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

This paper considers the attraction-repulsion chemotaxis system: ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w), 0=Δv+αu−βv, 0=Δw+γu−δw, subject to the non-flux boundary condition in a smooth bounded domain Ω⊂R2, with χ,ξ≥0, α,β,γ,δ>0. We establish the finite time blow-up conditions for nonradial solutions that the finite time blow-up occurs at x0∈Ω whenever ∫Ωu0(x)dx>8π/(χα−ξγ) with χα−ξγ>0, under ∫Ωu0(x)|x−x0|2dx 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 δ−β.