1. Introduction
The field of homogenization has its main source of inspiration in the problem of finding the macroscopic properties of strongly heterogeneous materials. Mathematically, the approach is to study a sequence of partial differential equations where a parameter ε associated with the length scales of the heterogeneities tends to zero. The sequence of solutions uε converges to the solution u to a so-called homogenized problem governed by a coefficient b, where b gives the effective property of the material and can be characterized by certain auxiliary problems called the local problems.

In this paper, we study the homogenization of the linear parabolic problem
(1)ρ(xε2)∂tuε(x,t)-∇·(a(xε1,tε12)∇uε(x,t)) =f(x,t) in ΩT,uε(x,t)=0 on ∂Ω×(0,T),uε(x,0)=u0(x) in Ω,
where ΩT=Ω×(0,T), Ω is an open, bounded set in ℝN with locally Lipschitz boundary, where both a and ρ possess unit periodicity in their respective arguments and the scales ε1, ε2, and ε12 fulfill a certain separatedness assumption.

The problem exhibits rapid spatial oscillations in ρ and spatial as well as temporal oscillations in a. Furthermore, there is a “mismatch” between the spatial scales in the sense that the frequency of the spatial oscillations in ρ(x/ε2) is higher than that of a(x/ε1,t/ε12). Since there are two spatial microscales represented in (1), one might expect two local problems with respect to one corrector each, see, for example, [1]. However, it is shown that no corrector corresponding to the scale emanating from ρ(x/ε2) appears in the local and homogenized problem and accordingly there is only one local problem appearing in the formulated theorem. Hence, the problem is not of a reiterated type. We prove by means of very weak multiscale convergence [2] that the corrector u2 associated with the gradient for the second rapid spatial scale y2 actually vanishes. Already, in [3, 4], it was observed that having more than one rapid temporal scale in parabolic problems does not generate a reiterated problem and in this paper we can see that nor does the addition of a spatial scale if it is contained in a coefficient that is multiplied with the time derivative of uε.

Thinking in terms of heat conduction, our result means that the heat capacity ρ may oscillate with completely different periodic patterns without making any difference for the homogenized coefficient as long as the arithmetic mean over one period is the same.

Parabolic homogenization problems for ρ≡1 have been studied for different combinations of spatial and temporal scales in several papers by means of techniques of two-scale convergence type with approaches related to the one first introduced in [5], see, for example, [2, 3, 6–8], and in, for example, [9–11], techniques not of two-scale convergence type are applied. Concerning cases where, as in (1) above, we do not have ρ≡1, Nandakumaran and Rajesh [12] studied a nonlinear parabolic problem with the same frequency of oscillation in time and space, respectively, in the elliptic part of the equation and an operator oscillating in space with the same frequency appearing in the temporal differentiation term. Recently, a number of papers have addressed various kinds of related problems where the temporal scale is not assumed to be identical with the spatial scale, see for example, [13, 14]. Up to the authors’ knowledge, this is the first study of this type of problems where the oscillations of the coefficient of the term including the time derivative do not match the spatial oscillations of the elliptic part.

Notation. We denote Yk=(0,1)N for k=1,…,n, Yn=Y1×⋯×Yn, yn=(y1,…,yn), dyn=dy1⋯dyn, Sj=S=(0,1) for j=1,…,m, Sm=S1×⋯×Sm, sm=(s1,…,sm), and dsm=ds1⋯dsm. Let εk(ε), k=1,…,n, and εj′(ε), j=1,…,m, be positive and go to zero when ε does. Furthermore, let F♯((0,1)M) be the space of all functions in Floc(ℝM) that are (0,1)M-periodic repetitions of some function in F((0,1)M).

2. Multiscale Convergence
A two-scale convergence was invented by Nguetseng [15] as a new approach for the homogenization of problems with fast oscillations in one scale in space. The method was further developed by Allaire [16] and generalized to multiple scales by Allaire and Briane [1]. To homogenize problem (1), we use the further generalization in the definition below, adapted to evolution settings, see, for example, [8].

Definition 1.
A sequence {uε} in L2(ΩT) is said to (n + 1,m + 1)-scale converge to u0∈L2(ΩT×Yn×Sm) if
(2)∫ΩTuε(x,t)v(x,t,xε1,…,xεn,tε1′,…,tεm′)dx dt ⟶∫ΩT∫Yn∫Smu0(x,t,yn,sm) ×v(x,t,yn,sm)dyndsmdx dt,
for any v∈L2(ΩT;C♯(Yn×Sm)). We write
(3)uε(x,t)⇀n+1,m+1u0(x,t,yn,sm).

Usually, some assumptions are made about how the scales are related to each other. We say that the scales in a list {ε1,…,εn} are separated if
(4)limε→0εk+1εk=0,
for k=1,…,n-1 and that the scales are well-separated if there exists a positive integer l such that
(5)limε→01εk(εk+1εk)l=0,
for k=1,…,n-1.

The concept in the following definition is used as an assumption in the proofs of the compactness results in Theorems 3 and 7. For a more technically formulated definition and some examples, see [17, Section 2.4].

Definition 2.
Let {ε1,…,εn} and {ε1′,…,εm′} be lists of well-separated scales. Consider all elements from both lists. If from possible duplicates, where by duplicates we mean scales which tend to zero equally fast, one member of each pair is removed and the list in order of magnitude of all the remaining elements is well separated, the lists {ε1,…,εn} and {ε1′,…,εm′} are said to be jointly well separated.

In the theorem below, which will be used in the homogenization procedure in Section 3, W21(0,T;H01(Ω),L2(Ω)) denotes all functions u∈ L2(0,T;H01(Ω)) such that ∂tu∈L2(0,T;H-1(Ω)), see, for example, [18, Chapter 23].

Theorem 3.
Let {uε} be a bounded sequence in W21(0,T;H01(Ω),L2(Ω)), and suppose that the lists {ε1,…,εn} and {ε1′,…,εm′} are jointly well separated. Then there exists a subsequence such that
(6)uε(x,t)⟶u(x,t) in L2(ΩT),uε(x,t)⇀u(x,t) in L2(0,T;H01(Ω)),(7)∇uε(x,t)⇀n+1,m+1∇u(x,t)+∑j=1n∇yjuj(x,t,yj,sm),
where u∈W21(0,T;H01(Ω),L2(Ω)), u1∈L2(ΩT×Sm;H♯1(Y1)/ℝ), and uj∈L2(ΩT×Yj-1×Sm;H♯1(Yj)/ℝ) for j=2,…,n.

Proof.
See [17, Theorem 2.74].

To treat evolution problems with fast time oscillations, such as (1), we also need the concept of very weak multiscale convergence, see, for example, [2, 5].

Definition 4.
A sequence {gε} in L1(ΩT) is said to (n+1,m+1)-scale converge very weakly to g0∈L1(ΩT×Yn×Sm) if
(8)∫ΩTgε(x,t)v(x,xε1,…,xεn-1) ×c(t,tε1′,…,tεm′)φ(xεn)dx dt⟶∫ΩT∫Yn∫Smg0(x,t,yn,sm)v(x,yn-1) ×c(t,sm)φ(yn)dyndsmdx dt,
for any v∈D(Ω,C♯∞(Yn-1)), c∈D(0,T;C♯∞(Sm)), and φ∈C♯∞(Yn)/ℝ, where
(9)∫Yng0(x,t,yn,sm)dyn=0.
We write
(10)gε(x,t)⇀n+1,m+1vwg0(x,t,yn,sm).

Remark 5.
The requirement (9) is imposed in order to ensure the uniqueness of the limit. For details, see [17, Proposition 2.26].

Remark 6.
The convergence in Definition 1 may take place only if {uε} is bounded in L2(ΩT) and hence also is a weakly convergent in L2(ΩT), at least up to suitable subsequences. For very weak multiscale convergence, this is not so. The main intention with the concept is to study sequences of the type {uε/εn}, which are in general not bounded in L2(ΩT). This requires a more restrictive class of test functions.

The theorem below is a key result for the homogenization procedure in Section 3.

Theorem 7.
Let {uε} be a bounded sequence in W21(0,T;H01(Ω),L2(Ω)), and assume that the lists {ε1,…,εn} and {ε1′,…,εm′} are jointly well separated. Then there exists a subsequence such that
(11)uε(x,t)εn⇀n+1,m+1vwun(x,t,yn,sm),
where, for n=1, u1∈L2(ΩT×Sm;H♯1(Y1)/ℝ) and, for n=2,3,…, un∈L2(ΩT×Yn-1×Sm;H♯1(Yn)/ℝ) are the same as those in Theorem 3.

Proof.
See [17, Theorem 2.54].

Remark 8.
For a sequence of solutions {uε} to (1), we may replace the requirement that {∂tuε} should be bounded in L2(0,T;H-1(Ω)) by the assumption that {uε} is bounded in L∞(ΩT) and still obtain (6), see [12, Lemmas 3.3 and (4.1)] and thereby also (7) and (11). The only difference is that u will belong to L2(0,T;H01(Ω)) instead of the space W21(0,T;H01(Ω),L2(Ω)). See also [13].

3. Homogenization
Let us now investigate the heat conduction problem
(12)ρ(xε2)∂tuε(x,t)-∇·(a(xε1,tε12)∇uε(x,t)) =f(x,t) in ΩT,uε(x,t)=0 on ∂Ω×(0,T),uε(x,0)=u0(x) in Ω,
which takes into consideration heat capacity oscillations. We assume that ρ∈C♯∞(Y2), is positive, f∈L2(ΩT), u0∈L2(Ω), and
(13)a(y1,s)ξ·ξ≥α|ξ|2
for some α>0, all (y1,s)∈ℝN+1, and all ξ∈ℝN, where a∈C♯(Y1×S)N×N. Moreover, we assume that {uε} is bounded in L∞(ΩT), see Remark 8, and that the lists {ε1,ε2} and {ε12} are jointly well separated. Note that this separatedness assumption implies, for example, that ε2 tends to zero faster than ε1, which means that we have a mismatch between the spatial scales in (12).

We give a homogenization result for this problem in the theorem below. In the proof, it is shown that the local problem associated with the slower spatial microscale is enough to characterize the homogenized problem; that is, the fastest spatial scale does not give rise to any corrector involved in the homogenization. We also prove that the second corrector u2 actually vanishes.

Theorem 9.
Let {uε} be a sequence of solutions to (12). Then
(14)uε(x,t)⇀u(x,t) in L2(0,T;H01(Ω)),(15)∇uε(x,t)⇀3,2∇u(x,t)+∇y1u1(x,t,y1,s)+∇y2u2(x,t,y2,s),
where u is the unique solution to
(16)(∫Y2ρ(y2)dy2)∂tu(x,t)-∇·(b∇u(x,t)) =f(x,t) in ΩT,u(x,t)=0 on ∂Ω×(0,T),u(x,0)=u0(x) in Ω,
with
(17)b∇u(x,t) =∫S∫Y1a(y1,s)(∇u(x,t)+∇y1u1(x,t,y1,s))dy1ds.
Here, u1∈L2(ΩT×S;H♯1(Y1)/ℝ) uniquely solves
(18)(∫Y2ρ(y2)dy2)∂su1(x,t,y1,s) -∇y1·(a(y1,s)(∇u(x,t)+∇y1u1(x,t,y1,s)))=0.
Furthermore, the corrector u2 vanishes.

Remark 10.
After a separation of variables, we can write the local problem as
(19)(∫Y2ρ(y2)dy2)∂szk(y1,s) -∑i,j=1N∂yi(aij(y1,s)(δjk+∂yjzk(y1,s)))=0
and the homogenized coefficient as
(20)bik=∫S∫Y1∑j=1N(aij(y1,s)(δjk+∂yjzk(y1,s)))dy1 ds,
where k=1,…,N and
(21)u1(x,t,y1,s)=∑k=1N∂xku(x,t)·zk(y1,s).

Remark 11.
Periodic homogenization problems of, for example, elliptic or parabolic type may be seen as special cases of the more general concepts of G-convergence, which gives a characterization of the limit problem but no suggestion of how to compute the homogenized matrix. Essential features of G-convergence for parabolic problems are that boundary conditions, and initial conditions are preserved in the limit. G-convergence for linear parabolic problems were studied already in [19] by Spagnolo and extended to the monotone case by Svanstedt in [20]. A treatment of this problem in a quite general setting is found in the recent work [21] by Paronetto.

Proof of Theorem <xref ref-type="statement" rid="thm10">9</xref>.
Following the procedure in Section 23.9 in [18], we obtain that {uε} is bounded in L2(0,T;H01(Ω)), see also [22]. Hence, (14) holds up to a subsequence. We proceed by studying the weak form of (12); that is,
(22)∫ΩT-ρ(xε2)uε(x,t)v(x)∂tc(t) +a(xε1,tε12)∇uε(x,t)∇v(x)c(t)dx dt =∫ΩTf(x,t)v(x)c(t)dx dt,
for all v∈H01(Ω) and c∈D(0,T). We pass to the limit by applying (6), taking into consideration Remark 8, and (7) with n=1 and m=1 and arrive at the homogenized problem
(23)∫ΩT∫S∫Y1-(∫Y2ρ(y2)dy2)u(x,t)v(x)∂tc(t) +a(y1,s)(∇u(x,t)+∇y1u1(x,t,y1,s)) ×∇v(x)c(t)dy1ds dx dt =∫ΩTf(x,t)v(x)c(t)dx dt.
To find the local problem associated with u1, let us again consider (22) in which we choose
(24)v(x)=ε1v1(x)v2(xε1); v1∈D(Ω), v2∈C♯∞(Y1)ℝ,(25)c(t)=c1(t)c2(tε12); c1∈D(0,T); c2∈C♯∞(S);
that is, we study
(26)∫ΩT-ρ(xε2)uε(x,t)v1(x)v2(xε1) ×(ε1∂tc1(t)c2(tε12)+ε1-1c1(t)∂sc2(tε12)) +a(xε1,tε12)∇uε(x,t) ·(ε1∇v1(x)v2(xε1)+v1(x)∇y1v2(xε1)) ×c1(t)c2(tε12)dx dt =∫ΩTf(x,t)ε1v1(x)v2(xε1)c1(t)c2(tε12)dx dt.
We first investigate the second term of the part of the expression containing time derivatives. We have
(27)∫ΩT-ρ(xε2)uε(x,t)v1(x)v2 ×(xε1)ε1-1c1(t)∂sc2(tε12)dx dt =∫ΩT-ε1-1uε(x,t)v1(x)v2 ×(xε1)(ρ(xε2)-∫Y2ρ(y2)dy2) ×c1(t)∂sc2(tε12)dx dt +∫ΩT-ε1-1uε(x,t)v1(x)v2(xε1) ×(∫Y2ρ(y2)dy2)c1(t)∂sc2(tε12)dx dt ⟶∫ΩT∫S∫Y1-(∫Y2ρ(y2)dy2)u1(x,t,y1,s) ×v1(x)v2(y1)c1(t) ×∂sc2(s)dy1ds dx dt,
where we have applied (11) with n=2 and m=1 and with n=1 and m=1, respectively, in the last step. The passage to the limit in the remaining part of (26) is a simple application of (7) with n=1 and m=1. This provides us with the weak form,
(28)∫ΩT∫S∫Y1-(∫Y2ρ(y2)dy2)u1(x,t,y1,s) ×v1(x)v2(y1)c1(t)∂sc2(s) +a(y1,s)(∇u(x,t)+∇y1u1(x,t,y1,s)) ·v1(x)∇y1v2(y1)c1(t)c2(s)dy1ds dx dt=0,
of the local problem (18). This means that u1, and thus also u, is uniquely determined and hence the entire sequence {uε} converges and not just the extracted subsequence.

This far, we have only used test functions oscillating with a period ε1, and hence we have not given the coefficient ρ(x/ε2) a fair chance to produce a second corrector u2. In order to do so, we use a slightly different set of test functions in (22). Again, we let c be as in (25), whereas v is chosen according to
(29)v(x)=ε2v1(x)v2(xε1)v~(xε2);iiiv1∈D(Ω), v2∈C♯∞(Y1),
where
(30)v~(y2)=v3(y2)-Kρ(y2); v3∈C♯∞(Y2),
with
(31)K=∫Y2ρ(y2)v3(y2)dy2.
Note that
(32)∫Y2ρ(y2)v~(y2)dy2=0,
which means that ρv~∈C♯∞(Y2)/ℝ. We get
(33)∫ΩT-ρ(xε2)uε(x,t)v1(x)v2(xε)v~(xε2) ×(ε2∂tc1(t)c2(tε12)+ε2ε12c1(t)∂sc2(tε12)) +a(xε1,tε12)∇uε(x,t) ·(ε2∇v1(x)v2(xε1)v~(xε2) +ε2ε1v1(x)∇y1v2(xε1)v~(xε2) +v1(x)v2(xε1)∇y2v~(xε2)) ×c1(t)c2(tε12)dx dt=∫ΩTf(x,t)ε2v1(x)v2(xε1) ×v~(xε2)c1(t)c2(tε12)dx dt,
and applying (11) with n=2 and m=1 together with (15), that is, (7) with n=2 and m=1, we achieve
(34)∫ΩT∫S∫Y2a(y1,s) ×(∇u(x,t)+∇y1u1(x,t,y1,s)+∇y2u2(x,t,y2,s)) ·v1(x)v2(y1)∇y2v~(y2)c1(t)c2(s)dy2ds dx dt=0.
Noting that a, u, and u1 are all independent of y2, (34) reduces to
(35)∫ΩT∫S∫Y2a(y1,s)∇y2u2(x,t,y2,s) ·v1(x)v2(y1)∇y2v~(y2)c1(t)c2(s)dy2ds dx dt=0.
Recalling (30), we have
(36)∫ΩT∫S∫Y2a(y1,s)∇y2u2(x,t,y2,s) ·v1(x)v2(y1)∇y2(v3(y2)-Kρ(y2)) ×c1(t)c2(s)dy2ds dx dt=0,
which after rearranging can be written as
(37)∫ΩT∫S∫Y2a(y1,s)∇y2u2(x,t,y2,s) ·v1(x)v2(y1)∇y2v3(y2)c1(t)c2(s)dy2ds dx dt =K∫ΩT∫S∫Y1(∑∫Y2a(y1,s)∇y2u2(x,t,y2,s) ·∇y2(1ρ(y2))dy2∑) ×v1(x)v2(y1)c1(t)c2(s)dy1ds dx dt.
If we replace v~ with 1/ρ in (33), let ε→0, and use (6) and (7) with n=2 and m=1, we find that
(38)∫ΩT∫S∫Y2a(y1,s)∇y2u2(x,t,y2,s) ·v1(x)v2(y1)∇y2(1ρ(y2))c1(t) ×c2(s)dy2ds dx dt=0.
This means that the right-hand side in (37) is zero. Applying several times the variational lemma on the remaining part, we obtain
(39)∫Y2a(y1,s)∇y2u2(x,t,y2,s)·∇y2v3(y2)dy2=0,
and hence the corrector u2 is zero.

Remark 12.
That u2 vanishes means that uε/ε2 tends to zero in the sense of very weak (3,2)-scale convergence. However, there might still be oscillations originating from the oscillations of ρ(x/ε2) that have an impact on uε. The possibility is that their amplitude is so small that the magnification by 1/ε2 is not enough for the oscillations to be recognized in the limit. In this sense, the concept of very weak multiscale convergence gives us a more precise idea of what a corrector equals zero means.