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:
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DOI: 10.5802/crmath.148
Classification: 35K35, 35A01, 35B44, 35B35, 92C17
Pan, Xu 1; Wang, Liangchen 1

1 School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, PR China
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Pan, Xu; Wang, Liangchen. 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. http://www.numdam.org/articles/10.5802/crmath.148/

[1] Bellomo, Nicola; Bellouquid, Abdelghani; Tao, Youshan; Winkler, Michael Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., Volume 25 (2015) no. 9, pp. 1663-1763 | DOI | MR | Zbl

[2] Haroske, Dorothee D.; Triebel, Hans Distributions, Sobolev Spaces, Elliptic Equations, EMS Textbooks in Mathematics, European Mathematical Society, 2008 | Zbl

[3] Horstmann, Dirk; Winkler, Michael Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equations, Volume 215 (2005) no. 1, pp. 52-107 | DOI | MR | Zbl

[4] Keller, Evelyn F.; Segel, Lee A. Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., Volume 26 (1970) no. 3, pp. 399-415 | DOI | MR | Zbl

[5] Kowalczyk, Remigiusz; Szymańska, Zuzanna On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., Volume 343 (2008) no. 1, pp. 379-398 | DOI | MR | Zbl

[6] Mizoguchi, Noriko; Souplet, Philippe Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 31 (2014) no. 4, pp. 851-875 corrigendum ibid. 36 (2019), no. 4, p. 1181 | DOI | Numdam | MR | Zbl

[7] Nirenberg, Louis An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 20 (1966), pp. 733-737 | Numdam | MR | Zbl

[8] Stinner, Christian; Tello, José Ignacio; Winkler, Michael Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., Volume 68 (2014) no. 7, pp. 1607-1626 | DOI | MR | Zbl

[9] Tao, Xueyan; Zhou, Shulin; Ding, Mengyao Boundedness of solutions to a quasilinear parabolic-parabolic chemotaxis model with nonlinear signal production, J. Math. Anal. Appl., Volume 474 (2019) no. 1, pp. 733-747 | MR | Zbl

[10] Tao, Youshan Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., Volume 381 (2011) no. 2, pp. 521-529 | MR | Zbl

[11] Tao, Youshan; Wang, Zhi-An Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., Volume 23 (2013) no. 1, pp. 1-36 | MR | Zbl

[12] Tao, Youshan; Winkler, Michael Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equations, Volume 252 (2012) no. 1, pp. 692-715 | MR | Zbl

[13] Wang, Liangchen; Li, Yuhuan; Mu, Chunlai Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., Volume 34 (2014) no. 2, pp. 789-802 | DOI | MR | Zbl

[14] Wang, Liangchen; Mu, Chunlai; Hu, Xuegang; Zheng, Pan Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant, J. Differ. Equations, Volume 264 (2018) no. 5, pp. 3369-3401 | DOI | MR | Zbl

[15] Wang, Liangchen; Mu, Chunlai; Zheng, Pan On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differ. Equations, Volume 256 (2014) no. 5, pp. 1847-1872 | DOI | MR | Zbl

[16] Winkler, Michael Does a “volume-filling effect” always prevent chemotactic collapse?, Math. Methods Appl. Sci., Volume 33 (2010) no. 1, pp. 12-24 | DOI | MR | Zbl

[17] Winkler, Michael; Tao, Youshan Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst., Volume 20 (2015) no. 9, pp. 3165-3183 | DOI | MR | Zbl

[18] Zhang, Qingshan; Liu, Xiaopan; Yang, Xiaofei Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals, J. Math. Phys., Volume 58 (2017) no. 11, 111504, 9 pages | MR | Zbl

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