Existence of weak solutions for a class of semilinear stochastic wave equations
type: Item
instance of: scholarly article
zbMATH Open document ID: 1251.60055
DOI: 10.1137/110826667
author: Carlo Marinelli
author: Lluís Quer-Sardanyons
published in: SIAM Journal on Mathematical Analysis
publication date: 2012-08-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1003.1024
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 35R60
zbMATH DE Number: 6070873
zbMATH Keywords: stochastic PDEs
zbMATH Keywords: nonlinear wave equations
zbMATH Keywords: monotone operators
zbMATH Keywords: infinite dimensional semimartingales
MaRDI profile type: Publication
OpenAlex ID: W2962944213
arXiv ID: 1003.1024
title: Existence of weak solutions for a class of semilinear stochastic wave equations
review text: The authors prove the existence of weak solutions (in the probabilistic sense) for the following semilinear wave equation: NEWLINE\[NEWLINE \frac{\partial^2 u}{\partial t^2}(x,\,t) - \Delta u(x,\,t) + \beta(u(x,\,t)) = \dot{\eta}(x,\,t),\quad (x,\,t)\in D\times [0,\,T] \subset{\mathbb R}^{d+1}, NEWLINE\]NEWLINE where \(\beta\) is a maximal monotone graph in \({\mathbb R}\times{\mathbb R}\) and \(\dot{\eta}\) is the time derivative of a stochastic integral with respect to a Hilbert space-valued martingale which may depend upon \(u\). Some auxiliary results on mild solutions of stochastic evolution equations with Lipschitz nonlinearities are also established.
reviewed by: Iulian Stoleriu
description: scientific article; zbMATH DE number 6070873