  Two-scale homogenization limits of parabolic cross-diffusion systems in a
heterogeneous medium with no-flux boundary conditions are proved. The
heterogeneity of the medium is reflected in the diffusion coefficients or by
the perforated domain. The diffusion matrix is of degenerate type and may be
neither symmetric nor positive semi-definite, but the diffusion system is
assumed to satisfy an entropy structure. Uniform estimates are derived from the
entropy production inequality. New estimates on the equicontinuity with respect
to the time variable ensure the strong convergence of a sequence of solutions
to the microscopic problems defined in perforated domains.

Analysis of PDEs