This study examines the effects of a logistic source on global solvability and stabilization in various models that generalize the following prototype \(\begin{aligned} \left\{ \begin{array}{l} n_{t}+u\cdot \nabla n =\Delta n -\chi \nabla \cdot ( n \nabla c)+\rho n -\mu n^2,\quad x\in \Omega , t>0,\\ u\cdot \nabla c=\Delta c-c+n,\quad x\in \Omega , t>0,\\ u_t+\nabla p+\kappa (u\cdot \nabla )u=\Delta u+n\nabla \phi ,\quad x\in \Omega , t>0,\\ \nabla \cdot u=0,\quad x\in \Omega , t>0, \end{array}\right. \qquad \qquad (*) \end{aligned}\) with no-flux boundary conditions for n and c, and no-slip boundary condition for u, in a bounded domain \(\Omega \subseteq {\mathbb {R}}^N (N\in \{2,3\})\) with a smooth boundary, where \( \phi \in W^{2,\infty }(\Omega ),\) and \(\chi > 0, \rho \in \mathbb {{R}},\) \(\mu > 0\) are given parameters. Additionally, assuming that \(N = 2,\kappa \in {\mathbb {R}}\) or \(N = 3,\kappa = 0.\) The study demonstrates that the corresponding initial boundary problem possesses a global classical solution, which is bounded on \(\Omega \times (0,\infty )\) under the explicit condition \(\mu \ge \frac{(N-2)_+\chi }{N} \) and suitable regularity assumptions on the initial data. To the best of our knowledge, this is the first attempt to study the boundedness of the system.
