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{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 \qquad \; + \xi u(u+1)^{m_3-1} \nabla w\right)+h(u) \\ v_t = \Delta v - f(u)v & \text{ in } \Omega \times (0, {T_{\text{max}}}), \\ w_t = \Delta w- g(u)w& \text{ in } \Omega \times (0, {T_{\text{max}}}), \end{cases} \end{align} ~~~~(\Diamond)$

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