We consider a quasilinear parabolic problem with time dependent coefficients oscillating rapidly in the space variable. The existence and uniqueness results are proved by using Rothe’s method combined with the technique of two-scale convergence. Moreover, we derive a concrete homogenization algorithm for giving a unique and computable approximation of the solution.

The main focus in this paper is on homogenization of the parabolic problem ${\partial }_t{u}^{\epsilon }-\nabla ·\left(a(x/\epsilon ,t/\epsilon ,t/{\epsilon }^r)\nabla u^{\epsilon }\right)=f$. Under certain assumptions on $a$, there exists a $G$-limit $b$, which we characterize by means of multiscale techniques for $r>0$, $r\ne 1$. Also, an interpretation of asymptotic expansions in the context of two-scale convergence is made.