In this paper, we study an attraction-repulsion chemotaxis with logistic damping, \(\begin{aligned} u_{t}&=\nabla \cdot (D(u)\nabla u)-\nabla \cdot (\chi u \nabla v) +\nabla \cdot (\xi u \nabla w)+au-bu^{\eta },\\ 0&=\Delta v+\alpha u-\beta v, \quad 0=\Delta w+\gamma u-\delta w, \end{aligned}\) for \(x\in \Omega , t>0\) , subject to null Neumann boundary conditions, where \(D(u)\ge c_{D}u^{m-1}\) . When the repulsion prevails over the attraction ( \(\xi \gamma >\chi \alpha \) ), the problem admits a globally bounded solution even if the logistic source was weak. When the attraction dominates the repulsion ( \(\xi \gamma <\chi \alpha \) ), the problem still possesses a globally bounded solution provided the logistic absorption is sufficiently strong in the sense of \(\eta >2\) , or the diffusion is sufficiently strong in the sense of \(m>1\) for the classical logistic source and suitable coefficient of absorption ( \(\eta =2\) and \(b=b^*\) ), or the coefficient of absorption is sufficiently large ( \(b>b^{*}\) ) for the classical logistic source, or the diffusion is sufficiently strong in the sense of \(m>2- {2}/{n}\) even if the logistic damping was weak. In the critical case \(\xi \gamma =\chi \alpha \) , strong logistic damping or strong diffusion ( \(\eta \ge 2\) or \(m>2-2/n\) ) lead to the global boundedness of solutions. For \(D(u)\equiv 1\) and \(n\ge 2\) , the convergence rates of global solutions are established in \(L^\infty \) -norm. The result answers the left question in J. Math. Anal. Appl. 455 (2017) 650–679.
