The main contribution of this paper is the homogenization of the linear parabolic equation∂tuε(x,t)-∇·(a(x/εq1,...,x/εqn,t/εr1,...,t/εrm)∇uε(x,t))=f(x,t)exhibiting an arbitrary finite number of both spatial and temporal scales. We briefly recall some fundamentals of multiscale convergence and provide a characterization of multiscale limits for gradients, in an evolution setting adapted to a quite general class of well-separated scales, which we name by jointly well-separated scales (see appendix for the proof). We proceed with a weaker version of this concept called very weak multiscale convergence. We prove a compactness result with respect to this latter type for jointly well-separated scales. This is a key result for performing the homogenization of parabolic problems combining rapid spatial and temporal oscillations such as the problem above. Applying this compactness result together with a characterization of multiscale limits of sequences of gradients we carry out the homogenization procedure, where we together with the homogenized problem obtainnlocal problems, that is, one for each spatial microscale. To illustrate the use of the obtained result, we apply it to a case with three spatial and three temporal scales withq1=1,q2=2, and0<r1<r2.

We study the homogenization of a parabolic equation with oscillations in both space and time in the coefficient a x/ε, t/ε 2 in the elliptic part and spatial oscillations in the coefficient ρ x/ε that is multiplied with the time derivative ∂ t u ε . We obtain a strange term in the local problem. This phenomenon appears as a consequence of the combination of the spatial oscillation in ρ x/ε and the temporal oscillation in a x/ε, t/ε 2 and disappears if either of these oscillations is removed.

We consider the homogenization of the linear parabolic problem ( / 2 ) ( , ) − ∇ ⋅ ( ( / 1 , / 2 1 )∇ ( , )) = ( , ) which exhibits a mismatch between the spatial scales in the sense that the coefficient ( / 1 , / 2 1 ) of the elliptic part has one frequency of fast spatial oscillations, whereas the coefficient ( / 2 ) of the time derivative contains a faster spatial scale. It is shown that the faster spatial microscale does not give rise to any corrector term and that there is only one local problem needed to characterize the homogenized problem. Hence, the problem is not of a reiterated type even though two rapid scales of spatial oscillation appear.

We first study the fundamental ideas behind two-scale convergence to enhance an intuitive understanding of this notion. The classical definitions and ideas are motivated with geometrical arguments illustrated by illuminating figures. Then a version of this concept, very weak two-scale convergence, is discussed both independently and briefly in the context of homogenization. The main features of this variant are that it works also for certain sequences of functions which are not bounded in L 2 (Ω) and at the same time is suited to detect rapid oscillations in some sequences which are strongly convergent in L 2 (Ω). In particular, we show how very weak two-scale convergence explains in a more transparent way how the oscillations of the governing coefficient of the PDE to be homogenized causes the deviation of the G-limit from the weak L 2 (Ω) N ×N -limit for the sequence of coefficients. Finally, we investigate very weak multiscale convergence and prove a compactness result for separated scales which extends a previous result which required well-separated scales.

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