Well-posedness, Regularity, and Strong Approximations of Superlinear Stochastic Reaction-Diffusion Equation
Abstract
This paper develops a general framework for the well-posedness, regularity, and strong approximation of the stochastic reaction--diffusion equation (SRDE) with superlinear drift and diffusion coefficients. We first extend the well-posedness results in \emph{W. Liu and M. Röckner, J. Funct. Anal., 2902--2922, 2010} and \emph{W. Liu, J. Differential Equations, 572--592, 2013} to the case of superlinear diffusion in the Gelfand triple V ↪ H ↪ V*, with V equipped with the norm |·|V, and derive a moment estimate by establishing a new Itô formula for |X|Vp with general p ≥ 2. We then apply this abstract result to the SRDE, establish higher spatial regularity Ḣ1+γ for any γ∈ [0,1] whenever the initial datum lies in the same Sobolev space, and obtain temporal Hölder regularity. Finally, we construct a family of tamed finite element methods (tamed-FEMs) for the SRDE under general assumptions on the tamed functions, derive their long-time unconditional stability, and establish optimal strong convergence rates. To our knowledge, this is the first strong approximation result for SPDEs with superlinear diffusion coefficients.
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BibTeX
@misc{liu2026wellposedness,
title = {{Well-posedness, Regularity, and Strong Approximations of Superlinear Stochastic Reaction-Diffusion Equation}},
author = {Zhihui Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16979},
eprint = {2609.16979},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16979},
note = {CC0},
}
RIS
TY - UNPB
TI - Well-posedness, Regularity, and Strong Approximations of Superlinear Stochastic Reaction-Diffusion Equation
AU - Zhihui Liu
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16979
UR - https://arxiv.org/abs/2609.16979
AB - This paper develops a general framework for the well-posedness, regularity, and strong approximation of the stochastic reaction--diffusion equation (SRDE) with superlinear drift and diffusion coefficients. We first extend the well-posedness results in \emph{W. Liu and M. Röckner, J. Funct. Anal., 2902--2922, 2010} and \emph{W. Liu, J. Differential Equations, 572--592, 2013} to the case of superlinear diffusion in the Gelfand triple V ↪ H ↪ V^*, with V equipped with the norm |·|_V, and derive a moment estimate by establishing a new Itô formula for |X|_V^p with general p ≥ 2. We then apply this abstract result to the SRDE, establish higher spatial regularity Ḣ^1+γ for any γ∈ [0,1] whenever the initial datum lies in the same Sobolev space, and obtain temporal Hölder regularity. Finally, we construct a family of tamed finite element methods (tamed-FEMs) for the SRDE under general assumptions on the tamed functions, derive their long-time unconditional stability, and establish optimal strong convergence rates. To our knowledge, this is the first strong approximation result for SPDEs with superlinear diffusion coefficients.
ER -