Influence of nonlinear production on the global solvability of an attraction‐repulsion chemotaxis system

This paper is dedicated to the attraction‐repulsion chemotaxis‐system ⋄$\Diamond$ ut=Δu−χ∇·(u∇v)+ξ∇·(u∇w)inΩ×(0,Tmax),0=Δv+f(u)−βvinΩ×(0,Tmax),0=Δw+g(u)−δwinΩ×(0,Tmax),\begin{equation} \hspace*{65pt}\left\{ \def\eqcellsep{&}\begin{array}{ll} u_{ t}=\Delta u -\chi \nabla \cdot (u\nabla v)+\xi \nabla \cdot (u\nabla w) & \text{in }\Omega \times (0,T_{\mathrm{max}}), \\ 0=\Delta v+f(u)-\beta v & \text{in } \Omega \times (0,T_{\mathrm{max}}), \\ 0=\Delta w+g(u)-\delta w & \text{in } \Omega \times (0,T_{\mathrm{max}}), \end{array} \right. \end{equation}defined in Ω, a smooth and bounded domain of Rn$\mathbb {R}^n$, with n≥2$n\ge 2$. Moreover, β,δ,χ,ξ>0$\beta ,\delta ,\chi ,\xi >0$ and f,g$f, g$ are suitably regular functions generalizing, for u≥0$u\ge 0$ and α, γ>0$\gamma >0$ the prototypes f(u)=αus$f(u)=\alpha u^s$, s>0$s>0$, and g(u)=γur$g(u)=\gamma u^r$, r≥1$r\ge 1$. We focus our analysis on the value Tmax∈(0,∞]$T_{\mathrm{max}}\in (0,\infty ]$, establishing the temporal interval of existence of solutions (u,v,w)$(u,v,w)$ to problem (⋄$\Diamond$). When zero‐flux boundary conditions are fixed, we prove the following results, all excluding chemotactic collapse scenarios under certain correlations between the attraction and repulsive effects describing the model. To be precise, for every α,β,γ,δ,χ>0$\alpha ,\beta ,\gamma ,\delta ,\chi >0$, and r>s≥1$r>s\ge 1$ (resp. s>r≥1$s>r\ge 1$), there exists ξ∗>0$\xi ^*>0$ (resp. ξ∗>0$\xi _*>0$) such that if ξ>ξ∗$\xi >\xi ^*$ (resp. ξ≥ξ∗$\xi \ge \xi _*$), any sufficiently regular initial datum u0(x)≥0$u_0(x)\ge 0$ (resp. u0(x)≥0$u_0(x)\ge 0$ enjoying some smallness assumptions) produces a unique classical solution (u,v,w)$(u,v,w)$ to problem (⋄$\Diamond$) which is global, i.e. Tmax=∞$T_{\mathrm{max}}=\infty$, and such that u, v and w are uniformly bounded. Conversely, the same conclusion holds true for every α,β,γ,δ,χ,ξ>0$\alpha ,\beta ,\gamma ,\delta ,\chi ,\xi >0$, 0