Martingale Solutions of Fractional Stochastic Reaction-Diffusion Equations Driven by Superlinear Noise
Bixiang Wang
TL;DR
This work establishes the existence of martingale solutions for a class of stochastic partial differential equations with a pseudo-monotone drift of polynomial growth and superlinear noise, without requiring local Lipschitz continuity. By formulating the problem in an abstract framework on a Gelfand triple and developing Galerkin approximations, the authors derive uniform estimates and prove tightness using the Skorokhod-Jakubowski representation in nonmetric spaces. They then apply the general result to fractional stochastic reaction-diffusion equations driven by superlinear noise, verifying the structural hypotheses under concrete growth conditions and obtaining existence (and, under additional assumptions, uniqueness) of martingale solutions with robust a priori bounds. The approach combines pseudo-monotone operator theory, compact embeddings, and a nonmetric Skorokhod representation to overcome non-Lipschitz drift and non-Lipschitz diffusion, offering a versatile methodology for a broad class of fractional SPDEs.
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.