BOUNDEDNESS VS. BLOW-UP IN A TWO-SPECIES CHEMOTAXIS SYSTEM WITH TWO CHEMICALS

We consider a model for two species interacting through chemotaxis in such a way that each species produces a signal which directs the respective motion of the other. Specifically, we shall be concerned with nonnegative solutions of the Neumann problem, posed in bounded domains Omega subset of R-n with smooth boundary, for the system {u(t) = Delta u - chi del.(u del v), x is an element of Omega, t > 0, 0 = Delta v - v + w, x is an element of Omega, t > 0, w(t) = Delta w - xi del.(w del z), x is an element of Omega, t > 0 0 = Delta z - z + u, x is an element of Omega, t > 0 with parameters chi is an element of {+/- 1} and xi is an element of {1}, thus allowing the interaction of either attraction-repulsion, or attraction-attraction, or repulsion-repulsion type. It is shown that in the attraction-repulsion case chi = 1 and xi = -1, if n <= 3 then for any nonnegative initial data u(0) is an element of C-0 ((Omega) over bar) and w(0) is an element of C-0 ((Omega) over bar), there exists a unique global classical solution which is bounded; in the doubly repulsive case when chi = xi = -1, the same holds true; in the attraction-attraction case chi = xi = 1, - if either n = 2 and integral(Omega) u(0) + integral(Omega) w(0) lies below some threshold, or n >= 3 and parallel to u(0)parallel to(L infinity)(Omega) and parallel to w(0)parallel to(L infinity)(Omega) are sufficiently small, then solutions exist globally and remain bounded, whereas - if either n = 2 and m is suitably large, or n >= 3 and m > 0 is arbitrary, then there exist smooth initial data u(0) and w(0) such that integral(Omega)u(0) + integral(Omega)w(0) = m and such that the corresponding solution blows up in finite time. In particular, these results demonstrate that the circular chemotaxis mechanism underlying (star) goes along with essentially the same destabilizing features as known for the classical Keller-Segel system in the doubly attractive case, but totally suppresses any blow-up phenomenon when only one, or both, taxis directions are repulsive.