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 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.
Related
- Superlinear nonlocal diffusion systems for multispecies populations: well-posedness and discrete chain rules
- Global Well-Posedness and Large Time Asymptotic Behavior of Strong Solutions to the Compressible Viscous Non-Resistive MHD Equations with Vacuum
- Global Well-Posedness and Asymptotic Behavior of Energy-Damped Subquintic Wave Equations
- Global Well-posedness and Asymptotic Analysis of a Damped Nonlinear Wave Equation with a Codimension-One Constraint
- Well-posedness, Regularity, and Strong Approximations of Superlinear Stochastic Reaction-Diffusion Equation
BibTeX
@misc{chen2026global,
title = {{Global Well-posedness and Asymptotic Extinction for Keller-Segel Systems with Superlinear Noise}},
author = {Yunfeng Chen and Tusheng Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36485},
eprint = {2610.36485},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36485},
}
RIS
TY - UNPB
TI - Global Well-posedness and Asymptotic Extinction for Keller-Segel Systems with Superlinear Noise
AU - Yunfeng Chen
AU - Tusheng Zhang
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36485
UR - https://arxiv.org/abs/2610.36485
AB - 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 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≥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 L^1 extinction and algebraic decay of fractional mass moments.
ER -