Boundedness and asymptotic stability in a chemotaxis model with indirect signal production and logistic source

This article concerns the chemotaxis-growth system with indirect signal production $$\displaylines{ u_t=\Delta u-\nabla\cdot(u\nabla v)+\mu u(1-u),\quad x\in \Omega,\; t>0,\cr 0=\Delta v-v+w,\quad x\in \Omega,\; t>0,\cr w_t=-\delta w+u,\quad x\in\Omega,\; t>0, }$$ on a smooth bounded domain \(\Omega\subset \mathbb{R}^n\) (\(n\geq1\) with homogeneous Neumann boundary condition, where the parameters \(\mu, \delta>0\). It is proved that if \(n\leq 2\) and \(\mu>0\), for all suitably regular initial data, this model possesses a unique global classical solution which is uniformly-in-time bounded. While in the case \(n\geq 3\), we show that if \(\mu\) is sufficiently large, this system possesses a global bounded solution. Furthermore, the large time behavior and rates of convergence have also been considered under some explicit conditions.