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

Caroline Bauzet (1) , Frédéric Lebon (1) , Asghar Maitlo (1)

1 LMA - Laboratoire de Mécanique et d'Acoustique [Marseille]

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. Trough a time-discretization of the equation and thanks to results on maximal monotone operator, 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.

Mots clés: multiplicative noise; additive noise; stochastic force; Itô integral; maximal monotone operator; Neumann condition; time discretization; heat equation; fixed point; Stochastic Barenblatt equation

Domaines: Equations aux dérivées partielles [math.AP]

Caroline Bauzet, Frédéric Lebon, Asghar Maitlo. The Neumann problem for a Barenblatt equation with a multiplicative stochastic force and a nonlinear source term. Nonlinear Differential Equations and Applications, 2019, ⟨10.1007/s00030-019-0567-5⟩. ⟨hal-01910747⟩