Semi-classical solutions for a nonlinear coupled elliptic-parabolic problem

Abstract

We give an existence result for a fully nonlinear system consisting of a parabolic equation strongly coupled with an elliptic one. It models in particular miscible displacement in porous media. To this aim, we adapt the tools of Ladyzenskaja, Solon-nikov and Uralćeva [27, 28] to the coupled nonlinear setting. Under some reasonable assumptions on the data, we state the existence of semi-classical solutions for the problem. We also give an existence result of weak solutions for a degenerate form of the problem.