Stochastic Allen-Cahn Equation with Logarithmic Potential

Abstract

We prove existence and uniqueness of a solution for the stochastic Allen-Cahn equation with logarithmic potential and multiplicative Wiener noise, under homogeneous Neumann boundary condition. The existence of a solution is obtained in the variational sense by means of an approximated equation involving a Yosida regularization of the nonlinearity. The noise is assumed to vanish at the extremal points of the physical relevant domain and to satisfy a suitable Lipschitz-continuity property, allowing to prove uniform estimates of the approximated solution. The passage to the limit is then carried out and continuous dependence on the initial datum of the solution is verified. Under an additional assumption, the existence of an analytically strong solution is proved. Finally, estimates for the derivatives of the logarithmic potential are derived, in view of the study of an optimal control problem associated to the stochastic Allen-Cahn equation.

arXiv commentTo be published in Nonlinear Analysis. Number of pages: 22. Main changes in the new version: added references, updated contact details

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Cited by in corpus (4)

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• Stochastic evolution equations with Wick-analytic nonlinearities
• Theoretical analysis of a finite-volume scheme for a stochastic Allen-Cahn problem with constraint
• Gradient flows and randomised thresholding: sparse inversion and classification