Global solvability and boundedness to a attraction–repulsion model with logistic source

Abstract

Abstract In this paper, we deal with an attraction–repulsion model with a logistic source as follows: { u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + ξ ∇ ⋅ ( u ∇ w ) + μ u q ( 1 − u ) in Q , v t = Δ v − α 1 v + β 1 u in Q , w t = Δ w − α 2 w + β 2 u in Q , $$\begin{aligned} \textstyle\begin{cases} {u_{t}} = \Delta u - \chi \nabla \cdot (u \nabla v) + \xi \nabla \cdot (u \nabla w) + \mu {u^{q}}(1 - u) &\text{in } Q , \\ {v_{t}} = \Delta v - {\alpha _{1}}v + {\beta _{1}}u &\text{in } Q , \\ {w_{t}} = \Delta w - {\alpha _{2}}w + {\beta _{2}}u & \text{in } Q , \end{cases}\displaystyle \end{aligned}$$ where Q = Ω × R + $Q = \Omega \times {\mathbb{R}^{+} }$ , Ω ⊂ R 3 $\Omega \subset {\mathbb{R}^{3}}$ is a bounded domain. We mainly focus on the influence of logistic damping on the global solvability of this model. In dimension 2, q can be equal to 1 (Math. Methods Appl. Sci. 39(2):289–301, 2016). In dimension 3, we derive that the problem admits a global bounded solution when q > 8 7 $q>\frac{8}{7}$ . In fact, we transfer the difficulty of estimation to the logistic term through iterative methods, thus, compared to the results in (J. Math. Anal. Appl. 2:448 2017; Z. Angew. Math. Phys. 73(2):1–25 2022) in dimension 3, our results do not require any restrictions on the coefficients.

Keywords

• Chemotaxis
• Global solvability
• Boundedness
• Logistic source
• Attraction–repulsion