Homogenization of nonlinear dissipative hyperbolic problems exhibiting arbitrarily many spatial and temporal scales

This paper concerns the homogenization of nonlinear dissipative hyperbolic 
problems 
\begin{gather*} 
\partial _{tt}u^{\varepsilon }\left( x,t\right) -\nabla \cdot \left( a\left( 
\frac{x}{\varepsilon ^{q_{1}}},\ldots ,\frac{x}{\varepsilon ^{q_{n}}},\frac{t 
}{\varepsilon ^{r_{1}}},\ldots ,\frac{t}{\varepsilon ^{r_{m}}}\right) \nabla 
u^{\varepsilon }\left( x,t\right) \right) \\ 
+g\left( \frac{x}{\varepsilon ^{q_{1}}},\ldots ,\frac{x}{\varepsilon ^{q_{n}} 
},\frac{t}{\varepsilon ^{r_{1}}},\ldots ,\frac{t}{\varepsilon ^{r_{m}}} 
,u^{\varepsilon }\left( x,t\right) ,\nabla u^{\varepsilon }\left( x,t\right) 
\right) =f(x,t) 
\end{gather*} 
where both the elliptic coefficient $a$ and the dissipative term $g$ are 
periodic in the $n+m$ first arguments where $n$ and $m$ may attain any 
non-negative integer value. The homogenization procedure is performed within 
the framework of evolution multiscale convergence which is a generalization 
of two-scale convergence to include several spatial and temporal scales. In 
order to derive the local problems, one for each spatial scale, the crucial 
concept of very weak evolution multiscale convergence is utilized since it 
allows less benign sequences to attain a limit. It turns out that the local 
problems do not involve the dissipative term $g$ even though the homogenized 
problem does and, due to the nonlinearity property, an important part of the 
work is to determine the effective dissipative term. A brief illustration of 
how to use the main homogenization result is provided by applying it to an 
example problem exhibiting six spatial and eight temporal scales in such a 
way that $a$ and $g$ have disparate oscillation patterns.


(Communicated by Dag Lukkassen)
Abstract. This paper concerns the homogenization of nonlinear dissipative hyperbolic problems ∂ttu ε (x, t) − ∇ · a x ε q 1 , . . . , x ε qn , t ε r 1 , . . . , t ε rm ∇u ε (x, t) +g x ε q 1 , . . . , x ε qn , t ε r 1 , . . . , t ε rm , u ε (x, t) , ∇u ε (x, t) = f (x, t) where both the elliptic coefficient a and the dissipative term g are periodic in the n + m first arguments where n and m may attain any non-negative integer value. The homogenization procedure is performed within the framework of evolution multiscale convergence which is a generalization of two-scale convergence to include several spatial and temporal scales. In order to derive the local problems, one for each spatial scale, the crucial concept of very weak evolution multiscale convergence is utilized since it allows less benign sequences to attain a limit. It turns out that the local problems do not involve the dissipative term g even though the homogenized problem does and, due to the nonlinearity property, an important part of the work is to determine the effective dissipative term. A brief illustration of how to use the main homogenization result is provided by applying it to an example problem exhibiting six spatial and eight temporal scales in such a way that a and g have disparate oscillation patterns.
1. Introduction. The framework of hyperbolic equations has applications in a wide range of areas such as, e.g., electromagnetics, acoustics, hyperbolic heat conduction and dynamic elasticity. Such physical phenomena may exhibit a multiscale behavior with respect to both space and time, for example in electromagnetism by a vibrating heterogeneous medium. Furthermore, if a physical system suffers from energy losses these may be accounted for by a so-called dissipative term. In this contribution we homogenize problems showing these three physical notions as we study the homogenization of hyperbolic problems having a nonlinear dissipative term and where there is an arbitrary number of both spatial and temporal scales involved. t ε rm , u ε (x, t) , ∇u ε (x, t) = f (x, t) in Ω T , u ε (x, t) = 0 on ∂Ω×(0,T ), (1) u ε (x, 0) = 0 in Ω, where ε > 0 tends to zero, 0 < q 1 < · · · < q n and 0 < r 1 < · · · < r m . Here where Ω is an open bounded subset of R N with C 2 boundary, and a and g are periodic with respect to the unit cube Y = (0, 1) N in R N in the n first variables and with respect to the interval S = (0, 1) in the following m variables. Observe that it is not necessarily the case that all the scales give rise to actual oscillations, i.e., a and g may have disparate oscillation patterns. In order to homogenize (1) we study the asymptotic behavior of the corresponding sequence of weak solutions {u ε } and derive the homogenized problem uniquely solved by the limit of this sequence. Along with the homogenized problem we obtain for each spatial scale present in a an associated local problem solved by socalled correctors. These generative local problems are either hyperbolic or elliptic and in the former case we say that we have resonance. It turns out that resonance occurs when the spatial oscillation mode in question coincides with some temporal oscillation mode, i.e., when the i-th spatial scale satisfies ε qi = ε rj for some j. Concerning the dissipative term g we find that it does not appear in the local problems and hence spatial scales only present in g merely give rise to non-generative local problems, i.e., problems with vanishing solutions.
1.2. Background and theoretical pivotals. For the homogenization of (1) we employ in a general sense the concept of multiscale convergence, a technique introduced in [18] for two spatial scales, formalized in [1] and generalized to an arbitrary number of spatial scales in [2]. In [14] a compactness result akin to multiscale convergence was proven in order to overcome the impediments appearing when there are rapid oscillations in time present as well, see also [24] for a related result. In [8] the notion of very weak multiscale convergence was introduced making it possible to handle fast temporal oscillations for any number of spatial scales. A linear parabolic problem with an arbitrary number of scales in both space and time was studied in [10] employing results, originally presented in [28], generalizing very weak multiscale convergence to the evolution setting. In the present paper we use a version of the compactness result concerning very weak evolution multiscale convergence found in [10] which together with a corresponding evolution multiscale compactness result for gradients are the key results in the homogenization of (1).
Homogenization of certain nonlinear hyperbolic problems has been performed, e.g., in [4], [26] and [32] having one rapid spatial scale, in [30] with two rapid spatial scales, in [25] having one rapid scale in both space and time, in [22] where two rapid spatial and one rapid temporal scale appear, and in [33] which involves one rapid spatial and two rapid temporal scales. The key homogenization tool in [33], making rapid temporal scales possible to treat, is a compactness result of very weak evolution multiscale convergence type. Since in the present paper we provide a general very weak evolution multiscale compactness result it is possible to formulate a homogenization result for nonlinear hyperbolic problems having an arbitrary number of spatial and temporal scales.
1.3. Novelties. The present paper deals with the homogenization of a hyperbolic problem that exhibits an arbitrary number of scales in both space and time. This particular combination has, up to the authors' knowledge, never been studied in detail before even though the parabolic setting has been investigated in [10]. Moreover, the evolution equation in (1) carries a somewhat tricky nonlinear dissipative term equipped with the full set of spatial and temporal scales. One of the main challenges in the homogenization procedure is the process of passing to the limit for the dissipative term since it requires nontrivial techniques, developed in this contribution, to handle.
1.4. Organization of the paper. This paper is organized in the following way. In Section 2 we introduce evolution multiscale convergence and its related very weak version and give some results which prove useful in the procedure of homogenizing (1). Section 3 begins by establishing existence, uniqueness and an a priori estimate for sequences of solutions to (1). The section proceeds by unraveling the relevant convergence properties of the dissipative term and is concluded by formulating and proving the main result of the paper, i.e., deriving the homogenized problem and the local problems for (1). In the final part, Section 4, we provide an illustrative example of the use of the general homogenization result given in this paper.

2.
Preliminaries. This section deals with evolution multiscale convergence and the related concept of very weak evolution multiscale convergence. We give some results essential for the homogenization of the hyperbolic problem (1) which is performed in Section 3.
2.1. Evolution multiscale convergence. Since we study the problem (1) exhibiting an arbitrary number of spatial and temporal scales we invoke the concept of evolution multiscale convergence, a generalization of Nguetseng's classical twoscale convergence [18]. We give the following definition, also exploited in, e.g., [9] and [10].
Proof. This follows readily from the corresponding result concerning an arbitrarily number of spatial scales as given in, e.g., Section 2 in [2].
Proof. This is a straightforward generalization of a special case of a classical result found in, e.g., Theorem 2 in [16].
Remark 2. The Propositions 1 and 2 and Theorem 2.3 actually hold also for mere joint separatedness whose definition amounts to an obvious modification of joint well-separatedness.
In order to characterize the evolution multiscale limit of bounded sequences of gradients in the context of hyperbolic problems we introduce a convenient function such that the first and second temporal derivatives belong to L 2 (Ω T ) and L 2 (0, T ; H −1 (Ω)), respectively. The norm of this Banach space is given by . Observe that throughout the present paper we assume that γ is given as above. We will later see how γ is connected to the hyperbolic problem studied in the present paper.
We are now prepared to formulate and prove the theorem below concerning in particular the evolution multiscale convergence of sequences of gradients where the so-called correctors emerge in the limit.

2.2.
Very weak evolution multiscale convergence. In [14] the homogenization of a linear parabolic problem with oscillations in one spatial and one temporal scale was studied. In that context a compactness result was presented which can be seen, to the best of the authors' knowledge, as the first result of very weak convergence type. This primordial version of very weak convergence was employed also in [11] and [12] where a spatial and a temporal scale, respectively, was added to the scales in [14]. In [8] very weak multiscale convergence was introduced where an arbitrary number of spatial scales is allowed. This elaborates an idea in [24] where a modified version of the result in [14] was presented.
In the process of homogenizing (1) we will need a convenient generalization, called very weak evolution multiscale convergence, of the space-exclusive concept in [8] which also takes rapid temporal oscillations into consideration in an explicit manner.

Remark 3. A unique very weak evolution multiscale limit is provided by requiring that
which is explained in detail in Proposition 2.26 in [28] for the case of very weak two-scale convergence. The generalization to very weak evolution multiscale convergence is straightforward.
The following compactness result concerning very weak evolution multiscale convergence will be a key result in the homogenization of (1). The result is a special case of Theorem 7 in [10] and Theorem 2.78 in [28]. Theorem 2.6. Let {u ε } be a bounded sequence in V (0, T ; H 1 0 (Ω) , L 2 (Ω)) and assume that the lists {ε 1 , . . . , ε n } and {ε 1 , . . . , ε m } are jointly well-separated. Then, up to a subsequence, Apart from being crucial in the homogenization procedure the theorem above illustrates the fundamental merit of the very weak evolution multiscale convergence in that it permits certain L 2 (Ω T )-unbounded sequences to attain a limit, possibly allowing new paths to non-trivial limits, something that ordinary evolution multiscale convergence does not.
3. Homogenization of the hyperbolic problem. We are now prepared to homogenize the nonlinear dissipative hyperbolic problem (1). The present section commences by a more precise formulation of the homogenization problem giving structure conditions on the coefficient of the elliptic term and on the nonlinear dissipative term. In the first Subsection we prove a crucial convergence result for sequences of dissipative terms and in the second Subsection we formulate and prove the main homogenization result.
Consider (1), i.e., the sequence of nonlinear dissipative hyperbolic problems where 0 < q 1 < · · · < q n , 0 < r 1 < · · · < r m and f ∈ L 2 (Ω T ). For the elliptic coefficient a we assume that there exists α > 0 such that the conditions are satisfied for all (y n , s m ) ∈ R nN × R m and all ξ ∈ R N . Moreover, suppose that for the dissipative term g there are positive constants C 0 , . . . , C 5 for which

Remark 4.
As an example of a non-trivial dissipative term satisfying (B1)-(B6) we have where ψ is non-negative and belongs to C 1 (Y n,m ). See Section 1 in [33] and Section 2 in [3] for similar examples, though in different settings, of dissipative terms.
in Ω which is a real reflexive Banach space with the same norm as V (0, T ; H 1 0 (Ω) , L 2 (Ω)), see, e.g., p. 104 in [20]. The weak formulation of (3) is thus that we search for so- (Ω) and c ∈ C(0, T ). We give the following existence and uniqueness theorem. Proof. The claim follows readily from Theorem 2.1 in [3].
Next, we give some crucial a priori estimates in the proposition below. Proof. Observe that Theorem 3.1 guarantees the existence of unique weak solutions in V 0 (0, T ; H 1 0 (Ω) , L 2 (Ω)) to (3). The a priori estimates follow readily from the results in the paragraph A priori estimates in Section 3 in [3].
Proof. Following along the same lines as in the proof of Theorem 4.1 in [30], the Aubin-Lions Lemma together with the a priori estimates of Proposition 3 imply that (5) holds up to a subsequence.

3.1.
A convergence result for the dissipative term. In the proof of the main result, i.e. Theorem 3.2 in Subsection 3.2, we need to establish a limit of evolution multiscale type for sequences of dissipative terms in order to obtain the homogenized problem. This work is carried out in the present preparative Subsection for the purpose of lightening the presentation of the proof of the main homogenization result. Moreover, the obtained convergence results in the present section may have some independent value within the theory of asymptotic analysis of nonlinear evolution multiscale functions.
To begin with we formulate the following preliminary proposition.
We are prepared to prove (6). Observe first that the final argument in the lefthand side of (6) is ∇φ( where the remainder κ ε is given according to Taking the auxiliary result (7) into consideration what remains to show is that Proceeding as in the verification of the auxiliary result we deduce that there exists a limit η 0 ∈ L 2 (Ω T × Y n,m ) such that, up to a subsequence, η ε (x, t) n+1,m+1 η 0 (x, t, y n , s m ) as ε → 0. We get, for every v ∈ L 2 (Ω T ; C (Y n,m )), where we have used (B6) and the Hölder Inequality. Observe that the sequences This together with the fact that ε qj and ε qj −qi appearing in the sums tend to zero for all i and j means that we arrive at Ω T Yn,m η 0 (x, t, y n , s m ) v (x, t, y n , s m ) dy n ds m dxdt = 0 as ε goes to zero, and hence η 0 = 0. The proof is complete.
Remark 5. It is not difficult to verify that Proposition 5 actually holds also for φ j ψ j replaced by any finite linear combination of such products.
We are now prepared to state the main result of this Subsection.
where u and u j , j = 1, . . . , n, are given according to Theorem 2.4. Then, up to a subsequence, ×w (x, t) dxdt for any w ∈ D(Ω T ).

Remark 6.
In practice, the assumption (11) amounts to excluding the possibility that The inequality (13) is a straightforward generalization of a result found in, e.g., Theorem 0.2 in [1].
Proof of Proposition 6. The sequence {g( x ε q n , t ε r m , u ε (x, t) , ∇u ε (x, t))} is clearly bounded in L 2 (Ω T ) and hence, by Theorem 2.3, up to a subsequence for some g 0 ∈ L 2 (Ω T ; C (Y n,m )). We need to identify the limit g 0 in terms of g as given in (12). By (B6) and the Hölder Inequality we get for any φ ∈ D (Ω T ) and, for each j = 1, . . . , n, any finite linear combination ψ j of products on the form φ j v 1,j · · · v j,j c 1,j · · · c m,j where φ j ∈ D (Ω T ), v i,j ∈ C ∞ (Y i ) for i = 1, . . . , j − 1, v j,j ∈ C ∞ (Y j ) /R and c k,j ∈ C ∞ (S k ) for k = 1, . . . , m.
In order to proceed we employ Evans's perturbed test function method, see [6], [7]. Introduce the arbitrary but fixed parameter θ > 0 and the sequences {φ µ } and {ψ µ j } of such test functions satisfying φ µ → u − θw in L 2 (0, T ; H 1 0 (Ω)) and ψ µ j → u j − θw j in L 2 (Ω T × Y j−1,m ; H 1 (Y j )/R) and a.e. in Ω T and Ω T × Y j,m , respectively, as µ → ∞ where w and w j belong to the same function spaces as φ and ψ j , respectively, for j = 1, . . . , n. Using the introduced sequences we investigate the asymptotic behavior when ε → 0 of the two norms in the last term of the left-hand side of (15). By the reverse and ordinary triangle inequalities, and since u ε → u in L 2(γ+1) (Ω T ) for some subsequence, see Theorem 2.4, it holds that u ε → u in L 2 (Ω T ) implying HOMOGENIZATION OF HYPERBOLIC PROBLEMS 639 as ε tends to zero and we have taken care of the first norm. Expanding the square of the second norm and using (11), Theorem 2.4 and Proposition 1 we deduce that, up to a sub sequence, Passing to the limit by letting ε → 0 in (15) using (16), (17), (14) and Proposition 5, taking Remark 5 into consideration, we get up to a subsequence. Observe that by (B2) we have the majorization and using Lebesgue's Generalized Majorized Convergence Theorem we get ×θw (x, t) dy n ds m dxdt as µ → ∞. Hence, (18) becomes and dividing by θ followed by letting θ tend to zero we obtain where we have used Lebesgue's Generalized Majorized Convergence Theorem in a similar manner as in the limit process with respect to µ above. Since w ∈ D(Ω T ) is arbitrary the inequality is in fact an equality and we have for any w ∈ D(Ω T ). In particular, the evolution multiscale convergence (14) holds with respect to test functions in the subspace D(Ω T ) of L 2 (Ω T ; C (Y n,m )) and thus the proof is complete.

3.2.
The main homogenization result. Before stating the main homogenization result we introduce the characteristic numbers d i and ρ i for i = 1, . . . , n defined with respect to the scale exponents 0 < q 1 < · · · < q n and 0 < r 1 < · · · < r m appearing in the problem (3) studied: • If q i < r 1 , then d i = m; if r j ≤ q i < r j+1 for some j = 1, . . . , m − 1, then d i = m − j; and if q i ≥ r m , then d i = 0. • If q i = r j for some j = 1, . . . , m we let ρ i = 1, otherwise ρ i = 0.
Remark 7. The number ρ i shows whether there is resonance (ρ i = 1) or not (ρ i = 0) with respect to the i-th spatial scale. The meaning of d i is simply that its value is the number of temporal scales "faster" than the spatial scale in question.
Remark 8. When d i = 0 in (22) the interpretation is that there is no local temporal integration involved and that there is no established independence of any local temporal variable. Analogously, there is no local spatial integration if i = n.
Remark 9. The local problems do not involve the dissipative term g. This implies that the correctors are independent of local spatial variables not present in a, i.e., those exclusive to g. In particular this means that the correctors associated with such local spatial variables vanish due to the mean value zero property. Hence, only local problems corresponding to local spatial scales present in a are generative, i.e., generates actual contribution to the system of local problems.
For solutions to (3) it is possible to verify assumption (11) of Proposition 6, utilized in the proof of Theorem 3.2, according to the lemma below. Lemma 3.3. Let {u ε } be the sequence of unique weak solutions in V 0 (0, T ; H 1 0 (Ω) , L 2 (Ω)) to (3). Then, up to a subsequence, where u and u j , j = 1, . . . , n, are given according to Theorem 2.4.
Proof. To begin with, by Theorem 3.1 there in fact exists a unique weak solution u ε ∈ V 0 (0, T ; H 1 0 (Ω) , L 2 (Ω)) to (3). Introduce ω ∈ D (Ω T ) and, for each index j = 1, . . . , n, let σ j be a finite linear combination of products on the form We investigate the right-hand side with respect to its limit as ε → 0 term by term and then add up the result. The first term. For the first term we have by the reverse triangle inequality. By condition (A3) and the weak formulation (4) we have In order to proceed with the inequality we must verify that see e.g. Section 24.3 in [34]. Let us introduce the space We have that {u ε } is bounded in V 0 (0, T ; H 1 0 (Ω) , L 2 (Ω)) due to Proposition 3 and, as noted in e.g. Subsection 2.2 in [22], that V 0 (0, T ; H 1 0 (Ω) , L 2 (Ω)) is continuously embedded in Λ (0, T, Ω) which implies that {u ε } is bounded in Λ (0, T, Ω). Hence,
Letting ε tend to zero and employing (27) we get for the right-hand side of (26) that its limit is equal to Fix some δ > 0. Then, for some ε 1 > 0, it holds that for all 0 < ε ≤ ε 1 (with respect to the extracted subsequence). Due to density we may choose ω and σ j , j = 1, . . . , n, such that Hence, by (25) and (30) we have for every 0 < ε ≤ ε 1 . The second term. For the second term on the right-hand side of (24) we have that there exists an ε 2 > 0 such that for any 0 < ε ≤ ε 2 by the fundamental convergence result of Proposition 1. The third term. For the third term we get by the reverse triangle inequality, (A3) and (30).
We also need the following lemma, henceforth referred to as the Periodic Generalized Variational Lemma.
Lemma 3.4. Let G be a non-empty open set in R and assume that w ∈ L 1 loc (G) and suppose that Proof. Suppose µ = 0. The statement in this case follows by the Generalized Variational Lemma, see e.g. Proposition 18.36 in [34], since the set of S-periodic repetitions of functions in D(S) is a mere subset of C ∞ (S). Suppose µ > 0. Since the space of all φ spans C ∞ (S) we have that the corresponding derivatives d ds φ will span C ∞ (S)/R. Hence, by induction, we have that the set of all d µ ds µ φ spans C ∞ (S)/R for any µ > 0. Then, the claim follows by Corollary 18.37 in [34] where C ∞ (S)/R is used instead of D(S)/R as the space of test functions similar to the argument in the case µ = 0.
We are now prepared to carry out the proof of the main result.
Our next aim is to extract the local problem (22) for each i = 1, . . . , n and the associated independencies with respect to the local variables. In order to do so fix i = 1, . . . , n and introduce the test functions v(x) = ε p v 1 (x)v 2 x ε q1 · · · v i+1 x ε qi , p > 0 and c(t) = c 1 (t)c 2 t ε r1 · · · c λ+1 t ε r λ , λ = 1, . . . , m with v 1 ∈ D (Ω) , v j ∈ C ∞ (Y j−1 ) for j = 2, . . . , i, v i+1 ∈ C ∞ (Y i ) /R, c 1 ∈ D (0, T ) and c l ∈ C ∞ (S l−1 ) for l = 2, . . . , λ + 1. We choose p and λ later. With these test functions used in (35) we obtain For i = 1 the number of temporal scales faster than the first, i.e. slowest, spatial scale ε is 6, i.e. d 1 = 6. The spatial scale in question coincides with a temporal scale which means that ρ 1 = 1. Furthermore, this spatial scale is present in a, i.e., the local problem is generative. In the same way d i , ρ i and the generative property for each i = 2, . . . , 5 are derived and we summarize the result in Table 1.  Table 1 We now have the information required in order to extract all local problems. Since the procedure is straightforward we only carry out the details for one of them. If we, e.g., consider i = 4 then a glance in Table 1 gives that the relevant characteristic numbers are d 4 = 3 and ρ 4 = 1 and that the local problem is generative. Hence, u 4 ∈ L 2 (Ω T × Y 3,4 ; H 1 (Y 4 )/R) and (22)  With the definition ofã in terms of a and using Table 1 again, equation (39) yields ∂ s4s4 u 4 x, t, y 2 , y 4 , s 1 , s 4 − ∇ y4 · S5 a y 2 , y 4 , s 1 , s 4 , s 5 ds 5 × ∇u (x, t) + ∇ y1 u 1 (x, t, y 1 , s 1 ) + ∇ y2 u 2 x, t, y 2 , s 1 +∇ y4 u 4 x, t, y 2 , y 4 , s 1 , = 0 where we have used the fact that a only depends on y 2 , y 4 , s 1 , s 4 and s 5 . Clearly, this means that the correctors only depend on these local variables and hence we have that u 4 ∈ L 2 (Ω T × Y 2 × S 1 × S 4 ; H 1 (Y 4 )/R). The two remaining generative local problems can be determined in a similar manner. What the illustrative example of this section demonstrates, apart from showing how to proceed in a special case, is that in spite of the fact that it may seem like that the oscillations of a and g match in the general homogenization result, it is possible to treat also problems with disparate oscillation patterns. In practice this is achieved by numbing out excessive oscillation modes.
Remark 11. An alternative approach in the study of problems of the type studied in the present paper involving several structure functions would have been to let the functions have formally different sets of local spatial and temporal scales but allowing any number of them to coincide. The reason we have chosen the method used here, i.e. to consider structure functions a and g with formally identical oscillation modes, is because it is more convenient to implement in practise.