Boundedness and finite-time blow-up in a quasilinear parabolic-elliptic-elliptic attraction-repulsion chemotaxis system

This paper deals with the quasilinear attraction-repulsion chemotaxis system {u(t) = del . ((u + 1)(m-1) del u - chi u(u + 1)(p-2)del v + xi u(u + 1)(q-2) del w) + f(u), 0 = Delta v + alpha u - beta v, 0 = Delta w + gamma u - delta w in a bounded domain Omega subset of R-n (n is an element of N) with smooth boundary partial derivative Omega, where m, p, q is an element of R, chi, xi, alpha, beta, gamma, delta > 0 are constants, and f is a function of logistic type such as f(u) = lambda u - mu u(kappa) with lambda, mu > 0 and kappa >= 1, provided that the case f(u) equivalent to 0 is included in the study of boundedness, whereas. is sufficiently close to 1 in considering blow-up in the radially symmetric setting. In the case that xi = 0 and f(u) = 0, global existence and boundedness have already been proved under the condition p < m+ 2/n. Also, in the case that m = 1, p = q = 2 and f is a function of logistic type, finite-time blow-up has already been established by assuming chi alpha - xi gamma > 0. This paper classifies boundedness and blow-up into the cases p < q and p > q without any condition for the sign of chi alpha - xi gamma and the case p = q with chi alpha - xi gamma < 0 or chi alpha - xi gamma > 0.