Diffusion in inhomogeneous media with periodic microstructures

Diffusion in inhomogeneous materials can be described by both the Fick and Fokker--Planck diffusion equations. Here, we study a mixed Fick and Fokker-Planck diffusion problem with coefficients rapidly oscillating both in space and time. We obtain macroscopic models performing the homogenization limit by means of the unfolding technique.


Introduction
The study of the motion of particles diffusing in a confined region is relevant in many different fields (see, for instance, the recent papers [8, 11-13, 15, 24] and the references therein). In several studies, it has been shown that the interaction of particles with the walls results into a diffusive coefficient depending on the space coordinates [20,21]. A rather natural microscopic counterpart is represented by the random walk models, with hopping probabilities depending on the site coordinates. Such kind of models have been, for instance, introduced in the study of wetting phenomena, in which the effect of competition between long range attraction and reflection at the wall is modeled [14]. We also mention that space dependent diffusion is also considered in some biological ionic channel models, to justify the selection of ionic species [2,5].
In the context of diffusion motion in inhomogeneous materials, due to the space dependence of the diffusion coefficient, the derivation of the macroscopic equation is not straightforward. Indeed, assuming that the flux is given either by −B∇u or −∇(cu), where B and c represent the diffusion coefficient and u the density field, gives rise to two different diffusion equations, known in the literature as the Fick and the Fokker-Planck diffusion laws [23,25,[27][28][29], respectively. In the recent paper [4], relying on a hydrodynamic limit computation, it has been proved that the two different choices mentioned above for the flux are connected to the microscopic structure of the inhomogeneity. Indeed, for local isotropic space inhomogeneities, the Fokker-Planck version of the flux is found, whereas when the space inhomogeneity is exclusively due to local anisotropy, the Fick expression is recovered. In mixed situations, the general flux structure −B∇(cu) is found and the corresponding general diffusion law u t = ∇ · [B∇(cu)] is obtained.
Here, we study such a mixed Fick and Fokker-Planck diffusion problem for inhomogeneous materials, whose diffusion properties are described by means of rapidly oscillating coefficients with respect to both space and time (see the initial-boundary value problem (2.4)-(2.6) below). We assume that such a material has an underlying periodic microstructure, whose characteristic length is of order ε α (ε and α being strictly positive real parameters), while its time oscillation has a period of order ε β , β being another strictly positive parameter.
As usual in this kind of very fast oscillating problems, the main purpose is to obtain a macroscopic model, overcoming the difficulties due to the intricate original geometry and appearing, for instance, in the numerical approach. To this purpose, we are led to let ε → 0, thus performing a homogenization limit. The resulting equation models the effective behavior of the medium in the macroscopic setting, keeping memory, in general, of the underlying periodic structure. However, the homogenization of the problem (2.4)-(2.6) seems to be a aac-ffp.tex -17 marzo 2020 too ambitious goal, without some further structural assumptions on the coefficients. For this reason, we shall confine our investigation to a particular case introduced in Section 2.2, where the capacitive coefficient in front of the time-derivative and the Fokker coefficient inside the spatial gradient are assumed to have a separate dependence on the time and space oscillating variables (similarly to the classical Fick case, treated in [1]), but admitting that the Fokker coefficient can be perturbed by a non-product additional coefficient of amplitude ε. We refer to this case as the weakly non-product case. However, as we will see in the sequel, when we consider the pure Fokker-Planck model (i.e., the diffusion matrix in front of the gradient term is the identity), with unit capacity, the perturbation does not play any role in the limit equation and disappears from the expression of the effective coefficients (see Subsections 2.2.2 and 2.2.3).
More precisely, in Subsection 2.2.2, by using the well-known two-scale expansion technique, introduced in [6], we formally show that the non-product perturbation does not affect the upscaled equation, when its amplitude ε is of the same order of the spatial oscillation period ε α (i.e. α = 1), as long as we assume the diffusion matrix B = I. This result is also rigorously proven in Section 4. It is rather natural to ask what would happen if such a perturbation were more intense with respect to the microscopic oscillation scale. This case will be considered in Subsection 2.2.3. However, since in the formal expansions we are obliged to deal only with integer powers, we cannot consider an exponent smaller than one for the ε amplitude of the non-product perturbation. Hence, we accelerate the microscopic spatial oscillations choosing the smaller oscillation period ε 2 . We will show that also in this case the small non-product perturbation does not affect the upscaled equation. However, we do not propose this as a general conclusion, since it could depend on the special choice of the diffusion matrix and the capacity coefficients.
In Section 4, we will rigorously prove that the same property holds also in the general mixed Fick and Fokker-Planck case, if the amplitude of the non-product perturbation in the Fokker coefficient is strictly smaller than the spatial oscillation period, i.e. α < 1.
At our knowledge, diffusion problems governed by Fick and/or Fokker-Planck laws depending on capacitive, diffusive and Fokker coefficients, highly oscillating with respect to time and space simultaneously are not considered in an extensive body of mathematical literature. Among the few results, we recall [1-3, 16-19, 26].
In particular, in [1] the authors have considered a homogenization problem in the framework of the standard heat equation, which is very close to the case analyzed in the present research. The main novelty of that paper, with respect to former literature, is not only the presence of a capacitive term, oscillating both in space and time, but also the fact that the homogenization problem has been solved under completely general assumptions on the space aac-ffp.tex -17 marzo 2020 Given a function w ∈ L 2 (Ω) (or w ∈ L 2 (Ω T )), we will denote by w 2 its L 2 (Ω)-norm (or L 2 (Ω T )-norm, respectively). Finally, γ will denote a strictly positive constant, which may vary from line to line.

The general problem
Consider the real functions a(x, t, y, τ ), c(x, t, y, τ ), and the n×n-matrix function B(x, t, y, τ ) with (x, t) ∈ Ω T and Q-periodic in (y, τ ). We assume that B ∈ L ∞ (Ω T × Q; R n×n ) is symmetric and satisfies the bounds for every ξ ∈ R n and almost every (x, t, y, τ ) ∈ Ω T × Q. We assume, also, that a, c ∈ L ∞ (Ω T × Q) satisfy the bounds for almost every (x, t, y, τ ) ∈ Ω T × Q. Moreover, we assume that a, c, B ij are Lipschitzcontinuous on Ω T × Q. Let α, β > 0 and set Given f ∈ L 2 (Ω T ) andū ∈ H 1 0 (Ω), we are interested in studying the family of mixed Fick and Fokker-Planck problems with oscillating coefficients Note that, in the case c = 1, the pure Fick problem is recovered, while in the case B = I, we obtain the pure Fokker-Planck problem. The terms a, B, c, and f will be respectively called capacity coefficient, diffusion matrix, Fokker coefficient, and source term. If we let v ε = c ε u ε , the above problem can be rewritten as the following Fick problem with linear lower order terms aac-ffp.tex -17 marzo 2020 We remark that, by [22,Chapter 4,Theorem 9.1], for every ε > 0 fixed, the problem (2.7)-(2.9) admits a unique solution v ε ∈ L 2 (0, T ; H 2 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)). Clearly this implies existence and uniqueness of the solution u ε ∈ L 2 (0, T ; H 1 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)) of the problem (2.4)-(2.6).
As we pointed out above, the homogenization of the previous problem in its full generality provides some very hard technical difficulties. For this reason, we shall treat only the special weakly non-product case described in the following Section 2.2.

The weakly non-product problem
Here we consider a special case in which the coefficients a and c of the problem (2.4)-(2.6) are factored in one term depending on (x, t, y) and another on (x, t, τ ); namely, the dependence on the micro-variables is separated. However, for the coefficient c, typical of the Fokker-Planck equation, we can admit a small general perturbation of the product part.
For later use, we set The main result of the paper is the following homogenization theorem, whose proof can be found in Subsection 4.3.

Formal expansions for the weakly non-product problem
In Section 4, we will prove rigorously the macroscopic equations for problem (2.14)- (2.16) in the case 0 < α ≤ 1 and β > 0. Namely, we will be able to homogenize the system in the case in which the spatial oscillations are not too fast with respect to the amplitude of the non-product perturbation in the Fokker coefficient appearing in the Fokker-Planck equation (2.14). However, before the rigorous approach, we first set up some formal expansions. More precisely, in Subsection 2.2.2, we consider, as an example, the case α = 1 and β = 2, which is indeed rigorously covered by Theorem 2.1. Moreover, in Subsection 2.2.3 we formally approach also some cases (with integer exponents) not covered by the theory developed in Section 4, that is to say, when the spatial oscillations are faster than the amplitude of the non-product perturbation in the Fokker coefficient (i.e., α > 1). In details, we consider the case α = 2 and β = 1, 2, 4, in which time oscillations are respectively slower, as fast as, and faster than spatial ones. Note that the case α = 2 and β = 4 corresponds to the natural parabolic scaling. In our formal expansions arguments, we assume that the diffusion matrix is B = Id, the capacity coefficients are a 1 = a 2 = 1, and the source term is f = 0. Note that this last assumption could be easily removed.

Space oscillations as fast as the perturbation amplitude
As mentioned above in this section, we formally study the case α = 1 and β = 2, which is covered by Theorem 2.1.
We let y = x/ε and τ = t/ε 2 and, by abusing the notation, we write the differential rules We then look for a solution of (2.14), using the formal expansion u ε (x, t) = u 0 (x, t, y, τ ) + εu 1 (x, t, y, τ ) + ε 2 u 2 (x, t, y, τ ) + · · · , (2.26) with u k a Q-periodic function with respect to (y, τ ). By replacing (2.26) in (2.14), we get aac-ffp.tex -17 marzo 2020 Thus, at order 1/ε 2 , we find the equation which must be solved assuming that v 0 is Q-periodic in (y, τ ). We prove, indeed, that v 0 does not depend on the microscopic variables: we first multiply (2.30) times v 0 and integrate on the microscopic cell By periodicity, the first and the third integral vanish, hence where we used (2.10). This implies that v 0 is constant with respect to y.
On the other hand, since both b 2 and v 0 do not depend on y, from (2.30) we immediately get that v 0 does not depend on τ as well. Note that, since v 0 (x, t) = b 1 (x, t, y)u 0 (x, t, y, τ ), we have that u 0 does not depend on τ .
We now consider the 1/ε order equation. From (2.14), (2.27), and (2.28), we have aac-ffp.tex -17 marzo 2020 Since b 2 and v 0 = b 1 u 0 do not depend on y, (2.33) simplifies to We now let v 1 (x, t, y, τ ) = b 1 (x, t, y)u 1 (x, t, y, τ ) and, from (2.34), we get We now look for a solution of the above equation in the factored form with ζ a Q-periodic function with respect to (y, τ ). By plugging (2.36) into (2.35), we get that ζ has to solve the equation We, finally, consider the ε 0 order equation, which will yield a compatibility condition providing an equation for u 0 . From (2.14), (2.27), and (2.28), we have which can be seen as an equation for u 2 . Hence, as usual, we introduce the function v 2 (x, t, y, τ ) = b 1 (x, t, y)u 2 (x, t, y, τ ) and rewrite (2.38) as for v 2 a Q-periodic function with respect to (y, τ ). Now, if we integrate (2.39) on Q, since b 1 does not depend on τ and b 2 does not depend on y, on the left hand side we find zero. Hence, we have the compatibility condition By the periodicity on Q and Gauss-Green formulas, we also have aac-ffp.tex -17 marzo 2020 Since, again by periodicity, where we have used that v 0 does not depend on y and τ . We have, finally, found an equation for v 0 . Indeed, we can deduce the equation that must be satisfied by the mean value of u 0 on the microscopic cell. If we let we can rewrite (2.43) as an equation for u, finding It is interesting to note that the non-product small correction εb ε in (2.14) does not play any role in the upscaled equation. Note that equation (2.43) (resp. (2.45)) coincides with the rigorous equations obtained in (4.18) (resp. (2.22)), once we have taken into account that, under the present assumptions, we have: i) the cell functions χ j in Theorem 4.3 are identically equal to zero; ii) the cell function ζ in Theorem 4.3, which is equal to the function ζ introduced in the equation (2.37) above, and the term ∇ y (b/b 1 ) disappear from the expressions of P eff in (4.21) and z eff in (4.22), due to the periodicity.

Space oscillations faster than the perturbation amplitude
In this section, we formally study the homogenization for the equation (2.14) in some cases not covered by the rigorous theory developed in Section 4. As mentioned above, we shall consider situations in which the spatial oscillation is faster than the amplitude of the nonproduct perturbation present in the Fokker coefficient.
We remark that, as we shall prove in Section 4 (for α < 1 or α = 1 and B = I) and as we found in Subsection 2.2.2 (for α = 1), the non-product perturbation εb ε appearing in (2.14) does not affect the upscaled equation. In the three cases discussed below, we shall see that this property is preserved in the case α = 2, namely, even when the spatial oscillation is fast, which is expected to reinforce the effect of the perturbation. We cannot conclude that this aac-ffp.tex -17 marzo 2020 is a general result for the scaling α > 1; indeed, it might depend on our peculiar choice of the diffusion matrix and the capacity coefficients in the formal computation.
We first consider the problem (2.14)-(2.16) for α = β = 2. Indeed, from the point of view of computations, such a case seems to be the most delicate among those discussed in this section. We then let y = x/ε 2 and τ = t/ε 2 and, by abusing the notation, we write the differential rules and look for a solution of (2.14), using the formal expansion (2.26). Differentiating in time, we are led again to (2.27), while differentiation in space yields where we took into account the powers of ε up to the order ε 0 . Thus, at order 1/ε 4 , we find the equation Recalling that b 2 does not depend on y (see (2.12)), from (2.48) we have that b 1 (x, t, y)u 0 (x, t, y, τ ) does not depend on y, thus we set v 0 (x, t, τ ) = b 1 (x, t, y)u 0 (x, t, y, τ ). We now consider the 1/ε 3 order equation. From (2.14), (2.27), and (2.47), we have where we have used that b 2 and v 0 do not depend on y. We now look for a solution of the above equation in the factored form By plugging (2.51) into (2.50) and using again that v 0 does not depend on y, we get that χ 1 has to solve the equation We now consider the 1/ε 2 order equation. From (2.14), (2.27), and (2.47), we have . Since the last two terms above integrate to zero on Y and v 0 does not depend on y, we have the compatibility condition Inserting, now, (2.51) in (2.54), we get the following equation for v 2 : We will look for a solution of the above equation in the factored form This leads to the equation for the unknown function χ 2 .
Next we consider the 1/ε order equation. From (2.14), (2.27), and (2.47) we have Then, we turn to the ε 0 order equation. From (2.14), (2.27), and (2.47), we have Since all the terms above but the first three on the left integrate to zero on Y, we have the compatibility condition where we have used that v 0 = b 1 u 0 and that v 0 and b 2 do not depend on y. Finally, by integrating over S, using the Q-periodicity of u 2 in (y, τ ) and the fact that both b 1 and v 0 do not depend on τ , we get for v 0 the equation The second case we consider here is the problem (2.14)-(2.16) for α = 2 and β = 4. We then let y = x/ε 2 and τ = t/ε 4 and, by abusing the notation, we write the differential rules and look for a solution of (2.14), using the formal expansion (2.26). By substituting (2.26) in (2.14), we get (2.67) and (2.47). Thus, at order 1/ε 4 , we find the equation Recalling that b 2 does not depend on y, from (2.68) we have that b 1 (x, t, y)u 0 (x, t, y, τ ) does not depend on y and τ , thus we set v 0 (x, t) = b 1 (x, t, y)u 0 (x, t, y, τ ). Indeed, we write (2.68) aac-ffp.tex -17 marzo 2020 as an equation for v 0 (which, clearly, has uniqueness) and note that v 0 , constant with respect to τ and y, solves such an equation. Now, we pass directly to the ε 0 order equation. From (2.14), (2.67), and (2.47), we have Integrating on Q, we find again (2.65) and, with the same arguments as those used above, we derive (2.45).
We now consider the 1/ε order equation. From (2.14), (2.71), and (2.47), we have By integrating on Y and using that v 0 does not depend on y and b 1 does not depend on τ , we arrive again to the compatibility condition (2.55), which implies that v 0 = v 0 (x, t).
We finally consider the ε 0 order equation. From (2.14), (2.71), and (2.47), we have Integrating on Q, we get once again (2.65) and, with the same arguments as those used above, we derive (2.45).

Preliminary results
In this Section, we always assume that α ≤ 1 and v ε is the solution to (2.21), under the assumptions listed in Subsection 2.2.

Estimates
We collect here some estimates that will be used in the sequel.
aac-ffp.tex -17 marzo 2020 Lemma 3.1. There exists γ > 0, depending on T, f 2 , v ε 2 and the structural constants of the problem, but independent of ε, such that Proof. Multiplying (2.17) by v ε /a ε 2 and integrating by parts, we obtain dx .

(3.2)
Under our assumptions on the sign of the coefficients, the left hand side of (3.2) can be bounded from below by the left hand side of (3.1). Again appealing to our assumptions and, in particular, to α ≤ 1, we see that all the functions appearing in the integrals on the right hand side of (3.2) are bounded by an absolute constant, with the exception of f , v ε , and ∇v ε . Then, by Young inequality, the right hand side of (3.2) can be bounded from above by where γ is independent of ε and δ > 0 can be chosen so that the gradient term can be absorbed into the left hand side. Finally, the result follows from the application of Gronwall lemma.
Taking into account that the initial datumū ε (and, therefore,v ε ) belongs not only to the space L 2 (Ω) (as needed in the previous estimate), but it is, indeed, in H 1 0 (Ω T ), we can obtain also some estimates for the time-derivative of the solution v ε , as stated in the next two lemmas. Lemma 3.2. There exists γ > 0, depending on T, f 2 , v ε 2 and the structural constants of the problem, but independent of ε, such that Proof. Let us multiply (2.17) times ∂v ε /∂t and integrate by parts to obtain (3.5) Under our assumptions on the sign of the coefficients, the left hand side of (3.5) can be bounded from below by the left hand side of (3.4). Next, we give estimates for each term I i . By Young inequality, we get Moreover, we calculate · ∇v ε dx ds (3.10) and thus, recalling that α ≤ 1, we obtain Again, an application of Young inequality gives For δ suitably small, we can absorb the terms in (3.12) multiplied by δ into the left hand side of (3.5). Then, the claim follows by applying Lemma 3.1.

aac-ffp.tex -17 marzo 2020
Proof. We select, as a test function in the integral formulation (2.21), the function We obtain Here, for any F = F (x, t), we denote by F (x, t) = F (x, t + h) its time shift. Let δ ∈ (0, T /2), 0 < h < δ/2, and assume that ϕ(x, t) = 0 for t < δ/2 and for t > T − δ/2. Using the formula (3.18) with ϕ(x, t) replaced with ϕ(x, t − h), and then changing variables to (x, t + h), but still keeping the old variable names, we obtain where ζ ∈ C 1 0 (δ/2, T − δ/2) is a nonnegative function such that ζ = 1 in (δ, T − δ) and |ζ ′ | ≤ γ/δ. On subtracting the two integral formulations (3.18) and (3.19), we obtain where, actually, only the estimation of I 1 requires a detailed calculation. Indeed, The term I 11 essentially equals the one estimated in the statement. The term I 12 is estimated, invoking the time regularity of a 1 , b 1 , by The integral I 13 can be bounded by means of the Hölder inequality as follows Clearly, the integrals I 2 , I 3 , I 4 and I 5 can be estimated by means of a similar device, once we remark that, owing to the assumed regularity in space of b 2 , b 1 , b, we get For example, the integral I 3 can be estimated by Finally, on collecting all the estimates above, we get (3.17).

Unfolding
In the sequel, we denote by [r] the integer part of r ∈ R and, for x ∈ R n , we define the vector with integer components [x] = ([x 1 ], . . . , [x n ]).

aac-ffp.tex -17 marzo 2020
Let us consider the tiling of R n given by the boxes ε α (ξ + Y), with ξ ∈ Z n . Following [1], We introduce also the space-time cell containing the point (x, t) as Definition 3.5. The time-periodic unfolding operator T ε of a Lebesgue measurable function w defined on Ω T is given by otherwise. (3.28) Note that, by definition, it easily follows that Definition 3.6. The space-time average operator M ε of a Lebesgue integrable function w defined on Ω T is given by otherwise.
For later use, we define the functional spaces (3.34)

Homogenization
In this section, u ε and v ε are the solutions of problem (2.14)-(2.16) and (2.17)-(2.19) in Subsection 2.2, and we assume all the hypotheses listed there. As in Section 3, we always assume α ≤ 1.
We remark that, in all the cases we deal with, the final structure of the macroscopic homogenized equation will be the same, though the coefficients in it have to be defined caseby-case. Results are presented in two subsections: Section 4.1 is devoted to the case β ≥ 2α (fast oscillations), while in Section 4.2 the case β < 2α (slow oscillations) is studied.
In each case we prove two theorems, the first states the homogenization result and gives the limit two-scale system, while the second one introduces the corrector factorization and the resulting single scale equation. For technical reasons, the uniqueness of the solutions of the two limit problems is dealt in the corollaries following the theorems.
where the cell functions χ j , j = 1, . . . , n, and ζ are Q-periodic, with null mean average over Q, and are the unique solutions of and Moreover, the system (4.6) and (4.7) can be written as the single scale equation 20) Proof We first note that, by classical results (see, i.e., [6, Chapter 1, Section 2.2]), equations (4.16) and (4.17) admit a unique Q-periodic solution with null mean average. Then, a standard computation shows that v 1 defined in (4.15) satisfies (4.7). Finally, inserting (4.15) into (4.6) and performing some algebraic computations we get equation (4.18).
In particular, the second equality in (4.20) can be obtained as follows. We first note that By summing the two equations above we get (4.20). Proof. First we note that the matrix B hom in (4.20) is made of two parts, the first one is symmetric and by standard calculations it is also positive definite. On the other hand, the second part, which is due to the presence of the derivative with respect to the microscopic time τ in the parabolic equation (4.16) for the cell functions χ j , is antisymmetric. However, the uniqueness for equation (4.18), complemented with (4.8) and (4.9), still follows by standard energy estimates, Gronwall and Young inequalities, taking into account that the antisymmetric part of the homogenized matrix B hom disappears in the energy estimate. Indeed, it is multiplied by the symmetric matrix (v −ṽ) x i (v −ṽ) x j , where v andṽ are two different solutions of (4.18). Thus, the estimation can be performed as usual.
To prove uniqueness for the problem (4.6)-(4.9), we assume that there exist two solutions (v, v 1 ) and (ṽ,ṽ 1 ). From Corollary 4.2 and Theorem 4.3 it follows that v 1 andṽ 1 are given as in (4.15) for v andṽ, respectively. By substituting these two representations of v 1 andṽ 1 in (4.6), it follows that both v andṽ satisfy (4.18). Thus, by uniqueness of the solution of (4.18), we have that v =ṽ and, therefore, we also have v 1 =ṽ 1 .  Moreover, the pair (v, v 1 ) is a weak solution of the two-scale problem (4.6), (4.8), (4.9), complemented with the microscale equation div y M S Bb 2 a 2 (∇v + ∇ y v 1 ) aac-ffp.tex -17 marzo 2020 Moreover, the system (4.6) and (4.24) can be written as the single scale equation (4.18), where q eff , P eff , and z eff are formally defined as in Theorem 4.3, and with χ and ζ being the solutions of (4.28) and (4.29).
Proof We first note that, by classical results (see, i.e., [  Proof. First we note that the matrix B hom in (4.30) is symmetric and, by standard calculations, it is also positive definite. Thus, the uniqueness for equation (4.18), complemented with (4.8) and (4.9), as usual follows by standard energy estimates, Gronwall and Young inequalities. The second part of the corollary can be proven as we did for Corollary 4.4.

Slow oscillations
In this section, we consider the remaining case β < 2α.
Corollary 4.11. Given v ∈ L 2 (0, T ; H 1 0 (Ω)), equation (4.32) admits a unique solution v 1 ∈ L 2 (Ω T × S; H 1 # (Y)) with Y v 1 dy = 0.   Similarly to the case β > 2α discussed above, we have the following corollary. Notice that, in the present case, i.e. β < 2α, the dependence of the cell functions χ and ζ on the microtime τ is only parametric (as well as on (x, t)), via the coefficients of the corresponding equations.

Proof of Theorem 2.1.
In the case β = 2α, by Theorems 4.1 and 4.3, we obtain that v ε ⇀ v weakly in L 2 (Ω T ), where v is the solution of (4.18). By using (3.1) and (3.17), it follows that the convergence is, indeed, strong in L 2 (Ω T ). Moreover, by the assumptions on b 1 , it follows that 1/b ε 1 ⇀ Y dy/b 1 weakly * in L ∞ (Ω T ). Therefore, The uniqueness of the solution u of (2.22)-(2.23) follows by the uniqueness for equation (4.18), complemented with the boundary and the initial conditions (4.8) and (4.9).
The cases β = 2α are treated in the same way, of course by appealing to Theorems 4.6 and 4.3, for β > 2α and Theorems 4.10 and 4.12, for β < 2α, respectively.

Some particular cases
Here we discuss some very special cases in which the upscaled equations take specific forms.
Incidentally, this is also the case when α < 1 even if b = 0, which means that the space oscillation is greater than the non-product perturbation.

Pure Fick case
In the case b 1 = b 2 = 1 and b = 0, as above we can fix α = 1 without loss of generality, In such a case, our problem is a particular case of the one studied in [1], with the time oscillation being a power of the space oscillation.
Moreover, the homogenized equation (2.22) reduces to [1, equation (7.4)], for any choice of β. In particular, when the capacity is independent of the macrovariables, the resulting equation turns to be the pure Fick equation Y a 1 dy S a −1 2 dτ u t − div Q (B eff /a 2 ) dy dτ S a −1 2 dτ ∇u = f, (4.41) where the capacity and the diffusion matrix appear mixed in the upscaled diffusion coefficient.

Pure Fokker-Planck case
If B is the identity matrix, it follows that χ is always identically zero, so that B eff = B, and by periodicity in Ω T , (4.43) which does not depend on the non-product perturbation b. We remark that this is also valid under the milder hypothesis that B does not depend on y.
The limit equation in the pure Fokker-Planck case has been written in the from (4.43) to make it as close as possible to the starting Fokker-Planck problem. However, it is possible to formally reduce it to a standard parabolic equation with lower order terms, in which the coefficients are expressed in terms of the mean value on Q of the coefficients of the original equation, i. e., a 1 , a 2 , b 1 , and b 2 .
Finally, we remark that in the very particular case in which the coefficients a 1 , a 2 , b 1 , and b 2 , do not depend on the macroscopic variables, the equation (4.43) becomes which shows that, even in such a special case, the capacity and the Fokker coefficient are mixed in the upscaled equation.