Stochastic Allen–Cahn equation with logarithmic potential

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