GLOBAL BOUNDEDNESS IN A QUASILINEAR TWO-SPECIES ATTRACTION-REPULSION CHEMOTAXIS SYSTEM WITH TWO CHEMICALS

This paper studies the quasilinear attraction-repulsion chemotaxis system of two-species with two chemicals u(t) = del . (D-1(u)del u) - del . (Phi(1)(u)del v), 0 = Delta v - v + w(gamma 1), w(t) = del . (D-2(w)del w) + del . (Phi(2)(w)del z), 0 = Delta z( )- z + u(gamma 2), subject to the homogeneous Neumann boundary conditions in a bounded domain Omega subset of R-N (N >= 2) with smooth boundary, where gamma(i) > 0, D-i, Phi(i) is an element of C-2[0,+infinity), D-i (s) >= (s + 1)(pi), Phi(i)(s) >= 0 for s >= 0, and Phi(i)(s) <= chi(i)s(qi) for s > s(0) with chi(i) > 0, p(i), q(i) is an element of R (i = 1, 2), s(0) > 1. It is shown that if gamma(1) < 2/N (or gamma(2) < 4/N with gamma(2) <= 1), the global boundedness of solutions are guaranteed by the self-diffusion dominance of u (or w) with p(1) > q(1) + gamma(1 )- 1- 2/N (or p(2) > q(2) + gamma(2) - 1 - 4/N); if p(j) >= q(i) + gamma(i) - 1 - 2/N, i, j = 1,2 (i.e. the self-diffusion of u and w are dominant), then the solutions are globally bounded; in particular, different from the results of the single-species chemotaxis system, for the critical case p(j) = q(i) + gamma(i )- 1 - 2/N, the global boundedness of the solutions can be obtained.