  In this contribution, a stochastic nonlinear evolution system under Neumann
boundary conditions is investigated. Precisely, we are interested in finding an
existence and uniqueness result for a random heat equation coupled with a
Barenblatt's type equation with a multiplicative stochastic force in the sense
of It\^o. In a first step we establish well-posedness in the case of an
additive noise through a semi-implicit time discretization of the system. 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.
