  We study the weak solvability of a system of coupled Allen-Cahn-like
equations resembling cross-diffusion which is arising as a model for the
consolidation of saturated porous media. Besides using energy like estimates,
we cast the special structure of the system in the framework of the
Leray-Schauder fixed point principle and ensure this way the local existence of
strong solutions to a regularised version of our system. Furthermore, weak
convergence techniques ensure the existence of weak solutions to the original
consolidation problem. The uniqueness of global-in-time solutions is guaranteed
in a particular case. Moreover, we use a finite difference scheme to show the
negativity of the vector of solutions.
