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 Δu - χ_1 \nabla \cdot (u \nabla w) + μ_1 u (1- u^{κ_1-1} - a_1 v^{λ_1-1}), &\quad x \in Ω,\ t>0,\\ % v_t = d_2 Δv - χ_2 \nabla \cdot (v \nabla w) + μ_2 v (1- a_2 u^{λ_2-1} - v^{κ_2-1}), &\quad x \in Ω,\ t>0,\\ % 0 = d_3 Δw + αu + βv - h(u,v,w), &\quad x \in Ω,\ t>0, \end{cases} \end{align*} where $\Omega \subset {\mathbb{R}}^n$ $(n\ge 2)$ 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.