Well-posedness for the coupling of a random heat equation with a multiplicative stochastic Barenblatt equation

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 random heat equation coupled with Barenblatt’s type multiplicative force the sense of Itô. first step, establish well-posedness case additive noise through semi-implicit time discretization system. second derivation continuous dependence estimates solution respect to data allows us show desired case.