Global boundedness and asymptotic stability of the attraction-repulsion chemotaxis system with nonlinear productions and logistic-type source on <inline-formula><tex-math id="M1">$ \mathbb{R}^N $</tex-math></inline-formula>

This paper deals with the Cauchy problem for the attraction-repulsion chemotaxis system$ \begin{align*} \left\{ \begin{array}{llll} u_t = \Delta u-\chi_1\nabla\cdot(u\nabla v)+\chi_2\nabla\cdot(u\nabla w) +u(a-bu^\gamma), \quad x\in\mathbb{R}^N, t>0, \\[0.2cm] 0 = \Delta v-\lambda_1v+\mu_1u^k, \quad x\in\mathbb{R}^N, t>0, \\[0.2cm] 0 = \Delta w-\lambda_2w+\mu_2u^l, \quad x\in\mathbb{R}^N, t>0, \end{array} \right. \end{align*} $where $ \chi_i\geq 0 $, $ \lambda_i, \mu_i>0 $ $ (i = 1, 2) $, and $ a, b, \gamma, k $, and $ l $ are positive constants. Here, $ N $ is a positive integer. We first give the global boundedness of classical solutions with nonnegative bounded and uniformly continuous initial functions for the cases of (1) $ k\leq \min\{l, \gamma\}, (2)\; l<k\leq\gamma, (3)\; \gamma<k\leq l $, and (4) $ k>\max\{\gamma, l\} $. Next, under appropriate assumptions on the parameters, the global asymptotic stability of the positive constant equilibria can be proven in the critical cases $ k = l = \gamma, k = l\neq\gamma $, and $ k = \gamma\neq l $. This result covers and extends the global boundedness and asymptotic behavior of solutions established in the previous paper for the attraction-repulsion chemotaxis system with linear secretions $ (k = l = 1) $ and logistic source $ (\gamma = 1) $.