Explicit lower bound of blow-up time in a fully parabolic chemotaxis system with nonlinear cross-diffusion

This paper detects the lower bounds of blow-up time of smooth solutions for the chemotaxis model { u(t) = Delta u - chi del . (u(u + 1)(m-1)del v), x is an element of B-1(0), t > 0, v(t) = Delta v - v + u, x is an element of B-1(0), t > 0, under homogeneous Neumann boundary conditions in a unit ball B-1(0) C R-3 centered at the origin, with positive constant chi and parameter m is an element of R. Under the assumption that (u(x, 0), v(x,0)) = (u(0)(vertical bar x vertical bar), v(0)(vertical bar x vertical bar)) is an element of C-0((B) over bar (1) (0)) x W-1,W-infinity (B-1(0)), it is shown that whenever m is an element of [2/3,2], the blow-up time of a classical solution to the corresponding initial boundary problem has an explicit lower bound measured in terms of chi, integral(B1 (0)) u(0)(p) and integral(B1(0)) vertical bar del(v0)vertical bar(2q) for appropriate p > 1 and q > 1. Here we underline that the global classical solution exists and is bounded if m < I, which leads to the assumption m > for addressing the properties of blow-up solutions. However, the question of lower bounds of blow-up time for the case m > 2 remains open due to technical reasons. (C) 2015 Elsevier Inc. All rights reserved.