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 = -(-Δ)^αu - χ_1 \nabla \cdot (u \nabla v_1) + χ_2 \nabla \cdot (u \nabla v_2) + a(x,t)u - b(x,t)u^γ, &x \in $\mathbb{R}$^N, , t > 0, \ &0 = Δv_1 - λ_1 v_1 + μ_1 u^k, &x \in $\mathbb{R}$^N, , t > 0, \ &0 = Δv_2 - λ_2 v_2 + μ_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: $\gamma \ge k+1$ 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 $\gamma \mathrm{\ne }k+1$.