×

Stochastic perturbation of the Allen-Cahn equation. (English) Zbl 0959.60047

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.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
PDFBibTeX XMLCite
Full Text: EuDML EMIS