  The present paper is concerned with a space-time homogenization problem for
nonlinear diffusion equations with periodically oscillating (in space and time)
coefficients. Main results consist of a homogenization theorem (i.e.,
convergence of solutions as the period of oscillation goes to zero) as well as
a characterization of homogenized equations. In particular, homogenized
matrices are described in terms of solutions to cell-problems, which have
different forms depending on the log-ratio of the spatial and temporal periods
of the coefficients. At a critical ratio, the cell problem turns out to be a
parabolic equation in microscopic variables (as in linear diffusion) and also
involves the limit of solutions, which is a function of macroscopic variables.
The latter feature stems from the nonlinearity of the equation, and moreover,
some strong interplay between microscopic and macroscopic structures can be
explicitly seen for the nonlinear diffusion. As for the other ratios, the cell
problems are always elliptic (in micro-variable only) and do not involve any
macroscopic variables, and hence, micro- and macrostructures are weakly
interacting each other. Proofs of the main results are based on the two-scale
convergence theory (for space-time homogenization). Furthermore, finer
asymptotics of gradients, diffusion fluxes and time-derivatives with certain
corrector terms will be provided, and a qualitative analysis on homogenized
matrices will be also performed.
