Abstract

We study the homogenization of a parabolic equation with oscillations in both space and time in the coefficient ğ‘Ž(𝑥/𝜀,𝑡/𝜀2) in the elliptic part and spatial oscillations in the coefficient 𝜌(𝑥/𝜀) that is multiplied with the time derivative 𝜕𝑡𝑢𝜀. 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 ğ‘Ž(𝑥/𝜀,𝑡/𝜀2) and disappears if either of these oscillations is removed.

1. Introduction

We study the homogenization ofğœŒî‚€ğ‘¥ğœ€î‚ğœ•ğ‘¡ğ‘¢ğœ€î‚€ğ‘Žî‚€ğ‘¥(𝑥,𝑡)−∇⋅𝜀,𝑡𝜀2∇𝑢𝜀𝑢(𝑥,𝑡)=𝑓(𝑥,𝑡)inΩ×(0,𝑇),𝜀𝑢(𝑥,0)=𝑔(𝑥)inΩ,𝜀(𝑥,𝑡)=0on𝜕Ω×(0,𝑇),(1.1) which contains oscillations in both space and time in the coefficient ğ‘Ž(𝑥/𝜀,𝑡/𝜀2) in the elliptic part and spatial oscillations in the coefficient 𝜌(𝑥/𝜀) that is multiplied with the time derivative 𝜕𝑡𝑢𝜀. The technique is an adaption of two-scale convergence to parabolic homogenization. To deal with the oscillations of 𝜌(𝑥/𝜀), we need to make a special choice of test functions for our approach to apply, which is the reason why an additional term is obtained 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 ğ‘Ž(𝑥/𝜀,𝑡/𝜀2) and disappears if either of these oscillations is removed. Understanding (1.1) in terms of physics, the coefficient 𝜌(𝑥/𝜀) means that the density and the heat capacity may follow a pattern of spatial heterogeneity similar to the thermal conductivity. It is worth noting that the strange term in the local problem appears with a coefficient 𝜌(𝑥/𝜀) with spatial oscillations with the same frequency as the heat conductivity coefficient but without the corresponding temporal oscillations. To the authors’ knowledge the physical interpretation of this phenomenon remains to be understood.

A related problem is studied by Nandakumaran and Rajesh in [1], with the temporal oscillations of the same frequency as the spatial ones and hence the resonance phenomenon in the local problem that we obtain for (1.1) does not appear; see also Remarks 3.3 and 3.4. They investigate𝜕𝑡𝜌𝑥𝜀,ğ‘¢ğœ€î‚î‚€ğ‘¥âˆ’âˆ‡â‹…ğ‘Žğœ€,𝑡𝜀,𝑢𝜀,∇𝑢𝜀=𝑓(𝑥,𝑡)inΩ×(0,𝑇),(1.2) with mixed boundary conditions under certain continuity and monotonicity assumptions on 𝜌 and ğ‘Ž. There will turn out to be a significant difference between the treatment of the cases where the speed of the temporal oscillations is governed by 𝜀, as in (1.2), and 𝜀2, which is considered in the main result of this paper. Simpler linear problems without temporal oscillations are found in, for example, [2, 3].

2. Two-Scale Convergence

Our main tools are some versions of two-scale convergence. Two-scale convergence was first introduced by Nguetseng in [4]. The definition below was established by Allaire in [5] and has become the standard way to define two-scale convergence. It is a slight modification of the original definition in [4].

Notation 1. 𝐹#(𝑌) means the space of all functions in 𝐹loc(ℝ𝑁) that are 𝑌-periodic repetitions of some function in 𝐹(𝑌). Ω is a bounded open set in ℝ𝑁 with a smooth boundary and Ω𝑇=Ω×(0,𝑇).

Definition 2.1. One says that a sequence {𝑢𝜀} in 𝐿2(Ω) two-scale converges to 𝑢0∈𝐿2(Ω×𝑌) if Ω𝑢𝜀(𝑥𝑥)𝑣𝑥,𝜀𝑑𝑥⟶Ω𝑌𝑢0(𝑥,𝑦)𝑣(𝑥,𝑦)𝑑𝑦𝑑𝑥(2.1) for any 𝑣∈𝐿2(Ω;𝐶#(𝑌)) when 𝜀→0. One writes 𝑢𝜀2⇀𝑢0.

Translating to the appropriate evolution setting we introduce the next variant.

Definition 2.2. One says that a sequence {𝑢𝜀} in 𝐿2(Ω𝑇) (2,2)-scale converges to 𝑢0∈𝐿2(Ω𝑇×𝑌×(0,1)) if Ω𝑇𝑢𝜀𝑥(𝑥,𝑡)𝑣𝑥,𝑡,𝜀,𝑡𝜀2𝑑𝑥𝑑𝑡⟶Ω𝑇10𝑌𝑢0(𝑥,𝑡,𝑦,𝑠)𝑣(𝑥,𝑡,𝑦,𝑠)𝑑𝑦𝑑𝑠𝑑𝑥𝑑𝑡(2.2) for any 𝑣∈𝐿2(Ω𝑇;𝐶#(𝑌×(0,1))) when 𝜀→0. One writes 𝑢𝜀(𝑥,𝑡)2,2⇀𝑢0(𝑥,𝑡,𝑦,𝑠).(2.3)

The somewhat weaker type of convergence defined next is an essential tool in the homogenization of (1.1) and under certain assumptions works without the requirement on boundedness in 𝐿2 which is necessary to obtain convergence up to a subsequence in usual two-scale convergence, see [6].

Definition 2.3. One says that a sequence {𝑤𝜀} in 𝐿1(Ω𝑇) (2,2)-scale converges very weakly to 𝑤0∈𝐿1(Ω𝑇×𝑌×(0,1)) if Ω𝑇𝑤𝜀(𝑥,𝑡)𝑣1(𝑥)𝑣2𝑥𝜀𝑐1(𝑡)𝑐2𝑡𝜀2⟶𝑑𝑥𝑑𝑡Ω𝑇10𝑌𝑤0(𝑥,𝑡,𝑦,𝑠)𝑣1(𝑥)𝑣2(𝑦)𝑐1(𝑡)𝑐2(𝑠)𝑑𝑦𝑑𝑠𝑑𝑥𝑑𝑡(2.4) for any 𝑣1∈𝐷(Ω), 𝑣2âˆˆğ¶âˆž#(𝑌)/ℝ, 𝑐1∈𝐷(0,𝑇), and 𝑐2âˆˆğ¶âˆž#(0,1) when 𝜀→0. One writes 𝑢𝜀(𝑥,𝑡)2,2⇀𝑣𝑤𝑢0(𝑥,𝑡,𝑦,𝑠).(2.5)

Let 𝑊12(0,𝑇;𝐻10(Ω),𝐿2(Ω)) be the space of all functions in 𝐿2(0,𝑇;𝐻10(Ω)) such that the time derivative belongs to 𝐿2(0,𝑇;𝐻−1(Ω)); see, for example, [7, Chapter  23]. For {𝑢𝜀} bounded in 𝑊12(0,𝑇;𝐻10(Ω),𝐿2(Ω)) we also have a characterization of the (2,2)-scale limit for the gradients ∇𝑢𝜀 and the corresponding very weak limit for {𝑢𝜀/𝜀}.

Theorem 2.4. Let {𝑢𝜀} be a bounded sequence in 𝑊12(0,𝑇;𝐻10(Ω),𝐿2(Ω)). Then, there exists a subsequence such that 𝑢𝜀(𝑥,𝑡)⟶𝑢(𝑥,𝑡)in𝐿2Ω𝑇,𝑢𝜀(𝑥,𝑡)⇀𝑢(𝑥,𝑡)in𝐿20,𝑇;𝐻10,(Ω)(2.6)∇𝑢𝜀(𝑥,𝑡)2,2⇀∇𝑢(𝑥,𝑡)+∇𝑦𝑢1(𝑥,𝑡,𝑦,𝑠),(2.7) where 𝑢∈𝑊12(0,𝑇;𝐻10(Ω),𝐿2(Ω)) and 𝑢1∈𝐿2(Ω𝑇×(0,1);𝐻1#(𝑌)/ℝ). Moreover 𝑢𝜀(𝑥,𝑡)𝜀2,2⇀𝑣𝑤𝑢1(𝑥,𝑡,𝑦,𝑠).(2.8)

Proof. The results in (2.7) and (2.8) can be seen as the period special case for the corresponding results in terms of Σ-convergence in [8]; see, for example, Defintion  3.1, Lemma  3.4 and Section  4.2 in [8]. We can also obtain (2.7) by a slight modification of the standard proof for bounded sequences in 𝐻1(Ω) if we observe (2.6), that is, that any bounded sequence in 𝑊12(0,𝑇;𝐻10(Ω),𝐿2(Ω)) contains a subsequence that converges strongly in 𝐿2(Ω𝑇); see, for example, [9]. In the same way (2.8) can be concluded from [6, Theorem  4].

Remark 2.5. Limits of the type in (2.8) appear in the proof of the homogenization result for (1.1) in Section 3. The important point here is to find a limit for a sequence {𝑢𝜀/𝜀}, where the denominator 𝜀 passes to zero, while the numerator 𝑢𝜀 does not. The reason why we assume that ∫𝑌𝑣2(𝑦)𝑑𝑦=0 in Definition 2.3 is that 𝑣2 has to be generated in a certain manner for the proof of (2.8) to work. This is not so with, for example, 𝑐2; see, for example, [6, 8, 9].

Remark 2.6. The results in Theorem 2.4 can also be obtained under the assumption that {𝑢𝜀} apart from being a sequence of solutions to (1.1) and hence bounded in 𝐿2(0,𝑇;𝐻10(Ω)) is also bounded in ğ¿âˆž(Ω𝑇). These conditions together imply that {𝑢𝜀} converges strongly in 𝐿2(Ω𝑇) up to a subsequence and hence the boundedness of {𝑢𝜀} in ğ¿âˆž(Ω𝑇) replaces the boundedness of {𝜕𝑡𝑢𝜀} in 𝐿2(0,𝑇;𝐻−1(Ω)); see Lemma  3.3 and (4.1) in [1]. The proof is then possible to perform in the same way as for {𝑢𝜀} bounded in 𝑊12(0,𝑇;𝐻10(Ω),𝐿2(Ω)). The only difference is that 𝑢 will belong to 𝐿2(0,𝑇;𝐻10(Ω)) instead of the space 𝑊12(0,𝑇;𝐻10(Ω),𝐿2(Ω)).

3. Homogenization

We develop a homogenization procedure for (1.1) and obtain the result in the theorem below. Omitting the rapid temporal oscillations, that is, replacing ğ‘Ž(𝑥/𝜀,𝑡/𝜀2) with ğ‘Ž(𝑥/𝜀), there are no important consequences of the appearance of 𝜌(𝑥/𝜀) and the local problem would be the same as for 𝜌=1. With the temporal oscillations the situation is, however, sometimes different from what it should have been with, for example, 𝜌=1. We need to apply (2.8) to find the local problem but encounter a difficulty in the sense that 𝜌𝑣 does not in general have average zero over 𝑌 for ğ‘£âˆˆğ¶âˆž#(𝑌) or even ğ‘£âˆˆğ¶âˆž#(𝑌)/ℝ. This necessitates a construction of special test functions to be used in the weak form (3.6) of (1.1) in the proof of Theorem 3.1. We assume that ğœŒâˆˆğ¶âˆž#(𝑌), 𝜌≥𝐶>0, 𝑓∈𝐿2(Ω𝑇), andğ‘Ž(𝑦,𝑠)𝑐⋅𝑐≥𝐴|𝑐|2,𝐴>0,(3.1) where ğ‘Žâˆˆğ¿âˆž#(𝑌×(0,1))𝑁×𝑁. It can be proven along the lines of the corresponding proof in [7, Section  23.9] that {𝑢𝜀} is bounded in the space 𝐿2(0,𝑇;𝐻10(Ω)); see also [2]. For, for example, 𝜌=1 it also holds that {𝜕𝑡𝑢𝜀} is bounded in 𝐿2(0,𝑇;𝐻−1(Ω)) and hence {𝑢𝜀} is bounded in the stronger space 𝑊12(0,𝑇;𝐻10(Ω),𝐿2(Ω)). Here, we instead make the physically quite natural assumption that {𝑢𝜀} is bounded in ğ¿âˆž(Ω𝑇); see [1] and the references therein.

Theorem 3.1. Let {𝑢𝜀} be a sequence of solutions to (1.1), where 𝜀→0, and assume that {𝑢𝜀} is bounded in ğ¿âˆž(Ω𝑇). Then, 𝑢𝜀⇀𝑢in𝐿20,𝑇;𝐻10,(Ω)(3.2) where 𝑢 is the solution to 𝑌𝜌(𝑦)𝑑𝑦𝜕𝑡𝑢(𝑥,𝑡)−∇⋅(𝑏∇𝑢(𝑥,𝑡))=𝑓(𝑥,𝑡)inΩ𝑇,𝑢(𝑥,0)=𝑔(𝑥)inΩ,𝑢(𝑥,𝑡)=0on𝜕Ω×(0,𝑇),(3.3) with 𝑏∇𝑢(𝑥,𝑡)=10î€œğ‘Œî€·ğ‘Ž(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇𝑦𝑢1(𝑥,𝑡,𝑦,𝑠)𝑑𝑦𝑑𝑠,(3.4) where 𝑢1∈𝐿2(Ω𝑇×(0,1);𝐻1#(𝑌)/ℝ) solves 𝜌(𝑦)𝜕𝑠𝑢1(𝑥,𝑡,𝑦,𝑠)âˆ’âˆ‡ğ‘¦â‹…î€·ğ‘Žî€·(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇𝑦𝑢1(𝑥,𝑡,𝑦,𝑠)=𝜌(𝑦)ğ‘Œî€·ğ‘Ž(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇𝑦𝑢1(𝑥,𝑡,𝑦,𝑠)⋅∇𝑦1𝜌.(𝑦)𝑑𝑦(3.5)

Remark 3.2. For 𝜌=1, the right-hand side of (3.5) is zero and hence the strange term in the local problem disappears. An interesting question is to investigate if there are ways to make the strange term disappear and obtain a more conventional local problem also when 𝜌 is oscillating. This would simplify the use of standard software for the solution of the local problem. See Remark 3.3, where this is discussed for the case where the temporal oscillations are of the same frequency as the spatial ones, and Remark 3.4.

Proof. Let us study the weak form Ω𝑇𝑥−𝜌𝜀𝑢𝜀(𝑥,𝑡)𝑣(𝑥)𝜕𝑡𝑥𝑐(𝑡)+ğ‘Žğœ€,𝑡𝜀2∇𝑢𝜀(=𝑥,𝑡)⋅∇𝑣(𝑥)𝑐(𝑡)𝑑𝑥𝑑𝑡Ω𝑇𝑓(𝑥,𝑡)𝑣(𝑥)𝑐(𝑡)𝑑𝑥𝑑𝑡,(3.6)𝑣∈𝐻10(Ω), 𝑐∈𝐷(0,𝑇) of (1.1). We apply (2.6) and (2.7), pass to the limit and arrive, up to a subsequence, at the homogenized problem Ω𝑇10𝑌−𝜌(y)𝑢(𝑥,𝑡)𝑣(𝑥)𝜕𝑡𝑐(𝑡)+ğ‘Ž(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇𝑦𝑢1=(𝑥,𝑡,𝑦,𝑠)⋅∇𝑣(𝑥)𝑐(𝑡)𝑑𝑦𝑑𝑠𝑑𝑥𝑑𝑡Ω𝑇𝑓(𝑥,𝑡)𝑣(𝑥)𝑐(𝑡)𝑑𝑥𝑑𝑡.(3.7) To find a local problem we choose 𝑣(𝑥)=𝜀𝑣1̃𝑣𝑥(𝑥)𝜀(3.8) in (3.6), where 𝑣1∈𝐷(Ω), ̃𝑣(𝑦)=𝑣2(𝐶𝑦)−𝜌(𝑦),𝑣2âˆˆğ¶âˆž#(𝑌),(3.9)𝐶=𝑌𝜌(𝑦)𝑣2(𝑦)𝑑𝑦.(3.10) Hence, we have 𝑌̃𝜌(𝑦)𝑣(𝑦)𝑑𝑦=0.(3.11) Further, we let 𝑐(𝑡)=𝑐1(𝑡)𝑐2𝑡𝜀2,𝑐1∈𝐷(0,𝑇),𝑐2âˆˆğ¶âˆž#(0,1)(3.12) and arrive at Ω𝑇𝑥−𝜌𝜀𝑢𝜀(𝑥,𝑡)𝑣1(̃𝑣𝑥𝑥)𝜀𝜀𝜕𝑡𝑐1(𝑡)𝑐2𝑡𝜀2+𝜀−1𝑐1(𝑡)𝜕𝑠𝑐2𝑡𝜀2𝑥+ğ‘Žğœ€,𝑡𝜀2∇𝑢𝜀(𝑥,𝑡)⋅𝜀∇𝑣1̃𝑣𝑥(𝑥)𝜀+𝑣1(𝑥)∇𝑦̃𝑣𝑥𝜀𝑐1(𝑡)𝑐2𝑡𝜀2=𝑑𝑥𝑑𝑡Ω𝑇𝑓(𝑥,𝑡)𝜀𝑣1̃𝑣𝑥(𝑥)𝜀𝑐1(𝑡)𝑐2𝑡𝜀2𝑑𝑥𝑑𝑡.(3.13) The choice of ̃𝑣 is motivated by the requirement that we should have (3.11) for (2.8) to be applicable. We let 𝜀→0 in (3.13) and apply (2.7) and (2.8) to î€œÎ©ğ‘‡ğ‘Žî‚€ğ‘¥ğœ€,𝑡𝜀2∇𝑢𝜀(𝑥,𝑡)⋅𝑣1(𝑥)∇𝑦̃𝑣𝑥𝜀𝑐1(𝑡)𝑐2𝑡𝜀2𝑑𝑥𝑑𝑡,Ω𝑇𝑥−𝜌𝜀𝑢𝜀(𝑥,𝑡)𝑣1̃𝑣𝑥(𝑥)𝜀𝜀−1𝑐1(𝑡)𝜕𝑠𝑐2𝑡𝜀2𝑑𝑥𝑑𝑡,(3.14) respectively. Noting that the rest of terms in (3.13) vanish, we obtain, up to a subsequence, Ω𝑇10𝑌−𝜌(𝑦)𝑢1(𝑥,𝑡,𝑦,𝑠)𝑣1̃(𝑥)𝑣(𝑦)𝑐1(𝑡)𝜕𝑠𝑐2(𝑠)+ğ‘Ž(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇𝑦𝑢1(𝑥,𝑡,𝑦,𝑠)⋅𝑣1(𝑥)∇𝑦̃𝑣(𝑦)𝑐1(𝑡)𝑐2(𝑠)𝑑𝑦𝑑𝑠𝑑𝑥𝑑𝑡=0.(3.15) Hence, observing that 𝑌−𝜌(𝑦)𝑢1−𝐶(𝑥,𝑡,𝑦,𝑠)𝜌(𝑦)𝑑𝑦=𝑌𝐶𝑢1(𝑥,𝑡,𝑦,𝑠)𝑑𝑦=0(3.16) and recalling (3.9), we arrive at Ω𝑇10𝑌−𝜌(𝑦)𝑢1(𝑥,𝑡,𝑦,𝑠)𝑣1(𝑥)𝑣2(𝑦)𝑐1(𝑡)𝜕𝑠𝑐2(𝑠)+ğ‘Ž(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇𝑦𝑢1(𝑥,𝑡,𝑦,𝑠)⋅𝑣1(𝑥)∇𝑦𝑣2(C𝑦)−𝑐𝜌(𝑦)1(𝑡)𝑐2(𝑠)𝑑𝑦𝑑𝑠𝑑𝑥𝑑𝑡=0.(3.17) We write Ω𝑇10𝑌−𝜌(𝑦)𝑢1(𝑥,𝑡,𝑦,𝑠)𝑣1(𝑥)𝑣2(𝑦)𝑐1(𝑡)𝜕𝑠𝑐2(𝑠)+ğ‘Ž(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇𝑦𝑢1(𝑥,𝑡,𝑦,𝑠)⋅𝑣1(𝑥)∇𝑦𝑣2(𝑦)𝑐1(𝑡)𝑐2(−𝑠)𝑑𝑦𝑑𝑠𝑑𝑥𝑑𝑡Ω𝑇10î‚µî€œğ‘Œî€·ğ‘Ž(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇𝑦𝑢1(𝑥,𝑡,𝑦,𝑠)⋅𝑣1(𝑥)∇𝑦𝐶𝜌(𝑦)𝑑𝑦⋅𝑐1(𝑡)𝑐2(𝑠)𝑑𝑠𝑑𝑥𝑑𝑡=0.(3.18) Applying repeatedly the variational lemma, we find that 10𝑌−𝜌(𝑦)𝑢1(𝑥,𝑡,𝑦,𝑠)𝑣2(𝑦)𝜕𝑠𝑐2(𝑠)+ğ‘Ž(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇𝑦𝑢1(𝑥,𝑡,𝑦,𝑠)⋅∇𝑦𝑣2(𝑦)𝑐2=(𝑠)𝑑𝑦𝑑𝑠10î‚µî€œğ‘Œî€·ğ‘Ž(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇𝑦𝑢1(𝑥,𝑡,𝑦,𝑠)⋅∇𝑦𝐶𝑐𝜌(𝑦)𝑑𝑦2(𝑠)𝑑𝑠(3.19) and, according to the definition (3.10) of 𝐶, 10𝑌−𝜌(𝑦)𝑢1(𝑥,𝑡,𝑦,𝑠)𝑣2(𝑦)𝜕𝑠𝑐2(𝑠)+ğ‘Ž(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇𝑦𝑢1(𝑥,𝑡,𝑦,𝑠)⋅∇𝑦𝑣2(𝑦)𝑐2=(𝑠)𝑑𝑦𝑑𝑠10𝑌𝜌(𝑦)ğ‘Œî€·ğ‘Ž(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇𝑦𝑢1(𝑥,𝑡,𝑦,𝑠)⋅∇𝑦1𝜌(𝑦)𝑑𝑦⋅𝑣2(𝑦)𝑐2(𝑠)𝑑𝑦𝑑𝑠(3.20) which is the weak form of (3.5).

Remark 3.3. We also briefly comment on case (1.2) originally studied in [1]. Restricting (1.2) to the linear setting studied in this paper means that we obtain ğœŒî‚€ğ‘¥ğœ€î‚ğœ•ğ‘¡ğ‘¢ğœ€î‚€ğ‘Žî‚€ğ‘¥(𝑥,𝑡)−∇⋅𝜀,𝑡𝜀∇𝑢𝜀(𝑥,𝑡)=𝑓(𝑥,𝑡)inΩ𝑇.(3.21) Introducing test functions corresponding to those used to find the local problem (3.5) in the weak form of (3.21), we arrive at Ω𝑇𝑥−𝜌𝜀𝑢𝜀(̃𝑣𝑥𝑥,𝑡)𝑣(𝑥)𝜀𝜀𝜕𝑡𝑐1(𝑡)𝑐2𝑡𝜀+𝑐1(𝑡)𝜕𝑠𝑐2𝑡𝜀𝑥+ğ‘Žğœ€,𝑡𝜀∇𝑢𝜀̃𝑣𝑥(𝑥,𝑡)⋅𝜀∇𝑣(𝑥)𝜀+𝑣1(𝑥)∇𝑦̃𝑣𝑥𝜀𝑐1(𝑡)𝑐2𝑡𝜀=𝑑𝑥𝑑𝑡Ω𝑇̃𝑣𝑥𝑓(𝑥,𝑡)𝜀𝑣(𝑥)𝜀𝑐1(𝑡)𝑐2𝑡𝜀𝑑𝑥𝑑𝑡.(3.22) Letting 𝜀 go to zero, we find, following the same procedure as in the proof of Theorem 3.1, that âˆ’âˆ‡ğ‘¦â‹…î€·ğ‘Žî€·(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇𝑦𝑢1(𝑥,𝑡,𝑦,𝑠)=𝜌(𝑦)ğ‘Œî€·ğ‘Ž(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇𝑦𝑢1(𝑥,𝑡,𝑦,𝑠)⋅∇𝑦1𝜌,(𝑦)𝑑𝑦(3.23) and hence it seems like a strange term has appeared also in the local problem for the homogenization of (3.21). However, we do not need very weak two-scale convergence to pass to the limit in (3.22), and hence we can replace ̃𝑣 with any 𝑣2âˆˆğ¶âˆž#(𝑌) and obtain the more conventional local problem âˆ’âˆ‡ğ‘¦â‹…î€·ğ‘Žî€·(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇𝑦𝑢1(𝑥,𝑡,𝑦,𝑠)=0(3.24) without any strange term. Observing that 1/ğœŒâˆˆğ¶âˆž#(𝑌) and hence is an admissible choice of the test function 𝑣2 in the weak form of (3.24), this means that the right-hand side in (3.23) is zero and hence (3.23) reduces to (3.24).

Remark 3.4. The question of obtaining a cancellation of the strange term for the homogenization of (1.1) similar to what we saw in Remark 3.3 is delicate. For {𝜕𝑡𝑢𝜀} bounded in 𝐿2(Ω𝑇), a such cancellation appears but under the present conditions of boundedness of {𝑢𝜀} in 𝐿2(0,𝑇;𝐻10(Ω)) and ğ¿âˆž(Ω𝑇) and strong convergence in 𝐿2(Ω𝑇) there are counterexamples. Hence, this far, we have only found ways to neutralize the strange term in (3.5) by means of nonstandard boundedness assumptions for (1.1). Forthcoming studies will address these questions in more detail.