Existence, Uniqueness, and Pathwise Regularity for Multidimensional Semilinear SPDEs with Locally Lipschitz Coefficients and Rough Initial Data
Abstract
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 the application of our theoretical results to linear stochastic partial differential equations, as well as to a broad class of models with non-globally Lipschitz nonlinearities. These include the stochastic Burgers, Allen--Cahn, Fisher--KPP, and Burgers--Fisher equations, alongside the Ginzburg--Landau system and models featuring exponential growth, such as the stochastic solid fuel ignition model.
Related
- Sharp non-uniqueness of weak solutions to the semilinear wave equation
- Existence of a separating norm for the semilinear heat equation
- Stable determination of damping and potential coefficients in a semilinear wave equation
- Interior regularity of mixed local-nonlocal parabolic semilinear equations
- Existence of infinitely many singular self-similar solutions for the semilinear heat equation
BibTeX
@misc{naik2026existence,
title = {{Existence, Uniqueness, and Pathwise Regularity for Multidimensional Semilinear SPDEs with Locally Lipschitz Coefficients and Rough Initial Data}},
author = {Jitendra Nath Naik and Lok Pati Tripathi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.13341},
eprint = {2607.13341},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.13341},
}
RIS
TY - UNPB
TI - Existence, Uniqueness, and Pathwise Regularity for Multidimensional Semilinear SPDEs with Locally Lipschitz Coefficients and Rough Initial Data
AU - Jitendra Nath Naik
AU - Lok Pati Tripathi
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2607.13341
UR - https://arxiv.org/abs/2607.13341
AB - 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 the application of our theoretical results to linear stochastic partial differential equations, as well as to a broad class of models with non-globally Lipschitz nonlinearities. These include the stochastic Burgers, Allen--Cahn, Fisher--KPP, and Burgers--Fisher equations, alongside the Ginzburg--Landau system and models featuring exponential growth, such as the stochastic solid fuel ignition model.
ER -