The Neumann problem for a Barenblatt equation with a multiplicative stochastic force and a nonlinear source term

Abstract

In this paper, we are interested in an existence and uniqueness result for a Barenblatt’s type equation forced by a multiplicative noise, with additionally a nonlinear source term and under Neumann boundary conditions. The idea to show such a well-posedness result is to investigate in a first step the additive case with a linear source term. Through a time-discretization of the equation and thanks to results on maximal monotone operators, one is able to handle the non-linearity of the equation and pass to the limit on the discretization parameter. This allows us to show existence and uniqueness of a solution in the case of an additive noise and a linear source term. In a second step, thanks to a fixed point procedure, one shows the announced result.