A refined criterion and lower bounds for the blow-up time in a parabolic-elliptic chemotaxis system with nonlinear diffusion

This paper deals with unbounded solutions to the following zero-flux chemotaxis system {u(t) = del center dot[(u + alpha)(m1-1 del u-)chi(u(u + alpha)m2-2 del u)] (x, t) epsilon Omega x (0, T-max), (lozenge) 0 = Delta v - M + u (x, t) epsilon Omega x (0, T-max) where alpha > 0, Omega is a smooth and bounded domain of R-n, with n >= 1, t is an element of (0, T-max) is T-max the blow-up time, and m(1), m(2) are real numbers. Given a sufficiently smooth initial data u(0) := u(x, 0) >= 0 and set M := 1/vertical bar Omega vertical bar integral(Omega) u(0)(x) dx, from the literature it is known that under a proper interplay between the above parameters m(1), m(2) and the extra condition integral(Omega) v(x, t) dx = 0, system (lozenge) possesses for any chi > 0 a unique classical solution which becomes unbounded at t NE arrow T-max. In this investigation we first show that for p(0) > n/2 (m(2) - m(1)) any blowing up classical solution in L-p0(Omega)-norm blows up also in L-infinity (Omega)-norm. Then we estimate the blow-up time Tmax providing a lower bound T-max (C) 2019 Elsevier Ltd. All rights reserved.