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Comptes Rendus. Mathématique
Partial differential equations
Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production
Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 161-168.

This paper deals with the chemotaxis system with nonlinear signal secretion

u t =·(D(u)u-S(u)v),xΩ,t>0,v t =Δv-v+g(u),xΩ,t>0,

under homogeneous Neumann boundary conditions in a bounded domain Ω n (n2). 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 s0 with C d ,C s >0 and α,β. The nonlinear signal secretion function g(s)C 1 ([0,)) is supposed to satisfy g(s)C g s γ foralls0 with C g ,γ>0. Global boundedness of solution is established under the specific conditions:

0<γ1andα+β<min1+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].

Received:
Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/crmath.148
Classification: 35K35,  35A01,  35B44,  35B35,  92C17
Xu Pan 1; Liangchen Wang 1

1. School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, PR China
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     title = {Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production},
     journal = {Comptes Rendus. Math\'ematique},
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     year = {2021},
     doi = {10.5802/crmath.148},
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Xu Pan; Liangchen Wang. Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production. Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 161-168. doi : 10.5802/crmath.148. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.148/

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