It is well known that solutions to deterministic Keller--Segel chemotaxis systems can blow up in finite time under certain conditions. To investigate whether noise can prevent such blow-up, we study parabolic--elliptic and fully parabolic Keller--Segel systems on smooth bounded domains with pointwise superlinear multiplicative Itô noise σ₀uʳ\, dWₜ, where W is a real Brownian motion. Our results show that superlinear multiplicative noise can prevent finite-time blow-up in Keller--Segel systems. For $r>3/2$, we prove global existence and pathwise uniqueness of nonnegative mild solutions in arbitrary spatial dimension, for every σ₀>0. At $r=3/2$, both systems are globally well-posed in two dimensions for every positive noise intensity; in arbitrary dimension, the parabolic--elliptic system is global under σ₀²≥2χ. For $r>3/2$ in the fully parabolic case, the proof combines a logarithmic estimate for the time derivative of the chemical signal with comparison of the local solution to the solution of a scalar stochastic differential equation. The two-dimensional case follows from mass quadratic variation, deterministic maximal regularity, and localized energy estimates. We also establish almost-sure L¹ extinction and algebraic decay of fractional mass moments.
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Yunfeng et al. (2026) studied this question.