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The present paper is concerned with a space-time homogenization problem fornonlinear 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 asa characterization of homogenized equations. In particular, homogenizedmatrices are described in terms of solutions to cell-problems, which havedifferent forms depending on the log-ratio of the spatial and temporal periodsof the coefficients. At a critical ratio, the cell problem turns out to be aparabolic equation in microscopic variables (as in linear diffusion) and alsoinvolves 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 beexplicitly seen for the nonlinear diffusion. As for the other ratios, the cellproblems are always elliptic (in micro-variable only) and do not involve anymacroscopic variables, and hence, micro- and macrostructures are weaklyinteracting each other. Proofs of the main results are based on the two-scaleconvergence theory (for space-time homogenization). Furthermore, finerasymptotics of gradients, diffusion fluxes and time-derivatives with certaincorrector terms will be provided, and a qualitative analysis on homogenizedmatrices will be also performed.

Space-time homogenization for nonlinear diffusion