Global weak martingale solutions to a stochastic two-sidedly degenerate aggregation-diffusion equation issued from biology

Abstract

The purpose of this paper is to establish the well-posedness of martingale (probabilistic weak) solutions to stochastic degenerate aggregation--diffusion equations arising in biological and public health contexts. The studied equation is of a stochastic degenerate parabolic type, featuring a nonlinear two-sidedly degenerate diffusion term accounting for repulsion, a locally Lipschitz reaction term representing competitive interactions, and a stochastic perturbation term capturing environmental noise and uncertainty in biological systems. The existence of martingale solutions is proved via an auxiliary nondegenerate stochastic system combined with the Faedo--Galerkin method. Convergence of approximate solutions is established through Prokhorov's compactness and Skorokhod's representation theorems, and uniqueness is obtained using a duality approach. Finally, numerical simulations are given to illustrate the impact of environmental noise on aggregation dynamics and the long-term behavior of the system, offering insights that may inspire medical innovation and predictive modeling in public health.