## Homogenization of parabolic equations with an arbitrary number of scales in both space and time.(English)Zbl 1406.35140

Summary: The main contribution of this paper is the homogenization of the linear parabolic equation $$\partial_tu^{\varepsilon}(x,t)-\nabla\cdot(a(x/\varepsilon^{q_1},\ldots,x/\varepsilon^{q_n},t/\varepsilon^{r_1},\ldots,t/\varepsilon^{r_m})\nabla u^{\varepsilon}(x,t))=f(x,t)$$ 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 obtain $$n$$ local 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 with $$q_1=1$$, $$q_2=2$$, and $$0<r_1<r_2$$.

### MSC:

 35K10 Second-order parabolic equations 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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