Chemotactic collapse in a parabolic-elliptic system of mathematical biology

Abstract

We study the blowup mechanism for a simplified system of chemotaxis. First, Moser's iteration scheme is applied and the blowup point of the solution is characterized by the behavior of the local Zygmund norm. Then, Gagliardo-Nirenberg's inequality gives ε0 > 0 satisfying lim supt↑Tmax ||u(t)|| L1(BR(x0)∩ω) ≥ ε0 for any blowup point x0 ∈ωand R > 0. On the other hand, from the study of the Green's function it appears that t → ||u(t)|| L1(BR(x0)∩ω) has a bounded variation. Those facts imply the finiteness of blowup points, and then, the chemotactic collapse at each blowup point and an estimate of the number of blowup points follow.