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

Abstract

AbstractWe establish criteria on the chemotactic sensitivity $$\chi $$ χ 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 $$\mathbb {R}^2$$ R 2 . We show that if $$\chi $$ χ 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 $$\chi $$ χ 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.