Critical mass phenomena in higher dimensional quasilinear Keller-Segel systems with indirect signal production

In this paper, we deal with quasilinear Keller-Segel systems with indirect signal production, {u(t) = del center dot ((u + 1)(m-1)del u) - del center dot (u del nu), x is an element of Omega, t > 0, 0 = Delta nu - mu(t) + w, x is an element of Omega, t > 0, w(t) + w = u, x is an element of Omega, t > 0, complemented with homogeneous Neumann boundary conditions and suitable initial conditions, where Omega subset of R-n (n >= 3) is a bounded smooth domain, m >= 1 and mu(t) := f(Omega)w(center dot, t) for t > 0. We show that in the case m >= 2 - 2/n, there exists M-c > 0 such that if either m > 2 - 2/n or f(Omega)u(0) < M-c, then the solution exists globally and remains bounded, and that in the case m <= 2 - 2/n, if either m < 2 - 2/n or M > 2(n/2) n(n-1)omega(n), then there exist radially symmetric initial data such that f(Omega)u(0) = M and the solution blows up in finite or infinite time, where the blow-up time is infinite if m = 2 - 2/n. In particular, if m = 2 - 2/n, there is a critical mass phenomenon in the sense that inf {M > 0 : there exists u(0) with f(Omega)u(0) = M such that the corresponding solution blows up in infinite time} is a finite positive number.