Emergence of large densities and simultaneous blow-up in a two-species chemotaxis system with competitive kinetics

Abstract

In this paper, a fully parabolic chemotaxis system for two species $ \begin{eqnarray*} \left\{\begin{array}{lll} u_t = \Delta u-\chi_1\nabla\cdot(u\nabla w)+\mu_1u(1-u-a_1v),\ \ \ &x\in \Omega,\ t>0,\\ v_t = \Delta v-\chi_2\nabla\cdot(v\nabla w)+\mu_2v(1-v-a_2u),\ \ &x\in \Omega,\ t>0,\\ w_t = \Delta w-w+u+v,\ \ &x\in \Omega,\ t>0 \end{array}\right. \end{eqnarray*} $ is considered associated with homogeneous Neumann boundary conditions in a smooth bounded domain $ \Omega\subset\mathbb{R}^n $, $ n\geq3 $, with parameters $ \chi_i, \mu_i, a_i>0 $, $ i = 1, 2 $. It is shown that for some low energy initial data, the influence of chemotactic cross-diffusion coupled with proliferation may force some solutions to exceed any given threshold. Further, it is proved that if blow-up happens in a two-species chemotaxis(-growth) system, it is simultaneous for both of the chemotactic species.