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

Neelima, David Šiška

Published 2017-05-29Version 1

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 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 $\mathscr{D}'$ that is open and compactly contained in $\mathscr{D}$. Analogous results are also obtained in weighted Sobolev spaces on the whole $\mathscr{D}$ using results of Krylov 1994.