Martingale Solutions of Fractional Stochastic Reaction-Diffusion Equations Driven by Superlinear Noise

TL;DR

This paper establishes the existence of martingale solutions for a class of fractional stochastic reaction-diffusion equations with superlinear noise, using advanced mathematical techniques to handle non-Lipschitz nonlinearities.

Contribution

It introduces a novel approach to prove existence of solutions for fractional stochastic PDEs with superlinear growth and non-Lipschitz coefficients, expanding the theoretical framework.

Findings

Existence of martingale solutions for fractional stochastic reaction-diffusion equations.

Application of pseudo-monotonicity and Skorokhod-Jakubowski theorems in this context.

Handling of superlinear noise without local Lipschitz conditions.

Abstract

In this paper, we prove the existence of martingale solutions of a class of stochastic equations with pseudo-monotone drift of polynomial growth of arbitrary order and a continuous diffusion term with superlinear growth. Both the nonlinear drift and diffusion terms are not required to be locally Lipschitz continuous. We then apply the abstract result to establish the existence of martingale solutions of the fractional stochastic reaction-diffusion equation with polynomial drift driven by a superlinear noise. The pseudo-monotonicity techniques and the Skorokhod-Jakubowski representation theorem in a topological space are used to pass to the limit of a sequence of approximate solutions defined by the Galerkin method.