Global existence of martingale solutions to stochastic keller-segel system with degenerate diffusion

TL;DR

This paper proves the global existence of martingale solutions for a stochastic degenerate Keller-Segel system with multiplicative noise, addressing challenges from degeneracy and non-coercivity using a fixed point approach.

Contribution

It establishes the first global existence result for martingale solutions to the stochastic degenerate Keller-Segel system with general nonnegative initial data.

Findings

Proved global existence of solutions under degenerate diffusion.

Developed a solution operator using Schauder fixed point theorem.

Addressed degeneracy and non-coercivity in the model.

Abstract

In this paper, we study the stochastic degenerate Keller-Segel system perturbed by linear multiplicative noise in a bounded domain $\mathcal{O}$. We establish the global existence of martingale solutions for this model with any nonnegative initial data in $H_{2}^{-1}(\mathcal{O})$. The main challenge in proving the existence of solutions arises from the degeneracy of the porous media diffusion and the lack of coercivity in the nonlinear chemotactic term. To overcome these difficulties, we construct a solution operator and apply the Schauder fixed point theorem within the variational framework.