Finite-time blow-up in the three-dimensional fully parabolic attraction-dominated attraction-repulsion chemotaxis system

Abstract

We show that the attraction-repulsion chemotaxis system \begin{equation*} \begin{cases} u_t = Δu - χ\nabla\cdot(u\nabla v_1) + ξ\nabla\cdot(u\nabla v_2)\\ \partial_t v_1 = Δv_1 - βv_1 + αu \\ \partial_t v_2 = Δv_2 - δv_2 + γu, \end{cases} \end{equation*} posed with homogeneous Neumann boundary conditions in bounded domains $\Omega =B_R\subset {\mathbb{R}}^3$, $R>0$, admits radially symmetric solutions which blow-up in finite time if it is attraction-dominated in the sense that $\chi \alpha -\xi \gamma >0$.