Chemotaxis can prevent thresholds on population density

Abstract

We define and (for q > n) prove uniqueness and an extensibility property of W '-solutions to ut = -∇ · (u ∇v) +?u - μu2 0 = Δv - v + u ∂vv|∂Ω = ∂vu|∂Ω = 0, u(0, ·) = u0, in balls in ℝn. They exist globally in time for μ ≥ 1 and, for a certain class of initial data, undergo finite-time blow-up if μ < 1. We then use this blow-up result to obtain a criterion guaranteeing some kind of structure formation in a corresponding chemotaxis system - thereby extending recent results of Winkler [26] to the higher dimensional (radially symmetric) case.