Shardlow, Tony Stochastic perturbation of the Allen-Cahn equation. (English) Zbl 0959.60047 Electron. J. Differ. Equ. 2000, Paper No. 47, 19 p. (2000). The Allen-Cahn equation is considered with small diffusion \(\varepsilon^2\) and Neumann conditions perturbed by a space time white noise of intensity \(\sigma\). In the limit, \(\sigma/\varepsilon^2\to 0\), solutions converge to the noise free problem in the \(L_2\)-norm. The simplest case is when the nonlinear \(f(u)\) is globally Lipschitz from \(L_2(0,1)\) to itself and the structure of the problem is preserved provided \(\sigma/\sqrt\varepsilon\to 0\) and Dirichlet boundary conditions are used. The constant variation formula and mild solutions are the basic facts for describing the behaviour of the stochastic Allen-Cahn P.D.E. by a system of stochastic differential equations. The analysis is given in detail and it is recommended to researchers interested in numerical analysis associated with asymptotic behaviour of perturbed P.D.E. Reviewer: Constantin Vârsan (Bucureşti) Cited in 6 Documents MSC: 60H15 Stochastic partial differential equations (aspects of stochastic analysis) Keywords:stochastic partial differential equations; asymptotics; Dirichlet boundary conditions PDFBibTeX XMLCite \textit{T. Shardlow}, Electron. J. Differ. Equ. 2000, Paper No. 47, 19 p. (2000; Zbl 0959.60047) Full Text: EuDML EMIS