Chemotaxis effect vs. logistic damping on boundedness in the 2-D minimal Keller-Segel model

We study the chemotaxis effect vs. logistic damping on boundedness for the well-known minimal Keller-Segel 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 domain Omega subset of R-2 with chi, mu > 0, nonnegative initial data u(0), v(0), and homogeneous Neumann boundary data. It is well known that this model allows only for global and uniform-in-time bounded solutions for any chi, mu > 0. Here, we carefully employ a simple and new method to regain its boundedness, with particular attention to how upper bounds of solutions qualitatively depend on chi and mu. More, precisely, it is shown that there exists C = C(u(0), v(0), Omega) > 0 such that parallel to u(., t)parallel to(L infinity(Omega)) <= C [1 + 1/mu + chi K (chi, mu)N(chi, mu)] and parallel to v(., t)parallel to(L infinity(Omega)) <= C [1 + 1/mu + chi(8/3)/mu K-8/3 (chi, mu)] =:CN(chi, mu) uniformly on [0,infinity), where K(chi, mu) = M (chi, mu) E (chi, mu), M(chi, mu) = 1 + 1/mu + root chi(1 + 1/mu(2)) and E(chi, mu) = exp[chi C-GN(2)/2 min{1, 2/chi}(4/mu parallel to u(0)parallel to(L1(Omega)) + 13/2 mu(2)vertical bar Omega vertical bar + parallel to del v(0)parallel to(L2(Omega)2))]. We notice that these upper bounds are increasing in chi, decreasing in mu, and have only one singularity at mu = 0, where the corresponding minimal model (removing the term u - mu u(2) in the first equation) is widely known to possess blow-ups for large initial data. (C) 2018 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.