Local and global pathwise solutions for a stochastically perturbed nonlinear dispersive PDE

摘要

In this paper, we consider the periodic Cauchy problem for a stochastically perturbed nonlinear dispersive partial differential equation with cubic nonlinearity, which involves the integrable Novikov equation arising from the shallow water wave theory as a special case. We first establish the existence and uniqueness of local pathwise solutions in Sobolev spaces H-s(T)(s > 3/2) with nonlinear multiplicative noise, where the key ingredients are the stochastic compactness method, the Skorokhod representation theorem and the Gyongy-Krylov characterization of convergence in probability. In the case of linear multiplicative noise, we investigate the conditions which lead to the blow-up phenomena and global existence of pathwise solution. Finally, we show that the linear multiplicative noise has a dissipative effect on the periodic peakon solutions to the associated deterministic Novikov equation. (C) 2020 Elsevier B.V. All rights reserved.