On global existence and blowup of solutions of stochastic Keller-Segel type equation

TL;DR

This paper investigates how stochastic perturbations affect the global existence or blowup of solutions in a Keller-Segel type equation, revealing divergence form noise promotes global solutions while non-divergence form noise can cause finite-time blowup.

Contribution

It provides new insights into the impact of different types of stochastic noise on solution behavior in Keller-Segel equations, including conditions for global existence and blowup.

Findings

Divergence form noise ensures global weak solutions for small initial data.

Non-divergence form noise can induce finite-time blowup with nonzero probability.

Results include continuous dependence on noise and local strong solutions.

Abstract

In this paper we consider a stochastic Keller-Segel type equation, perturbed with random noise. We establish that for special types of random pertubations (i.e. in a divergence form), the equation has a global weak solution for small initial data. Furthermore, if the noise is not in a divergence form, we show that the solution has a finite time blowup (with nonzero probability) for any nonzero initial data. The results on the continuous dependence of solutions on the small random perturbations, alongside with the existence of local strong solutions, are also derived in this work.