Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source

Khadijeh Baghaei

doi:10.5802/crmath.519

Partial differential equations

Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source

Khadijeh Baghaei ¹

¹ Pasargad Institute for Advanced Innovative Solutions, No.30, Hakim Azam St., North Shiraz St., Mollasadra Ave., Tehran, Iran

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1641-1652.

Abstract

We consider the 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} and ξ (v) = - (1 - α) γ^{'} (v),

where $k > 0$ and $α \in (0, 1) .$

We prove that the classical solutions to the above system are uniformly-in-time bounded provided that $k (1 - α) < \frac{4}{n + 5}$ and the initial value $v_{0}$ and $μ$ satisfy the following conditions:

\begin{matrix} 0 < ∥ v_{0} ∥_{L^{\infty} (Ω)} \leq {[\frac{4 [1 - k (1 - α)]}{k (n + 1) (1 - α)}]}^{\frac{1}{k}} - 1, \end{matrix}

and

μ > \frac{k n (1 - α) ∥ v_{0} ∥_{L^{\infty} (Ω)}}{(n + 1) (1 + ∥ v_{0} ∥_{L^{\infty} (Ω)})} .

This result improves the recent result obtained for this problem by Li and Lu (J. Math. Anal. Appl.) (2023).

Received: 2023-05-05
Revised: 2023-05-27
Accepted: 2023-05-29
Published online: 2023-11-17

DOI: 10.5802/crmath.519

Author's affiliations:

Khadijeh Baghaei ¹

¹ Pasargad Institute for Advanced Innovative Solutions, No.30, Hakim Azam St., North Shiraz St., Mollasadra Ave., Tehran, Iran

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2023__361_G10_1641_0,
     author = {Khadijeh Baghaei},
     title = {Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1641--1652},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.519},
     language = {en},
}

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Khadijeh Baghaei. Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1641-1652. doi : 10.5802/crmath.519. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.519/

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