Homogenization of stochastic semilinear parabolic equations with non-Lipschitz forcings in domains with fine grained boundaries
semilinear parabolic stochastic partial differential equations
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 \(\partial Q_T = (0,T) \times \partial \Omega\).
scientific article; zbMATH DE number 6329777