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

Jie Wu and Yujie Huang ()
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Jie Wu: College of Computer Science, Chengdu University, Chengdu 610106, China
Yujie Huang: Division of Mathematics, Sichuan University Jinjiang College, Meishan 620860, China

Mathematics, 2024, vol. 12, issue 8, 1-15

Abstract: In this paper, we consider the following two-dimensional chemotaxis system of attraction–repulsion with indirect signal production 𝜕 t u = Δ u − ∇ · χ 1 u ∇ v 1 + ∇ · ( χ 2 u ∇ v 2 ) , x ∈ R 2 , t > 0 , 0 = Δ v j − λ j v j + w , x ∈ R 2 , t > 0 , ( j = 1 , 2 ) , 𝜕 t w + δ w = u , x ∈ R 2 , t > 0 , u ( 0 , x ) = u 0 ( x ) , w ( 0 , x ) = w 0 ( x ) , x ∈ R 2 , where the parameters χ i ≥ 0 , λ i > 0 ( i = 1 , 2 ) and non-negative initial data ( u 0 ( x ) , w 0 ( x ) ) ∈ L 1 ( R 2 ) ∩ L ∞ ( R 2 ) . We prove the global bounded solution exists when the attraction is more dominant than the repulsion in the case of χ 1 ≥ χ 2 . At the same time, we propose that when the radial solution satisfies χ 1 − χ 2 ≤ 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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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