Refined existence and regularity results for a class of semilinear dissipative SPDEs

TL;DR

This paper establishes existence, uniqueness, and regularity results for a broad class of semilinear dissipative stochastic partial differential equations with minimal initial data assumptions and explores how solution properties depend on initial data and coefficients.

Contribution

It extends previous results by allowing merely measurable initial data and locally Lipschitz diffusion coefficients, providing quantitative moment estimates and regularity improvements.

Findings

Proves existence and uniqueness under minimal assumptions.

Establishes moment bounds depending on initial data integrability.

Shows regularity of solutions improves with initial data and coefficient regularity.

Abstract

We prove existence and uniqueness of solutions to a class of stochastic semilinear evolution equations with a monotone nonlinear drift term and multiplicative noise, considerably extending corresponding results obtained in previous work of ours. In particular, we assume the initial datum to be only measurable and we allow the diffusion coefficient to be locally Lipschitz-continuous. Moreover, we show, in a quantitative fashion, how the finiteness of the $p$-th moment of solutions depends on the integrability of the initial datum, in the whole range $p \in ]0,\infty[$. Lipschitz continuity of the solution map in $p$-th moment is established, under a Lipschitz continuity assumption on the diffusion coefficient, in the even larger range $p \in [0,\infty[$. A key role is played by an It\^o formula for the square of the norm in the variational setting for processes satisfying minimal integrability conditions, which yields pathwise continuity of solutions. Moreover, we show how the regularity of the initial datum and of the diffusion coefficient improves the regularity of the solution and, if applicable, of the invariant measures.