Approaching optimality in blow-up results for Keller-Segel systems with logistic-type dampening

Nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis system {ut = Delta u - del center dot (u del v) + lambda u - mu u(kappa), 0 = Delta v - (m) over bar (t) + u, (m) over bar (t) = 1 vertical bar Omega vertical bar integral(Omega) u(center dot, t) (*) in smooth bounded domains Omega subset of R-n, n >= 1, are known to be global in time if lambda >= 0, mu > 0 and kappa > 2. In the present work, we show that the exponent. = 2 is actually critical in the four- and higher dimensional setting. More precisely, if n >= 4, kappa is an element of (1, 2) and mu > 0 or n >= 5, kappa = 2 and mu is an element of (0, n - 4/n), for balls Omega subset of R-n and parameters lambda >= 0, m(0) > 0, we construct a nonnegative initial datum u(0) is an element of C-0 ((Omega) over bar) with integral(Omega) u(0) = m(0) for which the corresponding solution (u, v) of (*) blows up in finite time. Moreover, in 3D, we obtain finite-time blow-up for kappa is an element of (1, 3/2) (and lambda >= 0, mu > 0). As the corner stone of our analysis, for certain initial data, we prove that the mass accumulation function w(s, t) = integral(n root s)(0) rho(n-1) u(rho, t) d rho fulfills the estimate w(s) <= w/s. Using this information, we then obtain finite-time blow-up of u by showing that for suitably chosen initial data, s(0) and gamma, the function phi(t) = integral(s0)(0) s(-gamma) (s(0) - s)w(s, t) cannot exist globally.