CHEMOTACTIC AGGREGATION VERSUS LOGISTIC DAMPING ON BOUNDEDNESS IN THE 3D MINIMAL KELLER-SEGEL MODEL

摘要

We study chemotactic aggregation versus logistic damping on boundedness for the 3D minimal Keller-Segel (KS) model with logistic source, {u(t) = del. (del u - chi u del v) + u - mu u(2), x is an element of Omega, t 0; v(t) = Delta v - v + u, x is an element of Omega t 0}, in a smooth, bounded, but not necessarily convex domain Omega subset of R-3 with chi, mu 0, nonnegative initial data up, vo, and homogeneous Neumann boundary data. In a previous work [T. Xiang, T. Math. Anal. Appl., 459 (2018), pp. 1172-1200], the global boundedess of the above KS system in nonconvex domains is guaranteed under the explicit condition (*) mu theta(0)chi = 9/root 10-2 chi. As a continuation, under the critical condition (*), up to a scaling constant depending only on u(0), v(0), and Omega, we here establish explicit uniform-in-time upper bounds for the quantities parallel to u(.,t)parallel to L infinity(Omega) and parallel to v(., t) w(1), infinity(Omega) in terms of chi and mu; these bounds are defined for all chi = 0 and mu theta(0)chi, increasing in chi and decreasing in mu, and have only one singular line in it at mu = theta(0)chi. The corresponding 2D qualitative boundedness has been investigated [H. Jin and T. Xiang, C. R. Math. Acad. Sci. Paris, 356 (2018), pp. 875-885], wherein the only singular line of the upper bounds in it is shown to be mu = 0. By comparison, an interesting new feature is that the singular line it = 0 of the corresponding 2D upper bounds has moved up to mu = theta(0)chi in the 3D setting. It is worth mentioning that, in the absence of a logistic source, the corresponding classical KS model (by setting mu = 0 and removing the proliferation term +u in the first equation) is now well known to possess blow-up solutions for even small initial data [M. Winkler, T. Math. Pares Appl., 100 (2013), pp. 748-767].