Well-posedness result for a system of random heat equation coupled with a multiplicative stochastic Barenblatt equation

International audienceIn this paper, a stochastic nonlinear evolution system under Neumann boundary conditions is investigated. Precisely, we are interested in finding an existence and uniqueness result for a system of random heat equation coupled with a Barenblatt's type equation with a multiplicative stochastic force in the sense of Itô. To do so, we investigate in a first step the case of an additive noise through a semi-implicit in time discretization of the system. This allows us to show the well-posedness of the system in the additive case. In a second step, the derivation of continuous dependence estimates of the solution with respect to the data allows us to show the desired existence and uniqueness result for the multiplicative case