Homogenization of stochastic semilinear parabolic equations with non-Lipschitz forcings in domains with fine grained boundaries
type: Item
instance of: scholarly article
title: Homogenization of stochastic semilinear parabolic equations with non-Lipschitz forcings in domains with fine grained boundaries
zbMATH Open document ID: 1308.60080
author: Mamadou Sango
published in: Communications in Mathematical Sciences
publication date: 2014-08-13T00:00:00Z
reviewed by: Stefan Tappe
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 35B27
zbMATH DE Number: 6329777
zbMATH Keywords: semilinear parabolic stochastic partial differential equations
zbMATH Keywords: homogenization
zbMATH Keywords: perforated domains
MaRDI profile type: Publication
DOI: 10.4310/CMS.2014.V12.N2.A7
review text: The author studies the initial boundary value problem for stochastic partial differential equations of the form NEWLINE\[NEWLINE du^{(s)} = \Big[ \Delta u^{(s)} + f \Big( t, u^{(s)} \Big) \Big] dt + \sum_{k=1}^d g_k \Big( t, u^{(s)} \Big) dw_t^{(s) k} \;\;\text{ in \(Q_T^{(s)}\),}NEWLINE\]NEWLINE NEWLINE\[NEWLINE u^{(s)}(t,x) = 0 \;\;\text{ on \(\partial Q_T^{(s)}\),} NEWLINE\]NEWLINE NEWLINE\[NEWLINE u^{(s)}(0,x) = u_0(x) \;\;\text{ in \(\Omega^{(s)}\).} NEWLINE\]NEWLINE Here \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with \(n \geq 3\), and \(\Omega^{(s)}\) is a sequence of perforated domains in \(\Omega\). The equations are considered on the cylindrical domain \(Q_T^{(s)} = (0,T) \times \Omega^{(s)}\) with boundary \(\partial Q_T^{(s)} = (0,T) \times \partial \Omega^{(s)}\). The main result of the paper states that under suitable conditions a sequence of weak solutions converges to a solution of the stochastic initial boundary value problem NEWLINE\[NEWLINE du = [ \Delta u - c(x) u + f ( t, u ) ] dt + \sum_{k=1}^d g_k ( t, u ) dw_t^k \;\;\text{ in \(Q_T\),}NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(t,x) = 0 \;\;\text{ on \(\partial Q_T\),} NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(0,x) = u_0(x) \;\;\text{ in \(\Omega\),} NEWLINE\]NEWLINE where \(Q_T = (0,T) \times \Omega\) and \(\partial Q_T = (0,T) \times \partial \Omega\).
description: scientific article; zbMATH DE number 6329777