Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 58, pp. 1-17.

Title: Boundedness and asymptotic stability in a chemotaxis model with indirect signal production and logistic source
       
Authors: Xiaobing Ye (Chongqing Univ. of Posts and Telecom., Chongqing, China)
         Liangchen Wang (Chongqing Univ. of Posts and Telecom., Chongqing, China)

Abstract:  
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. 

Submitted February 15, 2021. Published August 02, 2022.
Math Subject Classifications: 92C17, 35K35, 35A01, 35B35.
Key Words: Chemotaxis; boundedness; asymptotic behavior; indirect signal production.