Lower bounds for blow-up in a model of chemotaxis

Many special cases of the classical Keller–Segel system for modeling chemotaxis have been investigated in the literature, and typically the solution of the governing equations will blow up at some finite time. However, the question of establishing lower bounds for this blow-up time has been largely ignored. This paper derives such a lower bound in a parabolic–parabolic model in both ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$.