Boundedness of Solutions for an Attraction–Repulsion Model with Indirect Signal Production

Wu, Jie; Huang, Yujie

doi:10.3390/math12081143

Open AccessArticle

Boundedness of Solutions for an Attraction–Repulsion Model with Indirect Signal Production

by

Jie Wu

^1,2 and

Yujie Huang

^3,*

¹

College of Computer Science, Chengdu University, Chengdu 610106, China

²

Visual Computing and Virtual Reality Key Laboratory of Sichuan Province, Sichuan Normal University, Chengdu 610068, China

³

Division of Mathematics, Sichuan University Jinjiang College, Meishan 620860, China

^*

Author to whom correspondence should be addressed.

Mathematics 2024, 12(8), 1143; https://doi.org/10.3390/math12081143

Submission received: 13 March 2024 / Revised: 3 April 2024 / Accepted: 3 April 2024 / Published: 10 April 2024

(This article belongs to the Special Issue Applications of Partial Differential Equations, 2nd Edition)

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Abstract

In this paper, we consider the following two-dimensional chemotaxis system of attraction–repulsion with indirect signal production

\{\begin{matrix} 𝜕_{t} u = Δ u - \nabla \cdot (χ_{1} u \nabla v_{1}) + \nabla \cdot (χ_{2} u \nabla v_{2}), & x \in R^{2}, t > 0, \\ 0 = Δ v_{j} - λ_{j} v_{j} + w, & x \in R^{2}, t > 0, (j = 1, 2), \\ 𝜕_{t} w + δ w = u, & x \in R^{2}, t > 0, \\ u (0, x) = u_{0} (x), w (0, x) = w_{0} (x), & x \in R^{2}, \end{matrix}

where the parameters

χ_{i} \geq 0, λ_{i} > 0 (i = 1, 2)

and non-negative initial data

(u_{0} (x), w_{0} (x)) \in L^{1} (R^{2}) \cap L^{\infty} (R^{2})

. We prove the global bounded solution exists when the attraction is more dominant than the repulsion in the case of

χ_{1} \geq χ_{2}

. At the same time, we propose that when the radial solution satisfies

χ_{1} - χ_{2} \leq \frac{2 π δ}{∥ u_{0} ∥_{L^{1} (R^{2})} + {∥ w_{0} ∥}_{L^{1} (R^{2})}}

, the global solution is bounded. During the proof process, we found that adding indirect signals can constrict the blow-up of the global solution.

Keywords: Keller–Segel; attraction–repulsion model; indirect signal production; radial solution; boundedness

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MDPI and ACS Style

Wu, J.; Huang, Y. Boundedness of Solutions for an Attraction–Repulsion Model with Indirect Signal Production. Mathematics 2024, 12, 1143. https://doi.org/10.3390/math12081143

AMA Style

Wu J, Huang Y. Boundedness of Solutions for an Attraction–Repulsion Model with Indirect Signal Production. Mathematics. 2024; 12(8):1143. https://doi.org/10.3390/math12081143

Chicago/Turabian Style

Wu, Jie, and Yujie Huang. 2024. "Boundedness of Solutions for an Attraction–Repulsion Model with Indirect Signal Production" Mathematics 12, no. 8: 1143. https://doi.org/10.3390/math12081143

APA Style

Wu, J., & Huang, Y. (2024). Boundedness of Solutions for an Attraction–Repulsion Model with Indirect Signal Production. Mathematics, 12(8), 1143. https://doi.org/10.3390/math12081143

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Boundedness of Solutions for an Attraction–Repulsion Model with Indirect Signal Production

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