Large densities in a competitive two-species chemotaxis system in the non-symmetric case

This paper deals with the two-species chemotaxis system with Lotka-Volterra competitive kinetics, \begin{align*} \begin{cases} u_t = d_1 \Delta u - \chi_1 \nabla \cdot (u \nabla w) + \mu_1 u (1 - u - a_1 v), & x\in\Omega,\ t>0,\\ v_t = d_2 \Delta v - \chi_2 \nabla \cdot (v \nabla w) + \mu_2 v (1 - a_2 u - v), & x\in\Omega,\ t>0,\\ 0 = d_3 \Delta w + \alpha u + \beta v - \gamma w, & x\in\Omega,\ t>0, \end{cases} \end{align*} under homogeneous Neumann boundary conditions and suitable initial conditions, where $\Omega \subset \mathbb{R}^n$ $(n \in \mathbb{N})$ is a bounded domain with smooth boundary, $d_1, d_2, d_3, \chi_1, \chi_2, \mu_1, \mu_2 > 0$, $a_1, a_2 \ge 0$ and $\alpha, \beta, \gamma > 0$. Under largeness conditions on $\chi_1$ and $\chi_2$, we show that for suitably regular initial data, any thresholds of the population density can be surpassed, which extends the previous results to the non-symmetric case. The paper contains a well-posedness result for the hyperbolic-elliptic limit system with $d_1=d_2=0$.