Uniform Large Deviation Principles of Fractional Reaction-Diffusion Equations Driven by Superlinear Multiplicative Noise on R^n

TL;DR

This paper establishes uniform large deviation principles for fractional stochastic reaction-diffusion equations on R^n with superlinear noise, addressing challenges from unbounded domains and nonlinear growth.

Contribution

It develops a framework for uniform large deviations in fractional reaction-diffusion equations with superlinear noise on unbounded domains, using weak convergence and tail estimates.

Findings

Proved Freidlin-Wentzell uniform large deviations for bounded initial data.

Established Dembo-Zeitouni uniform large deviations for compact initial data.

Overcame difficulties from superlinear growth and non-compact Sobolev embeddings.

Abstract

In this paper, we investigate the uniform large deviation principle of the fractional stochastic reaction-diffusion equation on the entire space R^n as the noise intensity approaches zero. The nonlinear drift term is dissipative and has a polynomial growth of any order. The nonlinear diffusion term is locally Lipschitz continuous and has a superlinear growth rate. By the weak convergence method, we establish the Freidlin-Wentzell uniform large deviations over bounded initial data as well as the Dembo-Zeitouni uniform large deviations over compact initial data. The main difficulties are caused by the superlinear growth of noise coefficients and the non-compactness of Sobolev embeddings on unbounded domains. The dissipativeness of the drift term and the idea of uniform tail-ends estimates of solutions are employed to circumvent these difficulties.