Blow-up for a stochastic model of chemotaxis driven by conservative noise on mathbb {R}^2

Abstract

We establish criteria on the chemotactic sensitivity chi for the non-existence of global weak solutions (i.e., blow-up in finite time) to a stochastic Keller–Segel model with spatially inhomogeneous, conservative noise on mathbb {R}^2. We show that if chi is sufficiently large then blow-up occurs with probability 1. In this regime, our criterion agrees with that of a deterministic Keller–Segel model with increased viscosity. However, for chi in an intermediate regime, determined by the variance of the initial data and the spatial correlation of the noise, we show that blow-up occurs with positive probability.