# Lp-estimates and regularity for SPDEs with monotone semilinearity

Neelima Neelima, David Siska

Research output: Contribution to journalArticlepeer-review

## 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. A typical example is the stochastic Ginzburg--Landau equation. The main result of this article are $L^p$-estimates for such equations. The $L^p$-estimates are subsequently employed in obtaining higher regularity. It is shown, under appropriate assumptions, that the solution is continuous in time with values in the Sobolev space $H^2(\mathscr{D}')$ and $L^2$-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 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.
Original language English 422–459 38 Stochastics and Partial Differential Equations: Analysis and Computations 8 11 Sep 2019 https://doi.org/10.1007/s40072-019-00150-w Published - 30 Jun 2020

• math.PR
• 60H15, 35R60

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