(s) ≤Cs s(1+s)β−1 for all s ≥ 0 with Cd ,Cs > 0 andα,β ∈R. The nonlinear signal secretion function g (s) ∈ C 1([0,∞)) is supposed to satisfy g (s) ≤ Cg sγ for all s ≥ 0 with Cg ,γ > 0. Global boundedness of solution is established under the specific conditions: 0 < γ≤ 1 and α+β< min { 1+ 1 n ,1+ 2 n −γ } . 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]. Mathematical subject classification (2010). 35K35, 35A01, 35B44, 35B35, 92C17. 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). Manuscript received 1st September 2020, revised 9th October 2020 and 9th December 2020, accepted 10th December 2020. ∗Corresponding author. ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 162 Xu Pan and Liangchen Wang