A nonlinear attraction-repulsion Keller–Segel model with double sublinear absorptions: criteria toward boundedness

This paper deals with the zero-flux attraction-repulsion chemotaxis model \begin{document}$ \begin{align} \begin{cases} u_t = \nabla \cdot \left((u+1)^{m_1-1}\nabla u - \chi u(u+1)^{m_2-1}\nabla v \right. & \text{ in } \Omega \times (0, {T_{\text{max}}}), \\ \left. \qquad \; + \xi u(u+1)^{m_3-1} w\right)+h(u) v_t \Delta f(u)v w_t w- g(u)w& \end{cases} \end{align} ~~~~(\Diamond)$\end{document} unknown $ (u, v, w)\! \!(u(x, t), v(x, w(x, t)) $. Here, x\!\in\! $, a bounded and smooth domain of {\mathbb R}^n ($ n\geq 1 $), t, \chi, >0 m_1, m_2, m_3 \in \mathbb{R} f(u), g(u) h(u) sufficiently regular functions generalizing prototypes f(u) K_1 u^{\alpha} K_2 u^{\gamma} k \mu u^{\beta} K_1, K_2, \beta>1 suitable \alpha, \gamma Besides, further initial data u(x, 0) u_0(x), v_0(x), w_0(x)\geq 0 are given, whereas {T_{\text{max}}} \infty] stands for maximal instant time up to which solutions system exist. We will derive relations between parameters involved (\Diamond)$ capable warrant that u, w global uniformly time. The article generalizes extends case nonlinear effects logistic perturbations some results recently developed [3] where, linear counterpart absence logistics, criteria towards boundedness established.