Asymptotic stability in a chemotaxis-competition system with indirect signal production

This paper deals with a fully parabolic inter-species chemotaxis-competition system with indirect signal production

$ \begin{eqnarray*} \label{1a} \left\{ \begin{split}{} &u_{t} = \text{div}(d_{u}\nabla u+\chi u\nabla w)+\mu_{1}u(1-u-a_{1}v), &(x,t)\in \Omega\times (0,\infty), \\ &v_{t} = d_{v}\Delta v+\mu_{2}v(1-v-a_{2}u), &(x,t)\in \Omega\times (0,\infty), \\ & w_{t} = d_{w}\Delta w-\lambda w+\alpha v, &(x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*} $

under zero Neumann boundary conditions in a smooth bounded domain $ \Omega\subset \mathbb{R}^{N} $ ($ N\geq 1 $), where $ d_{u}>0, d_{v}>0 $ and $ d_{w}>0 $ are the diffusion coefficients, $ \chi\in \mathbb{R} $ is the chemotactic coefficient, $ \mu_{1}>0 $ and $ \mu_{2}>0 $ are the population growth rates, $ a_{1}>0, a_{2}>0 $ denote the strength coefficients of competition, and $ \lambda $ and $ \alpha $ describe the rates of signal degradation and production, respectively. Global boundedness of solutions to the above system with $ \chi>0 $ was established by Tello and Wrzosek in [J. Math. Anal. Appl. 459 (2018) 1233-1250]. The main purpose of the paper is further to give the long-time asymptotic behavior of global bounded solutions, which could not be derived in the previous work.