Global boundedness of solutions to a chemotaxis consumption model with signal dependent motility and logistic source

Khadijeh Baghaei

doi:10.5802/crmath.605

Article de recherche - Équations aux dérivées partielles

Global boundedness of solutions to a chemotaxis consumption model with signal dependent motility and logistic source
[Limite globale des solutions d’un modèle de consommation de chimiotaxie avec motilité dépendante du signal et source logistique]

Khadijeh Baghaei

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1131-1145.

Résumé
Abstract

Cet article porte sur le système de chimiotaxie suivant :

\{\begin{matrix} u_{t} = \nabla \cdot (γ (v) \nabla u - u ξ (v) \nabla v) + μ u (1 - u), & x \in Ω, t > 0, \\ v_{t} = Δ v - u v, & x \in Ω, t > 0, \end{matrix}

sous des conditions aux limites homogènes de Neumann dans un domaine borné $Ω \subset ℝ^{n}, n \geq 2,$ avec une frontière lisse. Ici, les fonctions $γ (v)$ et $ξ (v)$ sont les suivantes :

γ (v) = {(1 + v)}^{- k} a n d ξ (v) = - (1 - α) γ^{'} (v),

où $k > 0$ and $α \in (0, 1) .$

Pour le système ci-dessus, nous prouvons que le problème de valeur limite initiale correspondant admet une unique solution classique globale qui est uniformément bornée en temps. Ce résultat est obtenu sous certaines conditions sur la valeur initiale $v_{0}$ et $μ$ et sans restriction sur $k$ et $α .$ Le résultat obtenu étend les résultats récents obtenus pour ce problème.

This paper deals with the following chemotaxis system:

\{\begin{matrix} u_{t} = \nabla \cdot (γ (v) \nabla u - u ξ (v) \nabla v) + μ u (1 - u), & x \in Ω, t > 0, \\ v_{t} = Δ v - u v, & x \in Ω, t > 0, \end{matrix}

under homogeneous Neumann boundary conditions in a bounded domain $Ω \subset ℝ^{n}, n \geq 2,$ with smooth boundary. Here, the functions $γ (v)$ and $ξ (v)$ are as:

γ (v) = {(1 + v)}^{- k} a n d ξ (v) = - (1 - α) γ^{'} (v),

where $k > 0$ and $α \in (0, 1) .$

For the above system, we prove that the corresponding initial boundary value problem admits a unique global classical solution which is uniformly-in-time bounded. This result is obtained under some conditions on initial value $v_{0}$ and $μ$ and without any restriction on $k$ and $α .$ The obtained result extends the recent results obtained for this problem.

Reçu le : 2023-08-09
Révisé le : 2023-12-20
Accepté le : 2024-01-01
Publié le : 2024-11-05

DOI : 10.5802/crmath.605

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2024__362_G10_1131_0,
     author = {Khadijeh Baghaei},
     title = {Global boundedness of solutions to a chemotaxis consumption model with signal dependent motility and logistic source},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1131--1145},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.605},
     language = {en},
}

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PB  - Académie des sciences, Paris
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Khadijeh Baghaei. Global boundedness of solutions to a chemotaxis consumption model with signal dependent motility and logistic source. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1131-1145. doi : 10.5802/crmath.605. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.605/

Bibliographie
Cité par

[1] Jaewook Ahn; Changwook Yoon Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing, Nonlinearity, Volume 32 (2019) no. 4, pp. 1327-1351 | DOI | MR | Zbl

[2] Khadijeh Baghaei Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source, C. R. Math. Acad. Sci. Paris, Volume 361 (2023), pp. 1641-1652 | DOI | MR | Zbl

[3] Khadijeh Baghaei; Ali Khelghati Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant, C. R. Math. Acad. Sci. Paris, Volume 355 (2017) no. 6, pp. 633-639 | DOI | Numdam | MR | Zbl

[4] Khadijeh Baghaei; Ali Khelghati Global existence and boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant and logistic source, Math. Methods Appl. Sci., Volume 40 (2017) no. 10, pp. 3799-3807 | DOI | MR | Zbl

[5] Martin Burger; Philippe Laurençot; Ariane Trescases Delayed blow-up for chemotaxis models with local sensing, J. Lond. Math. Soc., Volume 103 (2021) no. 4, pp. 1596-1617 | DOI | MR | Zbl

[6] Tobias Black; Michael Winkler Global weak solutions and absorbing sets in a chemotaxis–Navier-Stokes system with prescribed signal concentration on the boundary, Math. Models Methods Appl. Sci., Volume 32 (2022) no. 1, pp. 137-173 | DOI | MR | Zbl

[7] Xinru Cao; Johannes Lankeit Global classical small-data solutions for a three-dimensional chemotaxis Navier–Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differ. Equ., Volume 55 (2016) no. 4, 107, 39 pages | DOI | MR | Zbl

[8] Laurent Desvillettes; Yong-Jung Kim; Ariane Trescases; Changwook Yoon A logarithmic chemotaxis model featuring global existence and aggregation, Nonlinear Anal., Real World Appl., Volume 50 (2019), pp. 562-582 | DOI | MR | Zbl

[9] Kentarou Fujie; Jie Jiang Global existence for a kinetic model of pattern formation with density-suppressed motilities, J. Differ. Equations, Volume 269 (2020) no. 6, pp. 5338-5378 | DOI | MR | Zbl

[10] Kentaro Fujie; Jie Jiang Boundedness of classical solutions to a degenerate Keller–Segel type model with signal-dependent motilities, Acta Appl. Math., Volume 176 (2021), 3, 36 pages | DOI | MR | Zbl

[11] Kentaro Fujie; Takasi Senba Global boundedness of solutions to a parabolic-parabolic chemotaxis system with local sensing in higher dimensions, Nonlinearity, Volume 35 (2022) no. 7, pp. 3777-3811 | DOI | MR | Zbl

[12] Jiayi Han; Changchun Liu Global weak solution for a chemotaxis Navier–Stokes system with $p$ -Laplacian diffusion and singular sensitivity, Nonlinear Anal., Real World Appl., Volume 73 (2023), 103898, 24 pages | DOI | Zbl

[13] Hai-Yang Jin; Yong-Jung Kim; Zhi-An Wang Boundedness, stabilization, and pattern formation driven by density-suppressed motility, SIAM J. Appl. Math., Volume 78 (2018) no. 3, pp. 1632-1657 | DOI | MR | Zbl

[14] Hai-Yang Jin; Zhi-An Wang Critical mass on the Keller–Segel system with signal-dependent motility, Proc. Am. Math. Soc., Volume 148 (2020) no. 11, pp. 4855-4873 | DOI | MR | Zbl

[15] Ali Khelghati; Khadijeh Baghaei Boundedness of classical solutions for a chemotaxis model with rotational flux terms, Z. Angew. Math. Mech., Volume 98 (2018) no. 10, pp. 1864-1877 | DOI | MR | Zbl

[16] Ali Khelghati; Khadijeh Baghaei Boundedness of classical solutions for a chemotaxis system with general sensitivity function, Appl. Anal., Volume 98 (2019) no. 3, pp. 611-621 | DOI | MR | Zbl

[17] 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

[18] Philippe Laurençot Long term spatial homogeneity for a chemotaxis model with local sensing and consumption, Commun. Math. Sci., Volume 21 (2023) no. 6, pp. 1743-1750 | DOI | MR | Zbl

[19] Yan Li; Shuying Lu Global solutions to a chemotaxis-growth system with signal-dependent motilities and signal consumption, J. Math. Anal. Appl., Volume 521 (2023) no. 1, 126902, 17 pages | DOI | MR | Zbl

[20] Genglin Li; Michael Winkler Relaxation in a Keller–Segel-consumption system involving signal-dependent motilities, Commun. Math. Sci., Volume 21 (2023) no. 2, pp. 299-322 | DOI | MR | Zbl

[21] Genglin Li; Michael Winkler Refined regularity analysis for a Keller-Segel-consumption system involving signal-dependent motilities, Appl. Anal., Volume 103 (2024) no. 1, pp. 45-64 | DOI | MR | Zbl

[22] Xue Li; Liangchen Wang; Xu Pan Boundedness and stabilization in the chemotaxis consumption model with signal-dependent motility, Z. Angew. Math. Phys., Volume 72 (2021) no. 4, 170, 18 pages | DOI | MR | Zbl

[23] Dan Li; Jie Zhao Global boundedness and large time behavior of solutions to a chemotaxis-consumption system with signal-dependent motility, Z. Angew. Math. Phys., Volume 72 (2021) no. 2, 57, 20 pages | DOI | MR | Zbl

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

[25] Idan Tuval; Luis Cisneros; Christopher Dombrowski; Charles W. Wolgemuth; John O. Kessler; Raymond E. Goldstein Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, Volume 102 (2005) no. 7, pp. 2277-2282 | DOI | Zbl

[26] Youshan Tao; Michael Winkler Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equations, Volume 252 (2012) no. 3, pp. 2520-2543 | DOI | MR | Zbl

[27] Youshan Tao; Michael Winkler Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension, J. Differ. Equations, Volume 343 (2023), pp. 390-418 | DOI | MR | Zbl

[28] Liangchen Wang Global dynamics for a chemotaxis consumption system with signal-dependent motility and logistic source, J. Differ. Equations, Volume 348 (2023), pp. 191-222 | DOI | MR | Zbl

[29] Michael Winkler Stabilization in a two-dimensional chemotaxis-Navier–Stokes system, Arch. Ration. Mech. Anal., Volume 211 (2014) no. 2, pp. 455-487 | DOI | MR | Zbl

[30] Michael Winkler A result on parabolic gradient regularity in Orlicz spaces and application to absorption-induced blow-up prevention in a Keller–Segel-type cross-diffusion system, Int. Math. Res. Not., Volume 2023 (2023) no. 19, pp. 16336-16393 | DOI | MR | Zbl

[31] Michael Winkler Global generalized solvability in a strongly degenerate taxis-type parabolic system modeling migration-consumption interaction, Z. Angew. Math. Phys., Volume 74 (2023) no. 1, 32, 20 pages | DOI | MR | Zbl

[32] Jianping Wang; Mingxin Wang Boundedness in the higher-dimensional Keller–Segel model with signal-dependent motility and logistic growth, J. Math. Phys., Volume 60 (2019) no. 1, 011507, 14 pages | DOI | MR | Zbl

[33] Yamin Xiao; Jie Jiang Global existence and uniform boundedness in a fully parabolic Keller–Segel system with non-monotonic signal-dependent motility, J. Differ. Equations, Volume 354 (2023), pp. 403-429 | DOI | MR | Zbl

[34] Changwook Yoon; Yong-Jung Kim Global existence and aggregation in a Keller–Segel model with Fokker–Planck diffusion, Acta Appl. Math., Volume 149 (2017), pp. 101-123 | DOI | MR | Zbl

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