GLOBAL BOUNDEDNESS IN A QUASILINEAR FULLY PARABOLIC CHEMOTAXIS SYSTEM WITH INDIRECT SIGNAL PRODUCTION

In this paper we develop a new and convenient technique, with fractional Gagliardo-Nirenberg type inequalities inter alia involved, to treat the quasilinear fully parabolic chemotaxis system with indirect signal production: u(t) = del . (D(u) del u - S(u) del v), tau(1)v(t) = Delta v - a(1)v + b(1)w, tau(2)w(t) = Delta w - a(2)w + b(2)u, under homogeneous Neumann boundary conditions in a bounded domain Omega subset of R-n (n >= 1), where tau(i), a(i), b(i) > 0 (i = 1, 2) are constants, and the diffusivity D and the density-dependent sensitivity S satisfy D (s) >= a(0)(s + 1)(-alpha) and 0 <= S(s) <= b(0)(s + 1)(beta) for all s >= 0 with a(0), b(0) > 0 and alpha, beta is an element of R. It is proved that if alpha + beta < 3 and n = 1, or alpha + beta < 4/n with n >= 2, for any properly regular initial data, this problem has a globally bounded and classical solution. Furthermore, consider the quasilinear attraction-repulsion chemotaxis model: u(t) = del . (D(u) del u) - chi del . (u del z) + xi del . (u del w), z(t) = Delta z - rho z + mu u, w(t) = Delta w - delta w + gamma u, where chi, mu, xi, gamma, rho, delta > 0, and the diffusivity D fulfills D(s) >= c(0) (s + 1)(M-1) for any s >= 0 with c(0) > 0 and M is an element of R. As a corollary of the aforementioned assertion, it is shown that when the repulsion cancels the attraction (i.e. chi mu = xi gamma), the solution is globally bounded if M > -1 and n = 1, or M > 2 - 4/n with n >= 2. This seems to be the first result for this quasilinear fully parabolic problem that genuinely concerns the contribution of repulsion.