A fractional attraction-repulsion chemotaxis system with generalized logistic source and nonlinear productions

TL;DR

This paper investigates a fractional chemotaxis system with nonlinear sources, establishing global boundedness, asymptotic behavior, and spreading speeds of solutions under various conditions.

Contribution

It provides new results on the boundedness, asymptotics, and spreading speeds of solutions for a fractional chemotaxis model with generalized logistic sources.

Findings

Global boundedness of solutions in different parameter regimes

Asymptotic behavior characterized for specific cases

Spreading speed bounds and their dependence on parameters

Abstract

This paper studies a fractional attraction-repulsion system with generalized logistic source and nonlinear productions: \begin{equation*} \left\{ \begin{aligned} &u_t = -(-\Delta)^\alpha u - \chi_1 \nabla \cdot (u \nabla v) + \chi_2 \nabla \cdot (u \nabla w) + au - bu^\gamma, &x \in \mathbb{R}^N, \, t > 0, \\ &0 = \Delta v - \lambda_1 v + \mu_1 u^k, &x \in \mathbb{R}^N, \, t > 0, \\ &0 = \Delta w - \lambda_2 w + \mu_2 u^k, &x \in \mathbb{R}^N, \, t > 0. \end{aligned} \right. \end{equation*} We first establish the global boundedness of classical solutions with nonnegative bounded and uniformly continuous initial data in two different cases: $\gamma \geq k + 1$ and $\gamma < k + 1$, respectively. Next, we show the asymptotic behavior of the global solutions for both cases $\gamma = k + 1$ and $\gamma \neq k + 1$. Finally, we obtain the spreading speed of solutions. In particular, when $\gamma = k + 1$, the upper bound of the spreading speed increases monotonically with $k$. If the condition of balanced attraction-repulsion intensities is further specified, the spreading speed will be equal to $\frac{a}{N + 2\alpha}$.