CHEMOTAXIS CAN PREVENT THRESHOLDS ON POPULATION DENSITY

We define and (for q > n) prove uniqueness and an extensibility property of W-1,W-q-solutions to u(t) = -del . (u del v) + Ku - mu u(2) 0 = Delta v - v + u partial derivative(v)v\partial derivative Omega = partial derivative(v)u\partial derivative Omega = 0, u(0, .) = u(0), in balls in R-n. They exist globally in time for mu >= 1 and, for a certain class of initial data, undergo finite-time blow-up if mu < 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.