Finite-time blow-up criterion for a competing chemotaxis system.

This paper deals with the following attraction-repulsion chemotaxis system$ \begin{align} \begin{cases} u_t = \Delta u - \chi\nabla \cdot (u\nabla v)+\xi \nabla\cdot(u\nabla w), &x\in \Omega, t>0, \\ \tau_1v_t = \Delta v -\beta v+\alpha u, &x\in \Omega, t>0, \\ \tau_2 w_t = \Delta w-\delta w +\gamma u, &x\in \Omega, t>0, \\ u(x, t = 0) = u_0 (x) , \tau_1 v(x, t = 0) = \tau_1v_0 (x) , & x\in \Omega, \\ \tau_2 w(x, t = 0) = \tau_2 w_0 (x), & x\in\Omega, \end{cases} \end{align} $where the parameters $ \chi $, $ \xi $, $ \alpha $, $ \beta $, $ \gamma $ and $ \delta $ are positive, $ \tau_1, \tau_2 = 0, 1 $, and $ \Omega = B_1(0)\subset\mathbb{R}^2 $ is a unit ball supplemented with homogenous Neumann boundary conditions. By developing a differential inequality for the energy functional, we derive a precise criterion for finite-time blow-up of solution to the system with $ \tau_1 = 1, \tau_2 = 0 $ if attraction dominates (i.e. $ \theta = \chi\alpha-\xi\gamma>0 $). Moreover, finite-time blow-up solutions also are constructed for the system with $ \tau_1 = \tau_2 = 1 $ although the energy functional can not be expected to exist. [ABSTRACT FROM AUTHOR]