Existence, Uniqueness, and Pathwise Regularity for Multidimensional Semilinear SPDEs with Locally Lipschitz Coefficients and Rough Initial Data

We study multidimensional semilinear stochastic evolution equations driven by multiplicative noise and subject to rough initial data.

The drift and diffusion coefficients are assumed to take values in negative fractional order spaces and may exhibit temporal singularities at the initial time.

Our main results establish well-posedness and pathwise spatio-temporal regularity of the solutions when these coefficients are globally Lipschitz, and we prove the existence, uniqueness, and pathwise regularity of maximal local solutions when the coefficients are locally Lipschitz.

The local Lipschitz condition is formulated with respect to a specific time-weighted norm.

This enables us to apply our theoretical results to linear stochastic partial differential equations, as well as to models with non-globally Lipschitz nonlinearities such as the stochastic Burgers, Allen--Cahn, Fisher--KPP, Burgers--Fisher equations, and the Ginzburg--Landau system.

Furthermore, our framework accommodates singular initial data, such as the Dirac measure, in dimension $d=1$, and nonsmooth initial data in dimensions $d \in \{2,3\}$.