Persistence and boundedness in a two-species chemotaxis-competition system with singular sensitivity and indirect signal production

This paper deals with a two-species chemotaxis-competition system involving singular sensitivity and indirect signal production:

$ \begin{equation*} \begin{cases} u_{t} = \nabla\cdot(D(u)\nabla u)-\chi_1\nabla\cdot(\frac{u}{z^{k}}\nabla z)+\mu_1 u(1-u-a_1v), &x\in\Omega,\ t>0,\\ v_{t} = \nabla\cdot(D(v)\nabla v)-\chi_2\nabla\cdot(\frac{v}{z^{k}}\nabla z)+\mu_2 v(1-v-a_2 u), &x\in\Omega,\ t>0,\\ w_{t} = \Delta w-w+u+v,&x\in\Omega,\ t>0,\\ z_{t} = \Delta z-z+w,&x\in\Omega,\ t>0,\\ \end{cases} \end{equation*} $

where $ \Omega\subset R^{n} $ is a convex smooth bounded domain with homogeneous Neumann boundary conditions. The diffusion functions $ D(u), D(v) $ are assumed to fulfill $ D(u)\geq(u+1)^{\theta_1} $ and $ D(v)\geq(v+1)^{\theta_2} $ with $ \theta_1, \theta_2 > 0 $, respectively. The parameters are $ k\in (0, \frac{1}{2})\cup (\frac{1}{2}, 1] $, $ \chi_ {i} > 0, (i = 1, 2) $. Additionally, $ \mu_{i} $ should be large enough positive constants, and $ a_i $ should be positive constants which are less than the quantities associated with $ |\Omega| $. Through constructing some appropriate Lyapunov functionals, we can find the lower bounds of $ \int_{\Omega}u $ and $ \int_{\Omega}v $. This suggests that any occurrence of extinction, if it happens, will be localized spatially rather than affecting the population as a whole. Moreover, we demonstrate that the solution remains globally bounded if $ \min\{\theta_1, \theta_2\} > 1-\frac{2}{n+1} $ for $ n\geq2. $