Blow-up for a Stochastic Model of Chemotaxis Driven by Conservative Noise on $\mathbb{R}^2$

TL;DR

This paper investigates conditions under which solutions to a stochastic Keller--Segel chemotaxis model in two dimensions blow up in finite time, highlighting the influence of noise and chemotactic sensitivity on solution behavior.

Contribution

It establishes criteria for blow-up in a stochastic chemotaxis model, extending deterministic results to include the effects of conservative noise and spatial inhomogeneity.

Findings

Blow-up occurs with probability 1 when chemotactic sensitivity is sufficiently large.

In an intermediate regime, blow-up occurs with positive probability depending on initial data and noise correlation.

The criteria align with deterministic models with increased viscosity in the large sensitivity regime.

Abstract

We establish criteria on the chemotactic sensitivity $\chi$ for the non-existence of global weak solutions (i.e. \textit{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 \emph{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 \emph{blow-up} occurs with positive probability.