Stochastic pseudo-parabolic equation with logarithmic nonlinearity

This manuscript was dedicated to exploring the existence and uniqueness of global solutions pertaining to a specific class of stochastic pseudo-parabolic equations. These equations have logarithmic nonlinearity and are driven by Brownian motion. By utilizing the Galerkin method, Prokhorov's theorem, and Skorohod's embedding theorem, we proved the existence of a global solution in the weak probabilistic sense. Subsequently, by leveraging the uniqueness of solutions and applying the Yamada-Watanabe theorem, the global existence and uniqueness of a probability strong solution were established. A notable finding, in contrast to the stochastic heat equation, was that as the pseudo-parabolic coefficient $\mu$ increased, the growth condition imposed on the noise coefficient within the stochastic pseudo-parabolic equation could be significantly relaxed.