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

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.