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 = \Delta u - \chi\nabla\cdot(u\nabla v_1) + \xi\nabla\cdot(u\nabla v_2)\\ \partial_t v_1 = \Delta v_1 - \beta v_1 + \alpha u \\ \partial_t v_2 = \Delta v_2 - \delta v_2 + \gamma u, \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 $\chi\alpha-\xi\gamma>0$.