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 apply a new version of multiscale convergence named very weak multiscale convergence to find possible frequencies of oscillation in an unknown coefficient of a partial differential equation from its solution. We also use this notion to study homogenization of a certain linear parabolic problem with multiple spatial and temporal scales.
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