Global Well-posedness and Asymptotic Extinction for Keller-Segel Systems with Superlinear Noise

Abstract

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 $σ_0u^r\,\mathrm dW_t$, 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 $σ_0>0$. 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 $σ_0^2\ge2χ$. 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 $L^1$ extinction and algebraic decay of fractional mass moments.