Nonlocal logistics and nonlinear productions in an attraction-repulsion chemotaxis model: analysis of the global well-posedness

Abstract

This paper investigates a {{three-component}} chemotaxis system involving both attraction and repulsion effects, as well as a nonlocal logistic-type source term. Mathematically, if $u=u(x,t)$, $v = v(x,t)$ and $w = w(x,t)$ denote the cell distribution, and the attractive and the repulsive chemical signals, the model is then described by \begin{equation*} \begin{cases} u_t = \Delta u - \chi \nabla \cdot (u \nabla v) + \xi \nabla \cdot (u \nabla w) + a u^\alpha - b u^\alpha \int_\Omega u^\beta, & x \in \Omega, \ t > 0, \tau v_t = \Delta v - v + f(u), & x \in \Omega, \ t > 0, \tau w_t = \Delta w - w + g(u), & x \in \Omega, \ t > 0. \end{cases} \end{equation*} Here, $\Omega \subset \mathbb{R}^n$ ($n \geq 1$) is a bounded smooth domain, $\tau\in\{0,1\}$, $a,b,\alpha,\beta,\chi,\xi>0$, the production functions $f(u)$ and $g(u)$ are assumed to satisfy algebraic growth conditions of order $\ell$ and $\rho$, generalizing prototypes of the form $u^\ell$ and $u^\rho$, $\ell,\rho>0$. The work is devoted to proving the global existence and boundedness of classical solutions under a suitable balance between the signal production exponents $\ell, \rho$ and the nonlocal damping exponents $\alpha, \beta$, for regular enough initial data and zero-flux boundary restrictions. In this regard, two main theorems are established for the cases where the chemical signals satisfy either elliptic ($\tau=0$) or parabolic ($\tau=1$) partial differential equations, highlighting how sufficiently strong nonlocal damping prevents the formation of singularities in time. We extend the results obtained in [Chiyo et al., Appl. Math. Optim. 89:9 (2024)], where the fully parabolic ($\tau=1$) and only attraction version is studied. In our context, we establish well-posedness of the system and the long-time behavior of solutions.