Can chemotactic effects lead to blow-up or not in two-species chemotaxis-competition models?

Abstract: This paper deals with the two-species chemotaxis-competition models \begin{align*} \begin{cases} u_t = d_1 \Delta u - \chi_1 \nabla \cdot (u \nabla w) + \mu_1 u (1- u^{\kappa_1-1} - a_1 v^{\lambda_1-1}), &\quad 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^{\lambda_2-1} - v^{\kappa_2-1}), &\quad x \in \Omega,\ t>0,\ % 0 = d_3 \Delta w + \alpha u + \beta v - h(u,v,w), &\quad x \in \Omega,\ t>0, \end{cases} \end{align*} where $\Omega \subset \mathbb{R}^n$ $(n\ge2)$ is a bounded domain with smooth boundary, and $h=\gamma w$ or $h=\frac{1}{|\Omega|}\int_\Omega(\alpha u+ \beta v)\,dx$. In the case that $\kappa_1=\lambda_1=\kappa_2=\lambda_2=2$ and $h=\gamma w$, it is known that smallness conditions for the chemotacic effects lead to boundedness of solutions (Math.\ Methods Appl.\ Sci.; 2018; 41; 234--249). However, the case that the chemotactic effects are large seems not to have been studied yet; therefore it remains to consider the question whether the solution is bounded also in the case that the chemotactic effects are large. The purpose of this paper is to give a negative answer to this question.