Finite time blow-up of nonradial solutions in an attraction repulsion chemotaxis system

This paper considers the attraction-repulsion chemotaxis system: u(t) = Delta u - chi del. (u del v) + xi del . (u del w), 0 = Delta v + alpha u - beta v, 0 = Delta w-gamma u - delta w, subject to the non flux boundary condition in a smooth bounded domain Omega subset of R-2, with chi,xi >= 0, alpha, beta, gamma > 0. We establish the finite time blow-up conditions for nonradial solutions that the finite time blow-up occurs at x(0) is an element of Omega whenever integral(Omega)u(0)(x)dx > 8 pi/(chi alpha-xi gamma) with chi alpha - xi gamma > 0, under integral(Omega)u(0)(x)vertical bar x - x(0)vertical bar(2)dx sufficiently small. This does agree with the known blow-up conditions for radial solutions of the same model. The previous blow-up conditions for nonradial solutions are more complicated involving a classification to the sign of delta - beta. (C) 2016 Elsevier Ltd. All rights reserved.