A fractional attraction-repulsion chemotaxis system with time-space dependent growth source and nonlinear productions

This paper studies a fractional attraction-repulsion system with time-space dependent growth source and nonlinear productions: \begin{equation*} \left\{ \begin{aligned}\label{1.1} &u_t = -(-\Delta)^\alpha u - \chi_1 \nabla \cdot (u \nabla v_1) + \chi_2 \nabla \cdot (u \nabla v_2) + a(x,t)u - b(x,t)u^\gamma, &x \in \mathbb{R}^N, \, t > 0, \\ &0 = \Delta v_1 - \lambda_1 v_1 + \mu_1 u^k, &x \in \mathbb{R}^N, \, t > 0, \\ &0 = \Delta v_2 - \lambda_2 v_2 + \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. For a fixed $\gamma$, when $k$ exceeds the critical value $\gamma - 1$, a larger $b$ must be chosen to suppress the blow-up of the solution. Moreover, we show the persistence of the global solutions for both cases $\gamma = k + 1$ and $\gamma \neq k + 1$.