Global boundedness and stabilization of solutions for a chemotaxis system with acceleration and logistic source

Abstract This article is concerned with the following chemotaxis-growth system \begin{equation*}\begin{cases} u_t = \nabla\cdot \left(\nabla u - u\bf{w} \right) + \rho u - \mu u^\alpha ,&x \in \Omega ,\ t \gt 0,\\ v_{1,t} = \Delta v_1 - v_1 + u,&x \in \Omega ,\ t \gt 0,\\ v_{2,t} = \Delta v_2 - v_2 + v_1,&x \in \Omega ,\ t \gt 0,\\ \ldots \\ v_{k,t} = \Delta v_k - v_k + v_{k - 1},&x \in \Omega ,\ t \gt 0,\\ {\bf{w}}_t = \Delta {\bf{w}} - {\bf{w}} + \nabla v_k,&x \in \Omega ,\ t \gt 0, \end{cases} \end{equation*} under the homogeneous Neumann boundary condition for u , v i and the homogeneous Dirichlet boundary condition for $\bf{w}$ in a smooth bounded domain $\Omega \subset {\mathbb{R}^n}\left( {n \geqslant 1} \right),$ where ρ > 0, µ > 0, α > 1 and $i=1,\ldots,k$ . We reveal that when the index α , the spatial variable n , and the number of equations k satisfy certain relationships, the global solution of the system exists and converges to the constant equilibrium state in the form of exponential convergence.