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

Xu Pan; Liangchen Wang

doi:10.5802/crmath.148

Équations aux dérivées partielles

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

Xu Pan ¹ ; Liangchen Wang ¹

¹ School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, PR China

Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 161-168.

Résumé

This paper deals with the chemotaxis system with nonlinear signal secretion

\{\begin{matrix} u_{t} = \nabla \cdot (D (u) \nabla u - S (u) \nabla v), & x \in Ω, t > 0, \\ v_{t} = Δ v - v + g (u), & x \in Ω, t > 0, \end{matrix}

under homogeneous Neumann boundary conditions in a bounded domain $Ω \subset ℝ^{n}$ ( $n \geq 2$ ). The diffusion function $D (s) \in C^{2} ([0, \infty))$ and the chemotactic sensitivity function $S (s) \in C^{2} ([0, \infty))$ are given by $D (s) \geq C_{d} {(1 + s)}^{- α}$ and $0 < S (s) \leq C_{s} s {(1 + s)}^{β - 1}$ for all $s \geq 0$ with $C_{d}, C_{s} > 0$ and $α, β \in ℝ$ . The nonlinear signal secretion function $g (s) \in C^{1} ([0, \infty))$ is supposed to satisfy $g (s) \leq C_{g} s^{γ} for all s \geq 0$ with $C_{g}, γ > 0$ . Global boundedness of solution is established under the specific conditions:

0 < γ \leq 1 and α + β < min \{1 + \frac{1}{n}, 1 + \frac{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 $α + β < 1 + \frac{1}{n}$ , which is ignored in [9, Theorem 1.1].

Reçu le : 2020-09-01
Révisé le : 2020-10-09
Accepté le : 2020-12-10
Publié le : 2021-03-17

DOI : 10.5802/crmath.148

Classification : 35K35, 35A01, 35B44, 35B35, 92C17

Affiliations des auteurs :

Xu Pan ¹ ; Liangchen Wang ¹

¹ School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, PR China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2021__359_2_161_0,
     author = {Xu Pan and Liangchen Wang},
     title = {Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {161--168},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {2},
     year = {2021},
     doi = {10.5802/crmath.148},
     language = {en},
}

TY  - JOUR
AU  - Xu Pan
AU  - Liangchen Wang
TI  - Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 161
EP  - 168
VL  - 359
IS  - 2
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.148
LA  - en
ID  - CRMATH_2021__359_2_161_0
ER  -

%0 Journal Article
%A Xu Pan
%A Liangchen Wang
%T Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production
%J Comptes Rendus. Mathématique
%D 2021
%P 161-168
%V 359
%N 2
%I Académie des sciences, Paris
%R 10.5802/crmath.148
%G en
%F CRMATH_2021__359_2_161_0

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/

Bibliographie
Cité par

[1] Nicola Bellomo; Abdelghani Bellouquid; Youshan Tao; Michael Winkler 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] Dorothee D. Haroske; Hans Triebel Distributions, Sobolev Spaces, Elliptic Equations, EMS Textbooks in Mathematics, European Mathematical Society, 2008 | Zbl

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

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

[5] Remigiusz Kowalczyk; Zuzanna Szymańska 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] Noriko Mizoguchi; Philippe Souplet 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] Louis Nirenberg An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 20 (1966), pp. 733-737 | Numdam | MR | Zbl

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

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

[12] Youshan Tao; Michael Winkler 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] Liangchen Wang; Yuhuan Li; Chunlai Mu 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] Liangchen Wang; Chunlai Mu; Xuegang Hu; Pan Zheng 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] Liangchen Wang; Chunlai Mu; Pan Zheng 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] Michael Winkler 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] Michael Winkler; Youshan Tao 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] Qingshan Zhang; Xiaopan Liu; Xiaofei Yang 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

Cité par Sources :

Commentaires - Politique