Global well-posedness theory for a class of coupled parabolic-elliptic systems

We consider a fully coupled system consisting of a parabolic equation, with boundary and initial conditions, and an abstract elliptic equation in a variational form with time as a parameter. Such systems appear in applications related to the modeling of coupled diffusion and elastic deformation processes in inhomogeneous porous media within a quasi-static assumption. We establish the global existence, uniqueness, and continuous dependence on initial and boundary data of a weak solution to the system. The proof of this result involves the proposed pseudo-decoupling method which reduces the coupled system to an initial-boundary value problem for a single implicit equation and a refined approach to deriving a priori energy estimates based on component-wise contributions of system parameters to energy norms.