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

TL;DR

This paper establishes the well-posedness, including existence and uniqueness, for a coupled stochastic heat and Barenblatt equation with multiplicative noise, using discretization and dependence estimates.

Contribution

It extends previous results by proving well-posedness for a coupled nonlinear stochastic system with multiplicative noise.

Findings

Proved existence and uniqueness of solutions.

Developed semi-implicit discretization method.

Derived continuous dependence estimates.

Abstract

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.