A Strange Term in the Homogenization of Parabolic Equations with Two Spatial and Two Temporal Scales

Abstract

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

1. Introduction

We study the homogenization of𝜌𝑥𝜀𝜕^𝑡𝑢^𝜀(𝑥,𝑡)−∇⋅𝑎𝑥𝜀,𝜀𝑡²∇𝑢^𝜀𝑢(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 𝑎(𝑥/𝜀,𝑡/𝜀²) in the elliptic part and spatial oscillations in the coefficient 𝜌(𝑥/𝜀) that is multiplied with the time derivative 𝜕_𝑡𝑢^𝜀. 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 𝑎(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,𝑢^𝜀−∇⋅𝑎𝑥𝜀,𝜀𝑡,𝑢^𝜀,∇𝑢^𝜀=𝑓(𝑥,𝑡)inΩ×(0,𝑇),(1.2) with mixed boundary conditions under certain continuity and monotonicity assumptions on 𝜌 and 𝑎. 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 𝐹_loc(ℝ^𝑁) that are 𝑌-periodic repetitions of some function in 𝐹(𝑌). Ω is a bounded open set in ℝ^𝑁 with a smooth boundary and Ω_𝑇=Ω×(0,𝑇).

Definition 2.1. One says that a sequence {𝑢^𝜀} in 𝐿²(Ω) two-scale converges to 𝑢₀∈𝐿²(Ω×𝑌) if _Ω𝑢^𝜀(𝑥)𝑣𝑥,𝑥𝜀𝑑𝑥⟶_Ω_𝑌𝑢₀(𝑥,𝑦)𝑣(𝑥,𝑦)𝑑𝑦𝑑𝑥(2.1) for any 𝑣∈𝐿²(Ω;𝐶_#(𝑌)) when 𝜀→0. One writes 𝑢⇀^𝜀2𝑢₀.

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

Definition 2.2. One says that a sequence {𝑢^𝜀} in 𝐿²(Ω_𝑇) (2,2)-scale converges to 𝑢₀∈𝐿²(Ω_𝑇×𝑌×(0,1)) if _Ω𝑇𝑢^𝜀(𝑥,𝑡)𝑣𝑥,𝑡,𝑥𝜀,𝜀𝑡²𝑑𝑥𝑑𝑡⟶_Ω𝑇¹⁰_𝑌𝑢₀(𝑥,𝑡,𝑦,𝑠)𝑣(𝑥,𝑡,𝑦,𝑠)𝑑𝑦𝑑𝑠𝑑𝑥𝑑𝑡(2.2) for any 𝑣∈𝐿²(Ω_𝑇;𝐶_#(𝑌×(0,1))) when 𝜀→0. One writes 𝑢^𝜀(𝑥,𝑡)⇀^2,2𝑢₀(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 {𝑤^𝜀} in 𝐿¹(Ω_𝑇) (2,2)-scale converges very weakly to 𝑤₀∈𝐿¹(Ω_𝑇×𝑌×(0,1)) if _Ω𝑇𝑤^𝜀(𝑥,𝑡)𝑣₁(𝑥)𝑣₂𝑥𝜀𝑐¹(𝑡)𝑐₂𝑡⟶𝜀²𝑑𝑥𝑑𝑡_Ω𝑇¹⁰_𝑌𝑤₀(𝑥,𝑡,𝑦,𝑠)𝑣₁(𝑥)𝑣₂(𝑦)𝑐₁(𝑡)𝑐₂(𝑠)𝑑𝑦𝑑𝑠𝑑𝑥𝑑𝑡(2.4) for any 𝑣₁∈𝐷(Ω), 𝑣₂∈𝐶^∞#(𝑌)/ℝ, 𝑐₁∈𝐷(0,𝑇), and 𝑐₂∈𝐶^∞#(0,1) when 𝜀→0. One writes 𝑢^𝜀(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 {𝑢^𝜀} bounded in 𝑊¹²(0,𝑇;𝐻¹⁰(Ω),𝐿²(Ω)) we also have a characterization of the (2,2)-scale limit for the gradients ∇𝑢^𝜀 and the corresponding very weak limit for {𝑢^𝜀/𝜀}.

Theorem 2.4. Let {𝑢^𝜀} be a bounded sequence in 𝑊¹²(0,𝑇;𝐻¹⁰(Ω),𝐿²(Ω)). Then, there exists a subsequence such that 𝑢_𝜀𝑢^𝜀(𝑥,𝑡)⟶𝑢(𝑥,𝑡)in𝐿²Ω_𝑇,∇𝑢(𝑥_𝜀,𝑡)⇀𝑢(𝑥,𝑡)in𝐿²0,𝑇;𝐻¹⁰(Ω),(2.6)(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(Ω_𝑇×(0,1);𝐻^1#(𝑌)/ℝ). Moreover 𝑢^𝜀(𝑥,𝑡)𝜀^2,2⇀_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(Ω_𝑇); 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 {𝑢^𝜀/𝜀}, where the denominator 𝜀 passes to zero, while the numerator 𝑢^𝜀 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 {𝑢^𝜀} apart from being a sequence of solutions to (1.1) and hence bounded in 𝐿²(0,𝑇;𝐻¹⁰(Ω)) is also bounded in 𝐿^∞(Ω_𝑇). These conditions together imply that {𝑢^𝜀} converges strongly in 𝐿²(Ω_𝑇) up to a subsequence and hence the boundedness of {𝑢^𝜀} in 𝐿^∞(Ω_𝑇) replaces the boundedness of {𝜕_𝑡𝑢^𝜀} 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 {𝑢^𝜀} 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 𝑎(𝑥/𝜀,𝑡/𝜀²) with 𝑎(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 𝑣∈𝐶^∞#(𝑌) or even 𝑣∈𝐶^∞#(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 𝜌∈𝐶^∞#(𝑌), 𝜌≥𝐶>0, 𝑓∈𝐿²(Ω_𝑇), and𝑎(𝑦,𝑠)𝑐⋅𝑐≥𝐴|𝑐|²,𝐴>0,(3.1) where 𝑎∈𝐿^∞#(𝑌×(0,1))^𝑁×𝑁. It can be proven along the lines of the corresponding proof in [7, Section  23.9] that {𝑢^𝜀} is bounded in the space 𝐿²(0,𝑇;𝐻¹⁰(Ω)); see also [2]. For, for example, 𝜌=1 it also holds that {𝜕_𝑡𝑢^𝜀} is bounded in 𝐿²(0,𝑇;𝐻⁻¹(Ω)) and hence {𝑢^𝜀} is bounded in the stronger space 𝑊¹²(0,𝑇;𝐻¹⁰(Ω),𝐿²(Ω)). Here, we instead make the physically quite natural assumption that {𝑢^𝜀} is bounded in 𝐿^∞(Ω_𝑇); see [1] and the references therein.

Theorem 3.1. Let {𝑢^𝜀} be a sequence of solutions to (1.1), where 𝜀→0, and assume that {𝑢^𝜀} is bounded in 𝐿^∞(Ω_𝑇). Then, 𝑢^𝜀⇀𝑢in𝐿²0,𝑇;𝐻¹⁰(Ω),(3.2) where 𝑢 is the solution to _𝑌𝜌(𝑦)𝑑𝑦𝜕_𝑡𝑢𝑢((𝑥𝑥,𝑢,𝑡)𝑡()𝑥−,=0∇)0⋅=(o𝑏𝑔n∇(𝑥𝑢𝜕)Ω(𝑥,×i𝑡n)()0Ω=,,𝑇𝑓)(,𝑥,𝑡)inΩ_𝑇,(3.3) with 𝑏∇𝑢(𝑥,𝑡)=¹⁰_𝑌𝑎(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇_𝑦𝑢₁(𝑥,𝑡,𝑦,𝑠)𝑑𝑦𝑑𝑠,(3.4) where 𝑢₁∈𝐿²(Ω_𝑇×(0,1);𝐻^1#(𝑌)/ℝ) solves 𝜌(𝑦)=𝜕_𝑠𝑢𝜌₁(𝑦(𝑥),𝑡,𝑦_𝑌,𝑠)−∇_𝑦⋅𝑎(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇_𝑦𝑢₁(𝑥,𝑡,𝑦,𝑠)𝑎(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇_𝑦𝑢₁(𝑥,𝑡,𝑦,𝑠)⋅∇_𝑦𝜌1(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−𝜌𝑥𝜀𝑢^𝜀(𝑥,𝑡)𝑣(𝑥)𝜕_𝑡𝑐(𝑡)+𝑎𝑥𝜀,𝜀𝑡²∇𝑢^𝜀=(𝑥,𝑡)⋅∇𝑣(𝑥)𝑐(𝑡)𝑑𝑥𝑑𝑡_Ω𝑇𝑓(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)𝜕_𝑡𝑐(𝑡)+𝑎(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇_𝑦𝑢₁(𝑥,𝑡,𝑦,𝑠)⋅∇𝑣(𝑥)𝑐(𝑡)𝑑𝑦𝑑𝑠𝑑𝑥𝑑𝑡_Ω𝑇𝑓(𝑥,𝑡)𝑣(𝑥)𝑐(𝑡)𝑑𝑥𝑑𝑡.(3.7) To find a local problem we choose 𝑣(𝑥)=𝜀𝑣₁(𝑥)𝑣̃𝑥𝜀(3.8) in (3.6), where 𝑣₁∈𝐷(Ω), 𝑣̃(𝑦)=𝑣₂(𝑦)−𝐶𝜌(𝑦),𝑣²𝐶=_𝑌∈𝐶^∞#(𝑌),(3.9)𝜌(𝑦)𝑣₂(𝑦)𝑑𝑦.(3.10) Hence, we have _𝑌𝜌(𝑦)𝑣̃(𝑦)𝑑𝑦=0.(3.11) Further, we let 𝑐(𝑡)=𝑐₁(𝑡)𝑐₂𝑡𝜀²,𝑐¹∈𝐷(0,𝑇),𝑐₂∈𝐶^∞#(0,1)(3.12) and arrive at _Ω𝑇−𝜌𝑥𝜀𝑢^𝜀(𝑥,𝑡)𝑣₁(𝑥)𝑣̃𝑥𝜀𝜀𝜕^𝑡𝑐₁(𝑡)𝑐₂𝑡𝜀²+𝜀⁻¹𝑐₁(𝑡)𝜕_𝑠𝑐₂𝑡+𝑎𝑥𝜀²𝜀,𝜀𝑡²∇𝑢^𝜀(𝑥,𝑡)⋅𝜀∇𝑣₁(𝑥)𝑣̃𝑥𝜀+𝑣¹(𝑥)∇_𝑦𝑣̃𝑥𝜀𝑐¹(𝑡)𝑐₂𝑡=𝜀²𝑑𝑥𝑑𝑡_Ω𝑇𝑓(𝑥,𝑡)𝜀𝑣₁(𝑥)𝑣̃𝑥𝜀𝑐¹(𝑡)𝑐₂𝑡𝜀²𝑑𝑥𝑑𝑡.(3.13) The choice of 𝑣̃ 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 _Ω𝑇𝑎𝑥𝜀,𝜀𝑡²∇𝑢^𝜀(𝑥,𝑡)⋅𝑣₁(𝑥)∇_𝑦𝑣̃𝑥𝜀𝑐¹(𝑡)𝑐₂𝑡𝜀²𝑑𝑥𝑑𝑡,_Ω𝑇−𝜌𝑥𝜀𝑢^𝜀(𝑥,𝑡)𝑣₁(𝑥)𝑣̃𝑥𝜀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⋅−𝑣𝜌₁(𝑦)𝑢₁(𝑥,𝑡,𝑦,𝑠)𝑣₁(𝑥)𝑣̃(𝑦)𝑐₁(𝑡)𝜕_𝑠𝑐₂(𝑠)+𝑎(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇_𝑦𝑢₁(𝑥,𝑡,𝑦,𝑠)(𝑥)∇_𝑦𝑣̃(𝑦)𝑐₁(𝑡)𝑐₂(𝑠)𝑑𝑦𝑑𝑠𝑑𝑥𝑑𝑡=0.(3.15) Hence, observing that _𝑌−𝜌(𝑦)𝑢₁(𝑥,𝑡,𝑦,𝑠)−𝐶𝜌(𝑦)𝑑𝑦=_𝑌𝐶𝑢₁(𝑥,𝑡,𝑦,𝑠)𝑑𝑦=0(3.16) and recalling (3.9), we arrive at _Ω𝑇¹⁰_𝑌⋅−𝑣𝜌₁(𝑦)𝑢₁(𝑥,𝑡,𝑦,𝑠)𝑣₁(𝑥)𝑣₂(𝑦)𝑐₁(𝑡)𝜕_𝑠𝑐₂(𝑠)+𝑎(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇_𝑦𝑢₁(𝑥,𝑡,𝑦,𝑠)(𝑥)∇_𝑦𝑣²(𝑦)−C𝜌(𝑦)𝑐¹(𝑡)𝑐₂(𝑠)𝑑𝑦𝑑𝑠𝑑𝑥𝑑𝑡=0.(3.17) We write _Ω𝑇¹⁰_𝑌+𝑎−(𝜌𝑦(,𝑦𝑠))𝑢₁∇(𝑢𝑥,(𝑥𝑡,,𝑦𝑡,)𝑠+)𝑣∇_1𝑦(𝑥)𝑣₂(𝑦)𝑐₁(𝑡)𝜕_𝑠𝑐₂(𝑠)−𝑢₁(𝑥,𝑡,𝑦,𝑠)⋅𝑣₁(𝑥)∇_𝑦𝑣₂(𝑦)𝑐₁(𝑡)𝑐₂(𝑠)𝑑𝑦𝑑𝑠𝑑𝑥𝑑𝑡_Ω𝑇¹⁰_𝑌𝑎(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇_𝑦𝑢₁(𝑥,𝑡,𝑦,𝑠)⋅𝑣₁(𝑥)∇_𝑦𝜌𝐶(𝑦)𝑑𝑦⋅𝑐¹=0.(𝑡)𝑐₂(𝑠)𝑑𝑠𝑑𝑥𝑑𝑡(3.18) Applying repeatedly the variational lemma, we find that ¹⁰_𝑌+−𝑎𝜌((𝑦𝑦,)𝑠𝑢)₁(∇𝑥,𝑢𝑡(,𝑥𝑦,,𝑡𝑠))+𝑣₂∇(𝑦_𝑦)𝜕_𝑠𝑐₂(𝑠)=𝑢₁(𝑥,𝑡,𝑦,𝑠)⋅∇_𝑦𝑣₂(𝑦)𝑐₂(𝑠)𝑑𝑦𝑑𝑠₁₀_𝑌𝑎(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇_𝑦𝑢₁(𝑥,𝑡,𝑦,𝑠)⋅∇_𝑦𝜌𝐶(𝑦)𝑑𝑦𝑐²(𝑠)𝑑𝑠(3.19) and, according to the definition (3.10) of 𝐶, ¹⁰_𝑌+−𝑎𝜌((𝑦𝑦,)𝑠𝑢)₁(∇𝑥,𝑢𝑡(,𝑥𝑦,,𝑡𝑠))+𝑣₂∇(𝑦_𝑦)𝜕_𝑠𝑐₂(𝑠)=𝑢₁(𝑥,𝑡,𝑦,𝑠)⋅∇_𝑦𝑣₂(𝑦)𝑐₂(𝑠)𝑑𝑦𝑑𝑠₁₀_𝑌𝜌(𝑦)_𝑌𝑎(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇_𝑦𝑢₁(𝑥,𝑡,𝑦,𝑠)⋅∇_𝑦𝜌1(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 𝜌𝑥𝜀𝜕^𝑡𝑢^𝜀(𝑥,𝑡)−∇⋅𝑎𝑥𝜀,𝜀𝑡∇𝑢^𝜀(𝑥,𝑡)=𝑓(𝑥,𝑡)inΩ_𝑇.(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−𝜌𝑥𝜀𝑢^𝜀(𝑥,𝑡)𝑣(𝑥)𝑣̃𝑥𝜀𝜀𝜕^𝑡𝑐₁(𝑡)𝑐₂𝑡𝜀+𝑐¹(𝑡)𝜕_𝑠𝑐₂𝑡+𝑎𝑥𝜀𝜀,𝜀𝑡∇𝑢^𝜀(𝑥,𝑡)⋅𝜀∇𝑣(𝑥)𝑣̃𝑥𝜀+𝑣¹(𝑥)∇_𝑦𝑣̃𝑥𝜀𝑐¹(𝑡)𝑐₂𝑡=𝜀𝑑𝑥𝑑𝑡_Ω𝑇𝑓(𝑥,𝑡)𝜀𝑣(𝑥)𝑣̃𝑥𝜀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 −∇_𝑦=⋅𝜌𝑎(𝑦()𝑦,𝑠)_𝑌∇𝑢(𝑥,𝑡)+∇_𝑦𝑢₁(𝑥,𝑡,𝑦,𝑠)𝑎(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇_𝑦𝑢₁(𝑥,𝑡,𝑦,𝑠)⋅∇_𝑦𝜌1(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 𝑣̃ with any 𝑣₂∈𝐶^∞#(𝑌) and obtain the more conventional local problem −∇_𝑦⋅𝑎(𝑦,𝑠)∇𝑢(𝑥,𝑡)+∇_𝑦𝑢₁(𝑥,𝑡,𝑦,𝑠)=0(3.24) without any strange term. Observing that 1/𝜌∈𝐶^∞#(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 {𝜕_𝑡𝑢^𝜀} bounded in 𝐿²(Ω_𝑇), a such cancellation appears but under the present conditions of boundedness of {𝑢^𝜀} in 𝐿²(0,𝑇;𝐻¹⁰(Ω)) and 𝐿^∞(Ω_𝑇) and strong convergence in 𝐿²(Ω_𝑇) 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.