On non-periodic homogenization of time-dependent equations

The recently introduced needle problem approach for the homogenization of non-periodic problems was originally designed for the homogenization of elliptic problems. After a short review of the needle problem approach we demonstrate in this note how the stationary results can be transferred to time-dependent problems. The standard parabolic problem of the corresponding heat equation in a heterogeneous material is considered. Furthermore, we include an application to a hysteresis problem which appears in the theory of porous media.