FINITE-TIME BLOW-UP CRITERION FOR A COMPETING CHEMOTAXIS SYSTEM

This paper deals with the following attraction-repulsion chemotaxis system {u(t )= triangle u - chi del center dot (u del v) + xi del center dot (u del w), x is an element of ohm, t >0, tau(1)v(t)= triangle v - beta v+alpha u, x is an element of ohm, t >0, tau(2)w(t )= triangle w - delta w + gamma u, x is an element of ohm, t >0, u(x,t= 0) = u(0)(x), tau(1)v (x,t= 0) = tau(1)v(0)(x), x is an element of ohm, tau(2)w(x,t= 0) = tau(2)w(0)(x), x is an element of ohm, where the parameters chi, xi, alpha, beta, -y and delta are positive, tau(1), tau(2) = 0,1, and ohm = B-1(0) subset of 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-y > 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.