Boundedness and stabilization in a quasilinear chemotaxis model with nonlocal growth term and indirect signal production

In this paper, we study the following fully parabolic chemotaxis system with nonlocal growth and indirect signal production:

$ \begin{equation*} \left\{ \begin{array}{@{}l@{\quad}l} u_{t} = \nabla\cdot( D(u)\nabla u)-\nabla\cdot(u\nabla v)+f(u),&x\in\Omega,\,\,t>0,\\ v_{t} = \Delta v+w-v,&x\in\Omega,\,\,t>0,\\ w_{t} = \Delta w+u-w,&x\in\Omega,\,\,t>0,\\ \frac{\partial u}{\partial n} = \frac{\partial v}{\partial n} = \frac{\partial w}{\partial n} = 0, &x\in\partial\Omega,\,\,t>0, \\ u(x, 0) = u_{0}(x), v(x, 0) = v_{0}(x), w(x, 0) = w_{0}(x), &x\in\Omega, \\ \end{array}\right. \end{equation*} $

in a smooth bounded domain $ \Omega\subset\mathbb{R}^{n}(n\geq3) $, where $ D(u)\geq D_{1}u^{\gamma} $, $ f(u) = u(a_{0}-a_{1}u^{\sigma}+a_{2}\int_{\Omega}u^\sigma dx) $, $ D_{1}, a_{1}, a_{2} $ are positive constants, $ a_{0}, \gamma\in \mathbb{R} $, $ \sigma\geq\max\{1, -\gamma\} $ and $ a_{1}-a_{2}|\Omega| > 0 $. It is shown that the above system admits a globally bounded classical solution if $ \frac{4}{n}+\frac{\gamma}{\sigma} > \frac{3-\sigma}{1+\sigma} $. Furthermore, by the method of Lyapunov functionals, the global stability of steady states with convergence rates is established.