boundedness in a three-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion and logistic source

This article concerns the attraction-repulsion chemotaxis system with nonlinear diffusion and logistic source, $$\displaylines{ u_t=\nabla\cdot((u+1)^{m-1}\nabla u)-\nabla\cdot(\chi u\nabla v) +\nabla\cdot(\xi u\nabla w)+ru-\mu u^\eta, \cr x\in\Omega,\; t>0,\cr v_t=\Delta v+\alpha u-\beta v, \quad x\in\Omega, \; t>0,\cr w_t=\Delta w+\gamma u-\delta w, \quad x\in\Omega,\; t>0 }$$ under Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^3$ with smooth boundary. We show that if the diffusion is strong enough or the logistic dampening is sufficiently powerful, then the corresponding system possesses a global bounded classical solution for any sufficiently regular initial data. Moreover, it is proved that if $r=0$, $\beta>\frac{1}{2(\eta-1)}$ and $\delta>\frac{1}{2(\eta-1)}$ for the latter case, then $u(\cdot,t)\to 0$, $ v(\cdot,t)\to 0$ and $ w(\cdot,t)\to 0$ in $L^\infty(\Omega)$ as $ t \to \infty$.