# $L^p$-estimates and regularity for SPDEs with monotone semilinearity

## TL;DR

This paper establishes $L^p$-estimates and regularity results for semilinear SPDEs with monotone nonlinearities, including the stochastic Allen--Cahn and Ginzburg--Landau equations, facilitating numerical approximation analysis.

## Contribution

It provides new $L^p$-estimates and higher regularity results for a class of semilinear SPDEs with monotone nonlinearities, extending existing theory to weighted Sobolev spaces.

## Key findings

- Solutions are continuous in time with values in $H^2$ and $H^3$ Sobolev spaces.
- Solutions are $rac{1}{2} - rac{2}{q}$ Hölder continuous in time in weighted $L^q$-spaces.
- Results apply to equations with polynomial growth nonlinearities, including stochastic Allen--Cahn and Ginzburg--Landau.

## Abstract

Semilinear stochastic partial differential equations on bounded domains $\mathscr{D}$ are considered. The semilinear term may have arbitrary polynomial growth as long as it is continuous and monotone except perhaps near the origin. Typical examples are the stochastic Allen--Cahn and Ginzburg--Landau equations. The first main result of this article are $L^p$-estimates for such equations. The $L^p$-estimates are subsequently employed in obtaining higher regularity. This is motivated by ongoing work to obtain rate of convergence estimates for numerical approximations to such equations. It is shown, under appropriate assumptions, that the solution is continuous in time with values in the Sobolev space $H^2(\mathscr{D}')$ and $\ell^2$-integrable with values in $H^3(\mathscr{D}')$, for any compact $\mathscr{D}' \subset \mathscr{D}$. Using results from $L^p$-theory of SPDEs obtained by Kim~\cite{kim04} we get analogous results in weighted Sobolev spaces on the whole $\mathscr{D}$. Finally it is shown that the solution is H\"older continuous in time of order $\frac{1}{2} - \frac{2}{q}$ as a process with values in a weighted $L^q$-space, where $q$ arises from the integrability assumptions imposed on the initial condition and forcing terms.