Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains
The aim of the paper is to study the asymptotic behaviour of the stochastic reaction-diffusion equation \[ du^\varepsilon- \Delta u^\varepsilon dt=( F(t, x,u^\varepsilon ) + G(t,x))dt + u^\varepsilon\circ dW \] with boundary condition \(\frac{du^\varepsilon}{d\nu_\varepsilon}=0\) where \(t>\tau\) and \(W\) is a two-sided real-valued Wiener process. Here, \(F\) is a nonlinear function satisfying some growth conditions and boundedness of partial derivatives and \(G\) is a function in \(L^2_{\text{loc}}\). The set on which this equation is defined is an this smooth bounded domain \({\mathcal O}_\varepsilon\) in \(\mathbb R^{n+1}\). The authors investigate what happens to the family of cocycles for above equations as \(\varepsilon\to 0\) and consider the question ``does there exist a limiting cocycle?''
scientific article; zbMATH DE number 6668253
scientific article