Diffusion–reaction–dissolution–precipitation model in a heterogeneous porous medium with nonidentical diffusion coefficients: Analysis and homogenization

Abstract

We study a pore-scale model where two mobile species with different diffusion coefficients react and precipitate in the form of immobile species (crystal) on the surface of the solid parts in a porous medium. The reverse may also happen, i.e. the crystals may dissolute to give mobile species. The mathematical modeling of these processes will give rise to a coupled system of ordinary and partial differential equations. We first prove the existence of a unique nonnegative global weak solution and then upscale the model from microscale to macroscale.

Keywords

Reactive transport diffusion–reaction–dissolution–precipitation Langmuir reaction rates global existence homogenization asymptotic analysis two-scale convergence boundary unfolding

1. Introduction

Several problems in the fields of chemical engineering, material sciences, soil mechanics, reservoir flow, groundwater flow, etc. can be studied in the context of porous media flow, cf. [5,28,29,38,47,48,56,57]. Crystal dissolution and precipitation is one such flow phenomena other than diffusion, advection, and reaction that plays an important role when the minerals are present in the given porous medium. Dissolution and precipitation of minerals have been widely studied in recent years by mathematicians, engineers, hydrologists, etc., cf. [24,25,37,43–45,47,49,51,53,55,57] and references therein. A porous medium is heterogeneous in nature with porosity $θ ≪ 1$ and is composed of pore space and usually a connected solid matrix. It is important to model the transport processes in such a medium rigorously so that it depicts the real world situations as close as possible, therefore, in this context, the given porous medium may be treated as a multiscale domain, i.e. we describe the medium both at the micro scale and at the macro scale. The microscopic description of the medium models the transport processes rigorously, however, due to heterogeneity of the medium the micro scale models are not suited for conducting numerical simulations. We require the macroscopic description of the given porous medium where an averaged behavior of the domain and the transport processes are considered. This type of micro-macro scale modeling is widely considered in the literature, cf. [3,21,36,40,43] and references therein. The process to upscale from microscale to macroscale is called averaging/homogenization, cf. [1,14,21,41,42,44,56]. In our work, we shall use the periodic homogenization where we assume that the solid matrices are disconnected but distributed periodically in the given medium, see Section 2 for details. A natural porous medium has a complex geometry where the solid parts are connected and non-periodic but this periodicity assumption seems to be a good approximation to the actual domain.

The mobile species are usually present in the pore space whereas the crystals/minerals are present on the solid matrices/parts. The transportation of mobiles species is usually governed by diffusion, dispersion, or advection and is modeled by Fick’s law. The reaction amongst the mobile species is modeled by mass-action kinetics. The modeling of precipitation can be done with the help of mass action law and for the dissolution (adsorption) rate a multivalued function needs to be introduced. This idea has been explored in [25,53–55] and references therein. For equilibrium adsorption, a linear dissolution rate is proposed in [25]. The model proposed in [26,33] considered advection, diffusion, and reaction amongst mobile species where the authors have proved the existence of a global weak solution for a system of diffusion–reaction equations with identical diffusion coefficients. The existence theory in [26,33] resides on Schaefer’s fixed point Theorem and on the construction of a suitable Lyapunov functional which is used to obtain some a-priori estimates. The authors in [19] proved the global existence by considering a renormalized solution to circumvent the question of $L^{\infty}$ -boundness. The extension of these results from the case where all the species are mobile to a case where some of the species can be immobile can be found in [7,34,53,54]. The authors in [7] dealt with the adsorption reaction at the walls of the pore space and the existence of a global solution is proved. However, such a model with smooth rates is not suited for the case of precipitation-dissolution of minerals. To model the dissolution rate, we consider the multivalued discontinuous function as shown in [53,54]. For the case of one single kinetic reaction with one mobile and one immobile, the proof of the existence of a solution can be found in [24]. For a multispecies diffusion–reaction–dissolution system with identical diffusion coefficients, an existence theory is given in [31]. The homogenization of the model proposed in [31] is shown in [34].

In this paper, we consider a pore-scale model for a system of diffusion–reaction equations together with a dissolution and precipitation equation. The novelty of the current work is to consider the mobile species with nonidentical diffusion coefficients and Langmuir type reaction rate terms. In [46], the author has provided a survey on results related to the difficulties in the existence of a solution for a system of diffusion–reaction equations with different diffusion coefficients. Very recently, several results have been obtained on the existence of a global weak solution for a system of diffusion–reaction equations with certain restrictions on source terms, e.g., in [6,8,9,20]. However, for a system of diffusion–reaction–dissolution–precipitation model with nonidentical diffusion coefficients, the existence of solution still remains challenging. We will also obtain the macroscopic description of the pore scale model in this article.

We have organized the paper as follows: in Section 2, we introduce the periodic setting of the domain and the microscale model equations. Here we would like to mention that we have modeled the transport of mobile species in the pore space and dissolution and precipitation of minerals at the interfaces. In Section 3, we shall collect the mathematical tools required to analyze the model from Section 2. In Section 4, we shall prove one main Theorem of this paper, mainly about the existence and uniqueness of the solution. We employ Banach fixed point method to show the existence of a unique positive weak solution. Finally, in Section 5, we have obtained an anticipated upscaled model via asymptotic expansion and in Section 6, we derive the two-scale limit equations of the micromodel via two-scale convergence and periodic unfolding.

Fig. 1.

Mobile species in $Ω^{p}$ with crystal dissolution and precipitation on $Γ^{*}$ .

2. The model

To start with let us assume that $Ω \subset R^{n_{| n ⩾ 2}}$ be a (bounded domain) porous medium with a pore space $Ω^{p}$ and the union of solid parts $Ω^{s}$ such that $Ω : = Ω^{p} \cup Ω^{s}$ , ${\bar{Ω}}^{s} \cap Ω^{p} = ϕ$ , $Γ^{*} =$ union of boundaries of solid parts and $\partial Ω =$ outer boundary of Ω. Let $Y : = {(0, 1)}^{n} \subset R^{n}$ be a unit representative cell which is composed of a solid part $Y^{s}$ with boundary Γ and a pore part $Y^{p}$ such that $Y = Y^{s} \cup Y^{p}$ , ${\bar{Y}}^{s} \subset Y$ and ${\bar{Y}}^{s} \cap {\bar{Y}}^{p} = Γ$ . For $k \in Z^{n}$ , we define the translated sets as $Y_{k} : = Y + k$ and $Y_{k}^{λ} : = Y^{λ} + k$ for $λ \in {p, s}$ and $Γ_{k} : = Γ + k$ . Assume further that Ω is periodic (i.e. the solid parts in Ω are periodically distributed) and is covered by a finite union of the cells Y. To avoid technical difficulties, we postulate that the solid parts do not touch the boundary $\partial Ω$ , solid parts do not touch each other and solid parts do not touch the boundary of Y. Let $ε > 0$ be the scale parameter and Ω be covered by a finite union of translated versions of $ε Y_{k}$ cells such that $ε Y_{k} \subset Ω$ for $k \in Z^{n}$ , i.e. $Ω \subset ⋃_{k \in Z^{n}} ε Y_{k}$ , $Ω^{p} \subset ⋃_{k \in Z^{n}} ε Y_{k}^{p}$ , $Ω^{s} \subset ⋃_{k \in Z^{n}} ε Y_{k}^{s}$ and $Γ^{*} \subset ⋃_{k \in Z^{n}} ε Γ_{k}$ , see Fig. 1. We also define $Ω_{ε}^{p} : = ⋃_{k \in Z^{n}} {ε Y_{k}^{p} : ε Y_{k}^{p} \subset Ω}$ , $Ω_{ε}^{s} : = ⋃_{k \in Z^{n}} {ε Y_{k}^{s} : ε Y_{k}^{s} \subset Ω}$ , $Γ_{ε}^{*} : = ⋃_{k \in Z^{n}} {ε Γ_{k} : ε Γ_{k} \subset Ω}$ , $\partial Ω_{ε}^{p} : = \partial Ω \cup Γ_{ε}^{*}$ . We set $S : = [0, T)$ as the time interval for a $T > 0$ , $d y$ and $d x$ as volume elements in Y and Ω, and $d σ_{y}$ and $d σ_{x}$ as surface elements on Γ and $Γ_{ε}^{*}$ , respectively. The characteristic (indicator) function, $χ_{ε} (x) = χ (\frac{x}{ε})$ , of $Ω_{ε}^{p}$ in Ω is defined by $\begin{matrix} χ_{ε} (x) = \{\begin{matrix} 1 & if x \in Ω_{ε}^{p}, \\ 0 & if x \in Ω ∖ Ω_{ε}^{p} . \end{matrix} \end{matrix}$ Now, let two mobile species $M_{1}$ and $M_{2}$ be present in $Ω_{ε}^{p}$ whereas one immobile species (crystal) $M_{12}$ is present on $Γ_{ε}^{*}$ . The species $M_{1}$ , $M_{2}$ and $M_{12}$ are connected via following reaction: $\begin{matrix} (2.1) & M_{1} + M_{2} \leftrightarrow M_{12} on Γ_{ε}^{*} . \end{matrix}$ We make the assumptions that in the pore space no reactions amongst the mobile species are taking place and at the outer boundary no influx or outflux of $M_{1}$ and $M_{2}$ are present. However, as it can be seen in (2.1), we will impose a flux condition at the interface $Γ_{ε}^{*}$ . The relation (2.1) signifies that one molecule of each $M_{1}$ and $M_{2}$ will give one molecule of $M_{12}$ via precipitation which will be modeled by the reaction rate term which comes from Langmuir isotherm. In case of dissolution on $Γ_{ε}^{*}$ , one molecule of $M_{12}$ will dissolve to give one molecule of each $M_{1}$ and $M_{2}$ . The modeling of dissolution process is adopted from [25,53,54]. It is well-known that minerals have “constant activity”, i.e. the dissolution rate is constant if the mineral is present. If the mineral is absent, the dissolution rate can not be stronger than precipitation to maintain the non-negativity of the surface concentration, cf. [25,53,54]. This leads to a multi-valued dissolution term $R_{d} (w_{ε}) \in k_{d} ψ (w_{ε})$ , where $\begin{matrix} (2.2) & ψ (c) = \{\begin{matrix} {0} & if c < 0, \\ [0, 1] & if c = 0, \\ {1} & if c > 0 . \end{matrix} \end{matrix}$ Let the concentrations of $M_{1}$ , $M_{2}$ and $M_{12}$ be given by $u_{ε}$ , $v_{ε}$ and $w_{ε}$ , respectively. Then, the above problem (model) can be expressed as a coupled system of ODE-PDEs as: $\begin{array}{l} (2.3a) & \frac{\partial u_{ε}}{\partial t} + \nabla . (- {\bar{D}}_{1} \nabla u_{ε}) = 0 in S \times Ω_{ε}^{p}, \\ (2.3b) & - {\bar{D}}_{1} \nabla u_{ε} . \vec{n} = 0 on S \times \partial Ω, \\ (2.3c) & - {\bar{D}}_{1} \nabla u_{ε} . \vec{n} = ε \frac{\partial w_{ε}}{\partial t} on S \times Γ_{ε}^{*}, \\ (2.3d) & u_{ε} (0, x) = u_{I ε} (x) in Ω_{ε}^{p}, \\ (2.3e) & \frac{\partial v_{ε}}{\partial t} + \nabla . (- {\bar{D}}_{2} \nabla v_{ε}) = 0 in S \times Ω_{ε}^{p}, \\ (2.3f) & - {\bar{D}}_{2} \nabla v_{ε} . \vec{n} = 0 on S \times \partial Ω, \\ (2.3g) & - {\bar{D}}_{2} \nabla v_{ε} . \vec{n} = ε \frac{\partial w_{ε}}{\partial t} on S \times Γ_{ε}^{*}, \\ (2.3h) & v_{ε} (0, x) = v_{I ε} (x) in Ω_{ε}^{p}, \\ (2.3i) & \frac{\partial w_{ε}}{\partial t} = k_{d} (R (u_{ε}, v_{ε}) - z_{ε}) on S \times Γ_{ε}^{*}, \\ (2.3j) & z_{ε} \in ψ (w_{ε}) on S \times Γ_{ε}^{*}, \\ (2.3k) & w_{ε} (0, x) = w_{I ε} (x) on Γ_{ε}^{*}, \end{array}$ where $R : R^{2} \to [0, \infty)$ is defined by $\begin{matrix} R (u_{ε}, v_{ε}) = \{\begin{matrix} k \frac{k_{1} u_{ε} k_{2} v_{ε}}{{(1 + k_{1} u_{ε} + k_{2} v_{ε})}^{2}} & for u_{ε} > 0, v_{ε} > 0, \\ 0 & otherwise \end{matrix} \end{matrix}$ and $k = \frac{k_{f}}{k_{d}}$ . $k_{1}$ and $k_{2}$ are the Langmuir parameters for the mobile species $M_{1}$ and $M_{2}$ . The rate of dissolution is given by $R_{d} = k_{d} ψ (w_{ε})$ . Also, $k_{f}$ is the forward reaction rate constant for precipitation and $k_{d}$ denotes the dissolution rate constant. Let us denote the problem/model (2.3a)–(2.3k) by $(P_{ε})$ .

3. Mathematical preliminaries

Let $θ \in [0, 1]$ and $r, s \in R$ be such that $\frac{1}{r} + \frac{1}{s} = 1$ . Assume that $Ξ \in {Ω, Ω_{ε}^{p}}$ , then as usual $L^{r} (Ξ)$ , $H^{1, r} (Ξ)$ , $C^{θ} (\overline{Ξ})$ , ${(\cdot, \cdot)}_{θ, r}$ and ${[\cdot, \cdot]}_{θ}$ are the Lebesgue, Sobolev, Hölder, real- and complex-interpolation spaces respectively endowed with their standard norms. We denote $‖ f ‖_{Ξ}^{r} = \int_{Ξ} | f |^{r} d x$ and $‖ f ‖_{{(Ξ)}^{t}}^{r} = \int_{S \times Ξ} | f |^{r} d x d t$ . $C_{per}^{γ} (Y)$ denotes the set of all Y-periodic γ-times continuously differentiable functions in y for $γ \in N$ . In particular, $C_{per} (Y)$ is the space of all the Y-periodic continuous function in y. For a Banach space X, $X^{*}$ denotes its dual and the duality pairing is denoted by ${⟨ \cdot, \cdot ⟩}_{X^{*} \times X}$ . The symbols ↪, $↪ ↪$ and $\underset{↪}{d}$ denote the continuous, compact and dense embeddings, respectively. We define $L^{r} (Ξ) ↪ H^{1, s} {(Ξ)}^{*}$ as $\begin{matrix} {⟨ f, v ⟩}_{H^{1, s} {(Ξ)}^{*} \times H^{1, s} (Ξ)} = {⟨ f, v ⟩}_{L^{r} (Ξ) \times L^{s} (Ξ)} : = \int_{Ξ} f v d x for f \in L^{r} (Ξ), v \in H^{1, s} (Ξ) . \end{matrix}$ By Section 2, we note that ${lim}_{ε \to 0} ε | Γ_{ε}^{*} | = | Γ | \frac{| Ω |}{| Y |}$ . Since the surface area of $Γ_{ε}^{*}$ increases proportionally to $\frac{1}{ε}$ , i.e. $| Γ_{ε}^{*} | \to \infty$ as $ε \to 0$ , we introduce the $L^{r} (Γ_{ε}^{*}) - L^{s} (Γ_{ε}^{*})$ duality as $\begin{matrix} ⟨ ζ_{1}, ζ_{2} ⟩ : = ε \int_{Γ_{ε}^{*}} ζ_{1} ζ_{2} d σ_{x} for ζ_{1} \in L^{r} (Γ_{ε}^{*}), ζ_{2} \in L^{s} (Γ_{ε}^{*}) \end{matrix}$ and the space $L^{r} (S \times Γ_{ε}^{*})$ is furnished with the norm $\begin{matrix} ‖ ζ ‖_{{(Γ_{ε}^{*})}^{t}}^{r} : = \{\begin{matrix} ε \int_{S \times Γ_{ε}^{*}} | ζ (t, x) |^{r} d σ_{x} d t & for 1 ⩽ r < \infty, \\ {ess \sup}_{(t, x) \in S \times Γ_{ε}^{*}} | ζ (t, x) | & for r = \infty . \end{matrix} \end{matrix}$ From here and on, we take $r = 2$ . The Sobolev–Bochner spaces are given by: $U_{ε} : = {u_{ε} \in L^{2} (S; H^{1, 2} (Ω_{ε}^{p})) : \frac{\partial u_{ε}}{\partial t} \in L^{2} (S; H^{1, 2} {(Ω_{ε}^{p})}^{*})}$ , $V_{ε} : = {v_{ε} \in L^{2} (S; H^{1, 2} (Ω_{ε}^{p})) : \frac{\partial v_{ε}}{\partial t} \in L^{2} (S; H^{1, 2} {(Ω_{ε}^{p})}^{*})}$ , $W_{ε} : = {w_{ε} \in L^{2} (S; L^{2} (Γ_{ε}^{*})) : \frac{\partial w_{ε}}{\partial t} \in L^{2} (S; L^{2} (Γ_{ε}^{*}))}$ and $Z_{ε} : = {z_{ε} \in L^{\infty} (S \times Γ_{ε}^{*}) : 0 ⩽ z_{ε} ⩽ 1}$ , where $\frac{\partial}{\partial t}$ is the distributional time derivative and their respective norms can be defined as $‖ u ‖_{U_{ε}} : = ‖ u ‖_{L^{2} (S; H^{1, 2} (Ω_{ε}^{p}))} + ‖ \frac{\partial u}{\partial t} ‖_{L^{2} (S; H^{1, 2} {(Ω_{ε}^{p})}^{*})}$ and so on. The vector-valued function space $U_{ε} \times V_{ε} \times W_{ε} \times Z_{ε}$ is the solution space. A quadruple $(u_{ε}, v_{ε}, w_{ε}, z_{ε}) \in U_{ε} \times V_{ε} \times W_{ε} \times Z_{ε}$ is said to be a weak solution of (2.3a)–(2.3k) if $(u_{ε} (0), v_{ε} (0), w_{ε} (0)) = (u_{I ε}, v_{I ε}, w_{I ε}) \in L^{2} (Ω_{ε}^{p}) \times L^{2} (Ω_{ε}^{p}) \times L^{\infty} (Γ_{ε}^{*})$ and $\begin{array}{l} (3.1a) & {⟨ \frac{\partial u_{ε}}{\partial t}, ϕ ⟩}_{{(Ω_{ε}^{p})}^{t}} + {⟨ {\bar{D}}_{1} \nabla u_{ε}, \nabla ϕ ⟩}_{{(Ω_{ε}^{p})}^{t}} = - {⟨ \frac{\partial w_{ε}}{\partial t}, ϕ ⟩}_{{(Γ_{ε}^{*})}^{t}}, \forall ϕ \in L^{2} (S; H^{1, 2} (Ω_{ε}^{p})), \\ (3.1b) & {⟨ \frac{\partial v_{ε}}{\partial t}, θ ⟩}_{{(Ω_{ε}^{p})}^{t}} + {⟨ {\bar{D}}_{2} \nabla v_{ε}, \nabla θ ⟩}_{{(Ω_{ε}^{p})}^{t}} = - {⟨ \frac{\partial w_{ε}}{\partial t}, θ ⟩}_{{(Γ_{ε}^{*})}^{t}}, \forall θ \in L^{2} (S; H^{1, 2} (Ω_{ε}^{p})), \\ {⟨ \frac{\partial w_{ε}}{\partial t}, η ⟩}_{{(Γ_{ε}^{*})}^{t}} = k_{d} {⟨ R (u_{ε}, v_{ε}) - z_{ε}, η ⟩}_{{(Γ_{ε}^{*})}^{t}}, \\ (3.1c) & z_{ε} \in ψ (w_{ε}) a.e. on {(Γ_{ε}^{*})}^{t}, \forall η \in L^{2} (S; L^{2} (Γ_{ε}^{*})) . \end{array}$ For a $β \in R$ , let us define $β_{+} : = max {β, 0} ⩾ 0 and β_{-} : = max {- β, 0} ⩾ 0$ such that $β = β_{+} - β_{-}$ . We will make following assumptions for the sake of analysis:

$u_{I ε} (x), v_{I ε} (x), w_{I ε} (x) ⩾ 0$ .

$R (u_{ε}, v_{ε}) = 0$ for all $u_{ε} ⩽ 0$ , $v_{ε} ⩽ 0$ .

$R : R^{2} \to [0, \infty)$ is Locally Lipschitz in $R^{2}$ , as $\begin{matrix} | R (u_{ε}^{(1)}, v_{ε}) - R (u_{ε}^{(2)}, v_{ε}) | ⩽ L_{R} | u_{ε}^{(1)} - u_{ε}^{(2)} |, \end{matrix}$ where $L_{R} > 0$ is a constant. $L_{R} = sup L_{R} (u_{ε}, v_{ε})$ where $L_{R} (u_{ε}, v_{ε}) = k k_{1} k_{2} | v_{ε} | {(1 + k_{2} v_{ε})}^{2} + k_{1}^{2} u_{ε}^{(1)} u_{ε}^{(2)} |$ . Here we use the Langmuir isotherm to model the forward reaction phenomenon.

$u_{I ε}, v_{I ε} \in L^{2} (Ω_{ε}^{p})$ and $w_{I ε} \in L^{\infty} (Γ_{ε}^{*})$ .

${\bar{D}}_{1} = diag (D_{1}, D_{1}, \dots, D_{1})$ and ${\bar{D}}_{2} = diag (D_{2}, D_{2}, \dots, D_{2})$ , where $D_{1}$ , $D_{2}$ are positive constants.

We will now state the two main Theorems of this paper below: Theorem 3.1 for existence of solution of the micro problem and Theorem 3.2 for the homogenization of the micro problem $(P_{ε})$ .

Theorem 3.1.
Suppose the assumptions (A1)–(A5) hold true, then there exists a unique positive weak solution $(u_{ε}, v_{ε}, w_{ε}, z_{ε})$ of $(P_{ε})$ which satisfies $\begin{array}{l} (3.2a) & 0 ⩽ {‖ u_{ε} (t) ‖}_{Ω_{ε}^{p}} ⩽ M_{u}, 0 ⩽ {‖ v_{ε} (t) ‖}_{Ω_{ε}^{p}} ⩽ M_{v} a.e. in S \times Ω_{ε}^{p}, \\ (3.2b) & 0 ⩽ {‖ w_{ε} (t) ‖}_{Γ_{ε}^{}} ⩽ M_{w}, 0 ⩽ z_{ε} ⩽ 1 a.e. on S \times Γ_{ε}^{} and \\ {‖ u_{ε} (t) ‖}_{Ω_{ε}^{p}} + D_{1} {‖ \nabla u_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}} + {‖ \partial_{t} u_{ε} ‖}_{L^{2} (S; H^{1, 2} {(Ω_{ε}^{p})}^{})} + {‖ v_{ε} (t) ‖}_{Ω_{ε}^{p}} + D_{2} {‖ \nabla v_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}} \\ (3.2c) & + {‖ \partial_{t} v_{ε} ‖}_{L^{2} (S; H^{1, 2} {(Ω_{ε}^{p})}^{})} + {‖ w_{ε} (t) ‖}_{Γ_{ε}^{}} + {‖ \partial_{t} w_{ε} ‖}_{{(Γ_{ε}^{})}^{t}} ⩽ C, \end{array}$ for a.e. $t \in S$ , where C is a generic constant independent of ε.
Theorem 3.2.
Under the assumptions (A1)–(A5), there exist $(u_{0}, v_{0}, w_{0}, z_{0}) \in L^{2} (S; H^{1, 2} (Ω)) \times L^{2} (S; H^{1, 2} (Ω)) \times L^{2} (S; L^{2} (Ω \times Γ)) \times L^{\infty} (S \times Ω \times Γ)$ , as obtained in Lemma 6.2 , such that $(u_{0}, v_{0}, w_{0}, z_{0})$ is the unique solution of the problem $\begin{array}{l} (3.3a) & \frac{\partial u_{0}}{\partial t} - \nabla . (A \nabla u_{0}) + P (t, x) = 0 in S \times Ω, \\ (3.3b) & - A \nabla u_{0} . \vec{n} = 0 on S \times \partial Ω, \\ (3.3c) & u_{0} (0, x) = u_{I 0} (x) in Ω, \end{array}$ $\begin{array}{l} (3.4a) & \frac{\partial v_{0}}{\partial t} - \nabla . (B \nabla v_{0}) + P (t, x) = 0 in S \times Ω, \\ (3.4b) & - B \nabla v_{0} . \vec{n} = 0 on S \times \partial Ω, \\ (3.4c) & v_{0} (0, x) = v_{I 0} (x) in Ω, \end{array}$ $\begin{array}{l} (3.5a) & \frac{\partial w_{0}}{\partial t} = k_{d} (R (u_{0}, v_{0}) - z_{0}) in S \times Ω \times Γ, \\ (3.5b) & where z_{0} \in ψ (w_{0}) in S \times Ω \times Γ, \\ (3.5c) & w_{0} (0, x, y) = w_{I 0} (x, y) on Ω \times Γ, \end{array}$ satisfying the a-priori bound $\begin{array}{l} ‖ u_{0} ‖_{{(Ω)}^{t}} + ‖ \nabla u_{0} ‖_{{(Ω)}^{t}} + ‖ \partial_{t} u_{0} ‖_{L^{2} (S; H^{1, 2} {(Ω)}^{})} + ‖ v_{0} ‖_{{(Ω)}^{t}} + ‖ \nabla v_{0} ‖_{{(Ω)}^{t}} + ‖ \partial_{t} v_{0} ‖_{L^{2} (S; H^{1, 2} {(Ω)}^{})} \\ (3.6) & + ‖ w_{0} ‖_{{(Ω \times Γ)}^{t}} + ‖ \partial_{t} w_{0} ‖_{{(Ω \times Γ)}^{t}} + ‖ P ‖_{{(Ω)}^{t}} ⩽ C < \infty, \end{array}$ where $\begin{matrix} P (t, x) = \int_{Γ} \frac{1}{| Y^{p} |} \frac{\partial w_{0}}{\partial t} d σ_{y} \end{matrix}$ and the elliptic homogenized matrix $A = {(a_{i j})}_{1 ⩽ i, j ⩽ n}$ and $B = {(b_{i j})}_{1 ⩽ i, j ⩽ n}$ are defined by $\begin{matrix} a_{i j} = \int_{Y^{p}} \frac{D_{1}}{| Y^{p} |} (δ_{i j} + \sum_{i, j = 1}^{n} \frac{\partial l_{j}}{\partial y_{i}}) d y, b_{i j} = \int_{Y^{p}} \frac{D_{2}}{| Y^{p} |} (δ_{i j} + \sum_{i, j = 1}^{n} \frac{\partial l_{j}}{\partial y_{i}}) d y . \end{matrix}$ Moreover, $l_{j} \in H_{per}^{1, 2} (Y^{p})$ are the solutions of the cell problems $\begin{matrix} \{\begin{matrix} \nabla_{y} . (\nabla_{y} l_{j} + e_{j}) = 0 for all y \in Y^{p}, \\ (\nabla_{y} l_{j} + e_{j}) . \vec{n} = 0 on Γ, \\ y \mapsto l_{j} (y) is Y -periodic, \end{matrix} \end{matrix}$ for $j = 1, 2, \dots, n$ and for almost every $x \in Ω$ .

4. Proof of Theorem 3.1

This Section is devoted to establish the existence of a unique positive global in time weak solution of ( $P_{ε}$ ). We will employ some regularization technique and Banach’s fixed point argument for the existence of solution. Finally, we will conclude this Section by showing that the solution is unique. First, we start with the non-negativity of the concentrations (weak solution of ( $P_{ε}$ )) which is obtained in the next Lemma.

Lemma 4.1.
The weak solution of $(P_{ε})$ is nonnegative, i.e. $u_{ε}, v_{ε} ⩾ 0$ a.e. in $S \times Ω_{ε}^{p}$ and $w_{ε} ⩾ 0$ a.e. on $S \times Γ_{ε}^{}$ , respectively.*
Proof.
We choose test function $ϕ = -$ negative part of $u_{ε} = - {[u_{ε}]}_{-} \in L^{2} (S; H^{1, 2} (Ω_{ε}^{p}))$ . Then by (3.1a) $\begin{matrix} (4.1) & \frac{1}{2} {‖ {[u_{ε} (t)]}_{-} ‖}_{Ω_{ε}^{p}}^{2} + D_{1} {‖ \nabla {[u_{ε}]}_{-} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2} = \frac{1}{2} {‖ {[u_{I ε}]}_{-} ‖}_{Ω_{ε}^{p}}^{2} - {⟨ \frac{\partial w_{ε}}{\partial t}, - {[u_{ε}]}_{-} ⟩}_{{(Γ_{ε}^{})}^{t}} . \end{matrix}$ Since $u_{I ε} (x) ⩾ 0$ , ${[u_{I ε}]}_{-} = 0$ and relying on (A2) we get the estimate $\begin{array}{l} {⟨ \frac{\partial w_{ε}}{\partial t}, - {[u_{ε}]}_{-} ⟩}_{{(Γ_{ε}^{})}^{t}} & = k_{d} {⟨ R (u_{ε}, v_{ε}) - z_{ε}, - {[u_{ε}]}_{-} ⟩}_{{(Γ_{ε}^{})}^{t}} \\ (4.2) & = - k_{d} {⟨ z_{ε}, - {[u_{ε}]}_{-} ⟩}_{{(Γ_{ε}^{})}^{t}} ⩾ 0 . \end{array}$ (4.1) in combination with (4.2) gives that $\begin{matrix} \frac{1}{2} {‖ {[u_{ε} (t)]}_{-} ‖}_{Ω_{ε}^{p}}^{2} + D_{1} {‖ \nabla {[u_{ε}]}_{-} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2} ⩽ 0 ⟹ {[u_{ε} (t)]}_{-} = 0 . \end{matrix}$ Therefore, $u_{ε} (t) ⩾ 0$ for a.e. $t \in S$ . Similarly, we can show $v_{ε} (t) ⩾ 0$ for a.e. $t \in S$ . Next, we put $η = - X_{(0, t)} {[w_{ε}]}_{-} \in L^{2} (S; L^{2} (Γ_{ε}^{}))$ in (3.1c) which gives $\begin{array}{l} \frac{1}{2} {‖ {[w_{ε} (t)]}_{-} ‖}_{Γ_{ε}^{}}^{2} - \frac{1}{2} {‖ {[w_{I ε}]}_{-} ‖}_{Γ_{ε}^{}}^{2} & = k_{d} {⟨ R (u_{ε}, v_{ε}) - z_{ε}, - {[w_{ε}]}_{-} ⟩}_{{(Γ_{ε}^{})}^{t}} \\ = k_{d} {⟨ R (u_{ε}, v_{ε}), - {[w_{ε}]}_{-} ⟩}_{{(Γ_{ε}^{})}^{t}}, by the definition of z_{ε} . \end{array}$ Since $u_{ε}, v_{ε} ⩾ 0$ , $R (u_{ε}, v_{ε}) = k \frac{k_{1} k_{2} u_{ε} v_{ε}}{{(1 + k_{1} u_{ε} + k_{2} v_{ε})}^{2}} ⩾ 0$ which implies $R (u_{ε}, v_{ε}) (- {[w_{ε}]}_{-}) ⩽ 0$ . Hence, ${[w_{ε} (t)]}_{-} = 0$ , i.e. $w_{ε} (t) ⩾ 0$ for a.e. $t \in S$ . □
Lemma 4.2.
The concentrations* $u_{ε}$ and $v_{ε}$ satisfy $‖ u_{ε} (t) ‖_{Ω_{ε}^{p}} ⩽ M_{u}$ and $‖ v_{ε} (t) ‖_{Ω_{ε}^{p}} ⩽ M_{v}$ a.e. in $S \times Ω_{ε}^{p}$ , where the constants $M_{u}$ and $M_{v}$ are independent of δ, ε.
Proof.
We test (3.1a) with $ϕ = X_{(0, t)} u_{ε} \in L^{2} (S; H^{1, 2} (Ω_{ε}^{p}))$ and get $\begin{matrix} (4.3) & \frac{1}{2} {‖ u_{ε} (t) ‖}_{Ω_{ε}^{p}}^{2} - \frac{1}{2} {‖ u_{I ε} ‖}_{Ω_{ε}^{p}}^{2} + D_{1} ‖ \nabla u_{ε} ‖_{{(Ω_{ε}^{p})}^{t}}^{2} = - k_{d} {⟨ R (u_{ε}, v_{ε}) - z_{ε}, u_{ε} ⟩}_{{(Γ_{ε}^{})}^{t}} . \end{matrix}$ Since ${(\frac{k_{1} u_{ε} + k_{2} v_{ε}}{2})}^{2} ⩾ k_{1} u_{ε} k_{2} v_{ε}$ and ${(\frac{k_{1} u_{ε} + k_{2} v_{ε}}{1 + k_{1} u_{ε} + k_{2} v_{ε}})}^{2} ⩽ 1$ therefore $\begin{matrix} (4.4) & R (u_{ε}, v_{ε}) = k \frac{k_{1} u_{ε} k_{2} v_{ε}}{{(1 + k_{1} u_{ε} + k_{2} v_{ε})}^{2}} ⩽ k \frac{{(k_{1} u_{ε} + k_{2} v_{ε})}^{2}}{4 {(1 + k_{1} u_{ε} + k_{2} v_{ε})}^{2}} ⩽ \frac{k}{4} . \end{matrix}$ This gives $\begin{matrix} (4.5) & ε k_{d} \int_{Γ_{ε}^{}} | R (u_{ε}, v_{ε}) - z_{ε} | | u_{ε} | d σ_{x} ⩽ ε k_{d} \int_{Γ_{ε}^{}} (\frac{k}{4} + 1) | u_{ε} | d σ_{x} ⩽ \frac{ε k_{d}^{2}}{4 γ} {(1 + \frac{k}{4})}^{2} | Γ_{ε}^{} | + γ {‖ u_{ε} ‖}_{Γ_{ε}^{}}^{2}, \end{matrix}$ where γ is the Young’s inequality constant. Now to estimate the boundary term we use the trace inequality (A.3) and integrate both sides of (4.5) w.r.t. t to obtain $\begin{array}{l} | k_{d} {⟨ R (u_{ε}, v_{ε}) - z_{ε}, u_{ε} ⟩}_{{(Γ_{ε}^{})}^{t}} | & ⩽ \frac{k_{d}^{2} T}{4 γ} {(1 + \frac{k}{4})}^{2} sup_{ε > 0} ε | Γ_{ε}^{} | + C γ {{‖ u_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2} + ε^{2} {‖ \nabla u_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2}} \\ ⩽ C_{3} + C γ {‖ u_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2} + C γ ε^{2} {‖ \nabla u_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2}, \end{array}$ where $C_{3} (: = \frac{k_{d}^{2} T}{4 γ} {(1 + \frac{k}{4})}^{2} \frac{| Γ | | Ω |}{| Y |})$ and C are independent of ε and δ. As $0 < ε ≪ 1$ so (4.3) leads to $\begin{array}{l} {‖ u_{ε} (t) ‖}_{Ω_{ε}^{p}}^{2} + 2 D_{1} ‖ \nabla u_{ε} ‖_{{(Ω_{ε}^{p})}^{t}}^{2} ⩽ {‖ u_{I ε} ‖}_{Ω_{ε}^{p}}^{2} + 2 C_{3} + 2 C γ {‖ u_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2} + 2 C γ {‖ \nabla u_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2} \\ (4.6) & ⟹ {‖ u_{ε} (t) ‖}_{Ω_{ε}^{p}}^{2} + (2 D_{1} - 2 C γ) ‖ \nabla u_{ε} ‖_{{(Ω_{ε}^{p})}^{t}}^{2} ⩽ C_{4} + 2 C γ {‖ u_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2}, \end{array}$ where $C_{4} = 2 C_{3} + {‖ u_{I ε} ‖}_{Ω_{ε}^{p}}^{2}$ . Now for $γ \in (0, \frac{D_{1}}{C})$ we have $\begin{matrix} {‖ u_{ε} (t) ‖}_{Ω_{ε}^{p}}^{2} ⩽ C_{4} + 2 D_{1} {‖ u_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2} . \end{matrix}$ Then, Gronwall’s inequality yields $\begin{matrix} {‖ u_{ε} (t) ‖}_{Ω_{ε}^{p}} ⩽ M_{u} for a.e. t \in S . \end{matrix}$ Similarly, for $v_{ε}$ we obtain $\begin{matrix} {‖ v_{ε} (t) ‖}_{Ω_{ε}^{p}} ⩽ M_{v} for a.e. t \in S . \end{matrix}$ □
Lemma 4.3.
The concentrations* $u_{ε}$ and $v_{ε}$ satisfy $‖ \nabla u_{ε} ‖_{{(Ω_{ε}^{p})}^{t}} ⩽ {\bar{M}}_{u}$ and $‖ \nabla v_{ε} ‖_{{(Ω_{ε}^{p})}^{t}} ⩽ {\bar{M}}_{v}$ a.e. in $S \times Ω_{ε}^{p}$ , where the constants ${\bar{M}}_{u}$ and ${\bar{M}}_{v}$ are independent of δ, ε.
Proof.
Choosing $γ \in (0, \frac{D_{1}}{C})$ in (4.6), we are led to $\begin{array}{l} 2 (D_{1} - C γ) ‖ \nabla u_{ε} ‖_{{(Ω_{ε}^{p})}^{t}}^{2} ⩽ C_{4} + 2 D_{1} {‖ u_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2} \\ ⟹ 2 (D_{1} - C γ) ‖ \nabla u_{ε} ‖_{{(Ω_{ε}^{p})}^{t}}^{2} ⩽ C_{4} + C_{4} (e^{2 D_{1} t} - 1) \\ ⟹ ‖ \nabla u_{ε} ‖_{{(Ω_{ε}^{p})}^{t}} ⩽ {\bar{M}}_{u} . \end{array}$ Similarly, for $v_{ε}$ we get $\begin{matrix} ‖ \nabla v_{ε} ‖_{{(Ω_{ε}^{p})}^{t}} ⩽ {\bar{M}}_{v} . \end{matrix}$ □
Lemma 4.4.
The concentration $w_{ε}$ satisfies $‖ w_{ε} (t) ‖_{Γ_{ε}^{}} ⩽ M_{w}$ a.e. on* $S \times Γ_{ε}^{}$ , where the constant* $M_{w}$ is independent of δ, ε.
Proof.
We use $η = X_{(0, t)} w_{ε} \in L^{2} (S; L^{2} (Γ_{ε}^{}))$ in (3.1c) and integrate w.r.t. t to get $\begin{matrix} (4.7) & \frac{1}{2} {‖ w_{ε} (t) ‖}_{Γ_{ε}^{}}^{2} - \frac{1}{2} {‖ w_{I ε} ‖}_{Γ_{ε}^{}}^{2} ⩽ \frac{k_{d}^{2} T}{2} {(1 + \frac{k}{4})}^{2} sup_{ε > 0} ε | Γ_{ε}^{} | + \frac{1}{2} {‖ w_{ε} ‖}_{{(Γ_{ε}^{})}^{t}}^{2} . \end{matrix}$ Thus, (4.7) leads to $\begin{array}{l} {‖ w_{ε} (t) ‖}_{Γ_{ε}^{}}^{2} ⩽ {‖ w_{I ε} ‖}_{Γ_{ε}^{}}^{2} + k_{d}^{2} T {(1 + \frac{k}{4})}^{2} \frac{| Γ | | Ω |}{| Y |} + {‖ w_{ε} ‖}_{{(Γ_{ε}^{})}^{t}}^{2} \\ ⟹ {‖ w_{ε} (t) ‖}_{Γ_{ε}^{}}^{2} ⩽ C_{5} + {‖ w_{ε} ‖}_{{(Γ_{ε}^{})}^{t}}^{2}, where C_{5} = {‖ w_{I ε} ‖}_{Γ_{ε}^{}}^{2} + k_{d}^{2} T {(1 + \frac{k}{4})}^{2} \frac{| Γ | | Ω |}{| Y |} . \end{array}$ Then, Gronwall’s inequality implies $\begin{matrix} {‖ w_{ε} (t) ‖}_{Γ_{ε}^{}} ⩽ M_{w} for a.e. t \in S . \end{matrix}$ □
Lemma 4.5.
The concentration $w_{ε}$ satisfies $w_{ε} (t, x) ⩽ {\bar{M}}_{w}$ a.e. on $S \times Γ_{ε}^{}$ , where the constant* ${\bar{M}}_{w}$ is independent of δ, ε.
Proof.
By partial integration of (2.3i) and using (4.4) we obtain $\begin{array}{l} | w_{ε} (t, x) | ⩽ w_{I ε} (x) + k_{d} (1 + \frac{k}{4}) T ⩽ ‖ w_{I ε} ‖_{L^{\infty} (Γ_{ε}^{*})} + k_{d} (1 + \frac{k}{4}) T = {\bar{M}}_{w} for a.e. t \in S . \end{array}$ □

4.1. Existence of solution

As we can see in the system of equations (2.3a)–(2.3k), the multivaluedness of the dissolution rate term in (2.3i) creates the main difficulty in analysis. We tackle the multivaluedness by introducing a regularization parameter $δ > 0$ and in order to prove the existence of a weak solution we replace $ψ (w_{ε})$ by $ψ_{δ} (w_{ε})$ , where $\begin{matrix} (4.8) & ψ_{δ} (w_{ε}) = \{\begin{matrix} 0 & if w_{ε} < 0, \\ \frac{w_{ε}}{δ} & if w_{ε} \in (0, δ), \\ 1 & if w_{ε} > δ . \end{matrix} \end{matrix}$ For brevity of notation, we denote $u_{ε δ} = u_{ε}$ , $v_{ε δ} = v_{ε}$ and $w_{ε δ} = w_{ε}$ . First, we will establish the existence of solution of the regularized problem and then we perform $δ \to 0$ to obtain the existence of solution of (2.3a)–(2.3k). The regularized problem is given by Find $(u_{ε}, v_{ε}, w_{ε}) \in U_{ε} \times V_{ε} \times W_{ε}$ such that $(u_{ε} (0), v_{ε} (0), w_{ε} (0)) = (u_{I ε}, v_{I ε}, w_{I ε})$ and $\begin{array}{l} (4.9a) & {⟨ \frac{\partial u_{ε}}{\partial t}, ϕ ⟩}_{{(Ω_{ε}^{p})}^{T}} + {⟨ {\bar{D}}_{1}^{ε} \nabla u_{ε}, \nabla ϕ ⟩}_{{(Ω_{ε}^{p})}^{T}} = - {⟨ \frac{\partial w_{ε}}{\partial t}, ϕ ⟩}_{{(Γ_{ε}^{*})}^{T}}, \forall ϕ \in L^{2} (S; H^{1, 2} (Ω_{ε}^{p})), \\ (4.9b) & {⟨ \frac{\partial v_{ε}}{\partial t}, θ ⟩}_{{(Ω_{ε}^{p})}^{T}} + {⟨ {\bar{D}}_{2} \nabla v_{ε}, \nabla θ ⟩}_{{(Ω_{ε}^{p})}^{T}} = - {⟨ \frac{\partial w_{ε}}{\partial t}, θ ⟩}_{{(Γ_{ε}^{*})}^{T}}, \forall θ \in L^{2} (S; H^{1, 2} (Ω_{ε}^{p})), \\ (4.9c) & {⟨ \frac{\partial w_{ε}}{\partial t}, η ⟩}_{{(Γ_{ε}^{*})}^{T}} = k_{d} {⟨ R (u_{ε}, v_{ε}) - ψ_{δ} (w_{ε}), η ⟩}_{{(Γ_{ε}^{*})}^{T}}, \forall η \in L^{2} (S; L^{2} (Γ_{ε}^{*})) . \end{array}$ We consider the closed and convex sets $\begin{array}{l} F_{U_{ε}} = {u_{ε} \in U_{ε} : 0 ⩽ {‖ u_{ε} (t) ‖}_{Ω_{ε}^{p}} ⩽ M_{u} a.e. in {(Ω_{ε}^{p})}^{T}}, \\ F_{V_{ε}} = {v_{ε} \in V_{ε} : 0 ⩽ {‖ v_{ε} (t) ‖}_{Ω_{ε}^{p}} ⩽ M_{v} a.e. in {(Ω_{ε}^{p})}^{T}}, \\ F_{W_{ε}} = {w_{ε} \in W_{ε} : 0 ⩽ {‖ w_{ε} (t) ‖}_{Γ_{ε}^{*}} ⩽ M_{w} a.e. on {(Γ_{ε}^{*})}^{T}} . \end{array}$ Now, we show the existence of solution by using a nested (Banach) fixed point Theorem, cf. Theorem 3.1 in [10] or, Lemma 3.80 in [23]. The argument goes like this: for arbitrary $(u_{ε}, v_{ε}) \in F_{U_{ε}} \times F_{V_{ε}}$ , we consider (4.9c) with $w_{ε} (0, x) = w_{I ε} (x)$ , then by Picard–Lindelöf Theorem there exists a unique local solution $w_{ε} \in F_{W_{ε}}$ since the r.h.s. of (4.9c) is Lipschitz. Again, for $u_{ε} \in F_{U_{ε}}$ , (4.9b) with $v_{ε} (0, x) = v_{I ε} (x)$ has a unique solution $Z_{1} v_{ε} \in F_{V_{ε}}$ , where $Z_{1} : F_{V_{ε}} \to F_{V_{ε}}$ is a contraction map and has a fixed point in $F_{V_{ε}}$ . This will imply further the existence of solution $v_{ε}$ of (4.9b). Furthermore, for $u_{ε} \in F_{U_{ε}}$ , the equation (4.9a) has a fixed point $Z_{2} u_{ε} \in F_{U_{ε}}$ , where $Z_{2} : F_{U_{ε}} \to F_{U_{ε}}$ is a contraction map and has a fixed point in $F_{U_{ε}}$ . In other words, the fixed points of $Z_{1}$ and $Z_{2}$ are the solutions of the problem (4.9a)–(4.9c).

Lemma 4.6.
There exists a constant $C > 0$ , independent of δ and ε, such that $\begin{matrix} {‖ Z_{1} v_{ε} (t) ‖}_{Ω_{ε}^{p}} + D_{2} {‖ \nabla Z_{1} v_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}} + {‖ \partial_{t} Z_{1} v_{ε} ‖}_{L^{2} (S; H^{1, 2} {(Ω_{ε}^{p})}^{})} ⩽ C for a.e. t \in S . \end{matrix}$
Proof.
We fix an arbitrary $t \in S$ and test (4.9b) with $θ = X_{(0, t)} [Z_{1} v_{ε} - v_{I ε}]$ and the rest follows in the similar way as shown in [54], i.e. $\begin{matrix} (4.10) & {‖ Z_{1} v_{ε} (t) ‖}_{Ω_{ε}^{p}}^{2} + (4 D_{2} - 4 α_{1} D_{2}^{2}) {‖ \nabla Z_{1} v_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2} ⩽ C, \end{matrix}$ where $α_{1}$ is the Young’s inequality constant. Again, $\begin{matrix} | {⟨ \frac{\partial Z_{1} v_{ε}}{\partial t}, θ ⟩}_{Ω_{ε}^{p}} | ⩽ D_{2} {‖ \nabla Z_{1} v_{ε} (t) ‖}_{Ω_{ε}^{p}} ‖ \nabla θ ‖_{Ω_{ε}^{p}} + k_{d} \sqrt{\frac{| Γ | | Ω |}{| Y |}} ‖ θ ‖_{Γ_{ε}^{}} . \end{matrix}$ Now, applying trace inequality (A.3) and the inequality (4.10), the estimate for $\partial_{t} Z_{1} v_{ε}$ gives, $\begin{matrix} ⟹ {‖ Z_{1} v_{ε} (t) ‖}_{Ω_{ε}^{p}} + D_{2} {‖ \nabla Z_{1} v_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}} + {‖ \partial_{t} Z_{1} v_{ε} ‖}_{L^{2} (S; H^{1, 2} {(Ω_{ε}^{p})}^{})} ⩽ C . \end{matrix}$ □
Lemma 4.7.
Given* $(u_{ε}, v_{ε}) \in F_{U_{ε}} \times F_{V_{ε}}$ fixed, let $w_{ε}$ denote the solution of ( 4.9c ) then there exists a constant C, independent of δ and ε, such that $\begin{matrix} {‖ w_{ε} (t) ‖}_{Γ_{ε}^{}} + {‖ \frac{\partial w_{ε}}{\partial t} ‖}_{{(Γ_{ε}^{})}^{t}} ⩽ C for a.e. t \in S . \end{matrix}$
Proof.
We know by Lemma 4.4 $\begin{matrix} {‖ w_{ε} (t) ‖}_{Γ_{ε}^{}} ⩽ M_{w} . \end{matrix}$ We put $η = X_{(0, t)} \frac{\partial w_{ε}}{\partial t}$ in (4.9c), then $\begin{matrix} {‖ \frac{\partial w_{ε}}{\partial t} ‖}_{{(Γ_{ε}^{})}^{t}}^{2} ⩽ k_{d}^{2} {(1 + \frac{k}{4})}^{2} T sup_{ε > 0} ε | Γ_{ε}^{} | ⩽ k_{d}^{2} {(1 + \frac{k}{4})}^{2} T \frac{| Γ | | Ω |}{| Y |} . \end{matrix}$ The addition of the above two inequalities gives the desired result. □

Now, let ${v_{ε}}^{(1)}, {v_{ε}}^{(2)} \in F_{V_{ε}}$ with ${w_{ε}}^{(1)}, {w_{ε}}^{(2)} \in F_{W_{ε}}$ be the corresponding solution of (4.9c). In the next Lemma, we denote $\begin{matrix} N (v_{ε}, t) = {‖ v_{ε} ‖}_{L^{2} (S; H^{1, 2} (Ω_{ε}^{p}))}^{2} = {‖ v_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2} + {‖ \nabla v_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2} \end{matrix}$ and then by trace inequality (A.3), $\begin{matrix} (4.11) & {‖ v_{ε} ‖}_{{(Γ_{ε}^{})}^{t}}^{2} ⩽ C N (v_{ε}, t), for all v_{ε} \in L^{2} (S; H^{1, 2} (Ω_{ε}^{p})), \end{matrix}$ where C is independent of ε.
Lemma 4.8.
If ${w_{ε}}^{(1)}$ and ${w_{ε}}^{(2)}$ are two solutions of ( 4.9c ) and $w_{ε} = {w_{ε}}^{(1)} - {w_{ε}}^{(2)}$ then for a.e. $t \in S$ we have, $\begin{array}{l} {‖ w_{ε} ‖}_{{(Γ_{ε}^{})}^{t}}^{2} ⩽ C_{6} N (v_{ε}, t) (e^{t} - 1) and {‖ \frac{\partial w_{ε}}{\partial t} ‖}_{{(Γ_{ε}^{})}^{t}}^{2} ⩽ C_{7} (t) N (v_{ε}, t), \end{array}$ where $C_{6} = C k_{d}^{2} L_{R}^{2}$ , $C_{7} (t) = 2 C_{6} (1 + {(\frac{k_{d}}{δ})}^{2} (e^{t} - 1))$ , $C_{7} = {sup}_{t \in S} C_{7} (t)$ and $L_{R}$ is the Lipschitz constant.
Proof.
We use $η = X_{(0, t)} w_{ε}$ as test function in the variational formulation (4.9c) and simplify to get the inequality, $\begin{matrix} \frac{1}{2} \int_{0}^{t} \frac{\partial}{\partial s} {‖ w_{ε} (s) ‖}_{Γ_{ε}^{}}^{2} d s ⩽ k_{d} {⟨ {R (u_{ε}, v_{ε}^{(1)}) - R (u_{ε}, v_{ε}^{(2)})}, w_{ε} ⟩}_{{(Γ_{ε}^{})}^{t}} ⩽ k_{d} {⟨ L_{R} | v_{ε} |, w_{ε} ⟩}_{{(Γ_{ε}^{})}^{t}}, \end{matrix}$ as $ψ_{δ}$ is monotonically increasing and R is locally Lipschitz. Hence, $\begin{matrix} {‖ w_{ε} (t) ‖}_{Γ_{ε}^{}}^{2} ⩽ 2 k_{d} {⟨ L_{R} | v_{ε} |, w_{ε} ⟩}_{{(Γ_{ε}^{})}^{t}} . \end{matrix}$ By Young’s inequality and trace inequality (4.11), we have $\begin{matrix} 2 k_{d} {⟨ L_{R} | v_{ε} |, w_{ε} ⟩}_{{(Γ_{ε}^{})}^{t}} ⩽ k_{d}^{2} {L_{R}}^{2} C N (v_{ε}, t) + {‖ w_{ε} ‖}_{{(Γ_{ε}^{})}^{t}}^{2} . \end{matrix}$ Therefore, $\begin{matrix} {‖ w_{ε} (t) ‖}_{Γ_{ε}^{}}^{2} ⩽ C_{6} N (v_{ε}, t) + {‖ w_{ε} ‖}_{{(Γ_{ε}^{})}^{t}}^{2}, where C_{6} = k_{d}^{2} {L_{R}}^{2} C . \end{matrix}$ By Gronwall’s inequality, $\begin{matrix} {‖ w_{ε} (t) ‖}_{Γ_{ε}^{}}^{2} ⩽ C_{6} N (v_{ε}, t) e^{t} ⟹ {‖ w_{ε} ‖}_{{(Γ_{ε}^{})}^{t}}^{2} ⩽ C_{6} N (v_{ε}, t) (e^{t} - 1) . \end{matrix}$ Again, we test (4.9c) with $η = X_{(0, t)} \frac{\partial w_{ε}}{\partial t}$ and proceed in the similar way as above to obtain the estimate $\begin{matrix} {‖ \frac{\partial w_{ε}}{\partial t} ‖}_{{(Γ_{ε}^{})}^{t}}^{2} ⩽ C_{7} (t) N (v_{ε}, t), where C_{7} (t) = 2 C_{6} (1 + {(\frac{k_{d}}{δ})}^{2} (e^{t} - 1)) . \end{matrix}$ □
Lemma 4.9.
For a.e. $t \in S$ and a $μ > 0$ $\begin{array}{l} {‖ Z_{1} v_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2} ⩽ \frac{μ^{3} A (t)}{μ C_{8} + C_{9}} (e^{t (\frac{μ C_{8} + C_{9}}{μ^{2}})} - 1) N (v_{ε}, t) and \\ {‖ \nabla Z_{1} v_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2} ⩽ \frac{μ A (t)}{D_{2}} e^{t (\frac{μ C_{8} + C_{9}}{μ^{2}})} N (v_{ε}, t), \end{array}$ where $A (t) = \frac{C_{7} (t)}{2}$ with $C_{7} (t)$ from Lemma 4.8 and $C_{8}$ , $C_{9}$ are positive constants independent of δ and ε.
Proof.
We take (4.9b) for $Z_{1} v_{ε} = Z_{1} v_{ε}^{(1)} - Z_{1} v_{ε}^{(2)}$ and choose $θ = X_{(0, t)} Z_{1} v_{ε}$ as the test function. The rest follows like the previous lemma. □

Now, $\begin{array}{l} N (Z_{1} v_{ε}^{(1)} - Z_{1} v_{ε}^{(2)}, t) & = {‖ Z_{1} v_{ε}^{(1)} - Z_{1} v_{ε}^{(2)} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2} + {‖ \nabla Z_{1} v_{ε}^{(1)} - \nabla Z_{1} v_{ε}^{(2)} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2} \\ ⩽ \frac{μ^{3} A (t)}{μ C_{8} + C_{9}} (e^{(\frac{C_{8} μ + C_{9}}{μ^{2}}) t} - 1) N (v_{ε}^{(1)} - v_{ε}^{(2)}, t) \\ + \frac{μ A (t)}{D_{2}} e^{(\frac{C_{8} μ + C_{9}}{μ^{2}}) t} N (v_{ε}^{(1)} - v_{ε}^{(2)}, t) . \end{array}$ Inserting $t = μ^{2}$ gives, $\begin{array}{l} N (Z_{1} v_{ε}^{(1)} - Z_{1} v_{ε}^{(2)}, μ^{2}) ⩽ μ \overline{C} N (v_{ε}^{(1)} - v_{ε}^{(2)}, μ^{2}), \\ (4.12) & where \overline{C} = A (μ^{2}) [\frac{μ^{2} (e^{μ C_{8} + C_{9}} - 1)}{μ C_{8} + C_{9}} + \frac{e^{μ C_{8} + C_{9}}}{D_{2}}] . \end{array}$ In the estimates, the constants $C_{8}$ , $C_{9}$ are independent of the initial data, as T is an upper bound of t, $C_{7} (t)$ and $A (t)$ can be bounded by a constant independent of t. In (4.12) $\overline{C}$ does not depend on the initial data. So if, $0 < μ \overline{C} < 1$ that is, $0 < \overline{C} < \frac{1}{μ}$ then for μ small enough(but fixed) $Z_{1}$ is a contraction on the closed set $\begin{matrix} {v_{ε} \in L^{2} ([0, μ^{2}); H^{1, 2} (Ω_{ε}^{p})) : 0 ⩽ {‖ v_{ε} (t) ‖}_{Ω_{ε}^{p}} ⩽ M_{v}}, \end{matrix}$ and so by Banach’s fixed point Theorem it has a fixed point $v_{ε}$ in this set. By Lemma 4.6, $v_{ε} \in H^{1, 2} ([0, μ^{2}); H^{1, 2} {(Ω_{ε}^{p})}^{})$ and satisfy a-priori estimate. Also $v_{ε} \in C ([0, μ^{2}); L^{2} (Ω_{ε}^{p}))$ with $0 ⩽ ‖ v_{ε} (μ^{2}) ‖_{Ω_{ε}^{p}} ⩽ M_{v}$ . So, $v_{ε} (μ^{2})$ can be used as initial condition for extending the time interval of existence from $S_{1} : = [0, μ^{2})$ to $S_{2} : = [0, μ_{1}^{2}) \subseteq S$ , where $μ_{1} > μ$ . Since $μ_{1}$ is independent of the initial data, we proceed in the similar fashion as above and by bootstrapping argument we can obtain the existence of solution for a.e. $t \in S$ .
Lemma 4.10.
The fixed point* $v_{ε}$ of $Z_{1}$ belongs to $F_{V_{ε}}$ exists uniquely and satisfies $\begin{matrix} {‖ v_{ε} (t) ‖}_{Ω_{ε}^{p}} + D_{2} {‖ \nabla v_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}} + {‖ \partial_{t} v_{ε} ‖}_{L^{2} (S; H^{1, 2} {(Ω_{ε}^{p})}^{})} ⩽ C, \end{matrix}$ for a.e.* $t \in S$ , where the constant $C > 0$ does not depend on δ and ε.
Proof.
Let, $v_{ε}^{0} \in F_{V_{ε}}$ arbitrary. A sequence ${v_{ε}^{k}} \subset F_{V_{ε}} \cap H^{1, 2} (S; H^{1, 2} {(Ω_{ε}^{p})}^{})$ is constructed by the iteration $v_{ε}^{k} = Z_{1} (v_{ε}^{k - 1})$ , $k ⩾ 1$ . For $m > l$ we see that $\begin{matrix} N (v_{ε}^{m} - v_{ε}^{l}, t) \to 0 as m, l \to \infty . \end{matrix}$ So, ${v_{ε}^{k}}$ is a Cauchy sequence in $F_{V_{ε}}$ and $F_{V_{ε}}$ is a closed subspace of a complete metric space and hence complete. So there exists $v_{ε}$ such that $N (v_{ε}^{k}, t) \to N (v_{ε}, t)$ as $k \to \infty$ . Again, $\begin{matrix} N (Z_{1} v_{ε} - v_{ε}, t) ⩽ α N (v_{ε} - v_{ε}^{k - 1}, t) + N (v_{ε}^{k} - v_{ε}, t) \to 0 as k \to \infty . \end{matrix}$ This implies, $Z_{1} v_{ε} = v_{ε}$ . Therefore, $v_{ε}$ is a fixed point of $Z_{1}$ . If possible, let $p_{ε}$ is another fixed point of $Z_{1}$ . Now, $\begin{matrix} N (v_{ε} - p_{ε}, t) ⩽ α N (v_{ε} - p_{ε}, t) < N (v_{ε} - p_{ε}, t) as α = μ \overline{C} < 1 . \end{matrix}$ It’s a contradiction. So, $Z_{1}$ has a unique fixed point $v_{ε} \in F_{V_{ε}} \cap H^{1, 2} (S; H^{1, 2} {(Ω_{ε}^{p})}^{})$ . Again $v_{ε}$ is uniformly bounded in $H^{1, 2} (S; H^{1, 2} {(Ω_{ε}^{p})}^{})$ . So it has a weakly convergent subsequence ${v_{ε}^{k_{n}}}$ with $v_{ε} \in H^{1, 2} (S; H^{1, 2} {(Ω_{ε}^{p})}^{})$ as a weak limit. By construction all ${v_{ε}^{k_{n}}}$ satisfy the initial and boundary data and $\begin{matrix} (4.13) & {⟨ \frac{\partial v_{ε}^{k_{n}}}{\partial t}, θ ⟩}_{{(Ω_{ε}^{p})}^{t}} + {⟨ {\bar{D}}_{2} \nabla v_{ε}^{k_{n}}, \nabla θ ⟩}_{{(Ω_{ε}^{p})}^{t}} = - {⟨ \frac{\partial w_{ε}^{k_{n}}}{\partial t}, θ ⟩}_{{(Γ_{ε}^{})}^{t}}, \forall θ \in L^{2} (S; H^{1, 2} (Ω_{ε}^{p})), \end{matrix}$ and $w_{ε}^{k_{n}}$ satisfies $\begin{matrix} (4.14) & {⟨ \frac{\partial w_{ε}^{k_{n}}}{\partial t}, η ⟩}_{{(Γ_{ε}^{})}^{t}} = k_{d} {⟨ R (u_{ε}, v_{ε}^{k_{n} - 1}) - ψ_{δ} (w_{ε}^{k_{n}}), η ⟩}_{{(Γ_{ε}^{})}^{t}}, \forall η \in L^{2} (S; L^{2} (Γ_{ε}^{})), \end{matrix}$ with $w_{I ε}$ as initial data. By Lemma 4.8 the sequence ${w_{ε}^{k_{n}}}$ converges strongly to a limit $w_{ε}$ in $H^{1, 2} (S; L^{2} (Γ_{ε}^{}))$ . We can pass to the limit in (4.13) and (4.14) since R and $ψ_{δ}$ are continuous and obtain $(v_{ε}, w_{ε})$ is a solution of the regularized problem.

Uniqueness: Let $(v_{ε 1}, w_{ε 1})$ and $(v_{ε 2}, w_{ε 2})$ be two solutions of the regularized problem related with the equations (4.9b) and (4.9c). Take $v_{ε} = v_{ε 1} - v_{ε 2}$ and $w_{ε} = w_{ε 1} - w_{ε 2}$ . Now by Lemma 4.8, $\begin{matrix} {‖ w_{ε} (t) ‖}_{Γ_{ε}^{}}^{2} ⩽ k_{d}^{2} {L_{R}}^{2} {‖ v_{ε} ‖}_{{(Γ_{ε}^{})}^{t}}^{2} + {‖ w_{ε} ‖}_{{(Γ_{ε}^{})}^{t}}^{2} . \end{matrix}$ We apply Gronwall’s inequality and integrate w.r.t. t to get $\begin{matrix} (4.15) & {‖ w_{ε} ‖}_{{(Γ_{ε}^{})}^{t}}^{2} ⩽ k_{d}^{2} {L_{R}}^{2} {‖ v_{ε} ‖}_{{(Γ_{ε}^{})}^{t}}^{2} (e^{t} - 1), for a.e. t \in S . \end{matrix}$ Now, replacing θ by $v_{ε}$ in (4.9b), we have $\begin{matrix} \frac{1}{2} {‖ v_{ε} (t) ‖}_{Ω_{ε}^{p}}^{2} + D_{2} {‖ \nabla v_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2} ⩽ | k_{d} {⟨ R (u_{ε}, v_{ε}) - ψ_{δ} (w_{ε}), v_{ε} ⟩}_{{(Γ_{ε}^{})}^{t}} |, \end{matrix}$ where $\begin{matrix} | k_{d} {⟨ R (u_{ε}, v_{ε}) - ψ_{δ} (w_{ε}), v_{ε} ⟩}_{{(Γ_{ε}^{})}^{t}} | ⩽ \frac{k_{d}^{2} L_{R}}{δ} \sqrt{e^{t} - 1} ‖ v_{ε} ‖_{{(Γ_{ε}^{})}^{t}}^{2}, by (4.15) . \end{matrix}$ Now by the trace inequality (A.4) $\begin{matrix} ‖ v_{ε} ‖_{Γ_{ε}^{}}^{2} ⩽ C_{1} ‖ v_{ε} ‖_{Ω_{ε}^{p}}^{2} + C_{2} ‖ v_{ε} ‖_{Ω_{ε}^{p}} ‖ \nabla v_{ε} ‖_{Ω_{ε}^{p}} . \end{matrix}$ Hence, $\begin{matrix} {‖ v_{ε} (t) ‖}_{Ω_{ε}^{p}}^{2} + 2 D_{2} {‖ \nabla v_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2} ⩽ 2 C_{10} C_{1} ‖ v_{ε} ‖_{{(Ω_{ε}^{p})}^{t}}^{2} + \frac{C_{2} C_{10}^{2}}{2 μ} ‖ v_{ε} ‖_{{(Ω_{ε}^{p})}^{t}}^{2} + 2 C_{2} μ ‖ \nabla v_{ε} ‖_{{(Ω_{ε}^{p})}^{t}}^{2}, \end{matrix}$ where $C_{10} = \frac{k_{d}^{2} L_{R}}{δ} \sqrt{e^{T} - 1}$ . Setting $μ = \frac{D_{2}}{C_{2}}$ implies ${‖ v_{ε} (t) ‖}_{Ω_{ε}^{p}}^{2} ⩽ C_{11} {‖ v_{ε} ‖}_{{(Ω_{ε}^{p})}^{t}}^{2}$ , where $C_{11} = (2 C_{10} C_{1} + \frac{C_{2}^{2} C_{10}^{2}}{2 D_{2}})$ . Therefore, Gronwall’s inequality gives, $\begin{matrix} {‖ v_{ε} (t) ‖}_{Ω_{ε}^{p}}^{2} = 0 ⟹ v_{ε 1} (t) = v_{ε 2} (t) . \end{matrix}$ By (4.15), $\begin{matrix} {‖ w_{ε} (t) ‖}_{{(Γ_{ε}^{})}^{t}}^{2} = 0 ⟹ w_{ε 1} (t) = w_{ε 2} (t) . \end{matrix}$ □

Now to show the existence of solution of (4.9a) and (4.9c), we define a fixed point operator $Z_{2} : F_{U_{ε}} \to F_{U_{ε}}$ and proceed in the similar way as before. This can be summarized in the following Lemma: Similarly, for $u_{ε}$ we will get,
Lemma 4.11.
For each* $δ > 0$ , the problem ( 4.9a )–( 4.9c ) has a unique solution $(u_{ε δ}, w_{ε δ}) \in U_{ε} \times W_{ε}$ which satisfies $\begin{array}{l} {‖ u_{ε δ} (t) ‖}_{Ω_{ε}^{p}} + D_{1} {‖ \nabla u_{ε δ} ‖}_{{(Ω_{ε}^{p})}^{t}} + {‖ \partial_{t} u_{ε δ} ‖}_{L^{2} (S; H^{1, 2} {(Ω_{ε}^{p})}^{})} + {‖ v_{ε δ} (t) ‖}_{Ω_{ε}^{p}} + D_{2} {‖ \nabla v_{ε δ} ‖}_{{(Ω_{ε}^{p})}^{t}} \\ + {‖ \partial_{t} v_{ε δ} ‖}_{L^{2} (S; H^{1, 2} {(Ω_{ε}^{p})}^{})} + {‖ w_{ε δ} (t) ‖}_{Γ_{ε}^{}} + {‖ \partial_{t} w_{ε δ} ‖}_{{(Γ_{ε}^{})}^{t}} ⩽ C, for a.e. t \in S, \end{array}$ where the constant $C > 0$ is independent of δ and ε.

Now we have to send $δ \to 0$ . For each $δ > 0$ , Lemma 4.11 provides the necessary estimates to pass the limit. Let, $z_{ε δ} \in L^{\infty} ({Γ_{ε}^{}}^{t})$ defined by, $z_{ε δ} (t, x) = ψ_{δ} (w_{ε δ} (t, x))$ a.e. in ${(Γ_{ε}^{})}^{t}$ . Now by compactness argument there exists a subsequence and $(u_{ε}, v_{ε}, w_{ε}, z_{ε}) \in U_{ε} \times V_{ε} \times W_{ε} \times Z_{ε}$ such that for $δ \to 0$
$u_{ε δ} ⇀ u_{ε}$ weakly in $L^{2} (S; H^{1, 2} (Ω_{ε}^{p}))$ ,

$\frac{\partial u_{ε δ}}{\partial t} ⇀ \frac{\partial u_{ε}}{\partial t}$ weakly in $L^{2} (S; H^{1, 2} {(Ω_{ε}^{p})}^{})$ ,

$v_{ε δ} ⇀ v_{ε}$ weakly in $L^{2} (S; H^{1, 2} (Ω_{ε}^{p}))$ ,

$\frac{\partial v_{ε δ}}{\partial t} ⇀ \frac{\partial v_{ε}}{\partial t}$ weakly in $L^{2} (S; H^{1, 2} {(Ω_{ε}^{p})}^{})$ ,

$w_{ε δ} ⇀ w_{ε}$ weakly in $L^{2} (S; L^{2} (Γ_{ε}^{}))$ ,

$\frac{\partial w_{ε δ}}{\partial t} ⇀ \frac{\partial w_{ε}}{\partial t}$ weakly in $L^{2} ({Γ_{ε}^{}}^{t})$ ,

$z_{ε δ} \overset{}{⇀} z_{ε}$ weakly in $L^{\infty} ({Γ_{ε}^{}}^{t})$ .
Proof of Theorem 3.1.
All bounds are inherited from Lemma 4.11 and by weak convergences (i)–(vi). $u_{ε}$ , $v_{ε}$ and $w_{ε}$ satisfy equation (3.1a), (3.1b) and the ODE (3.1c). However, the second part of (3.1c) needs attention. We first consider the behavior of $u_{ε δ}$ on ${(Γ_{ε}^{})}^{t}$ . By the a-priori estimates of Lemma 4.11 and by Corollary 4 and Lemma 9 of [52] we get, for $s \in (0, 1)$ , $\begin{array}{l} u_{ε δ} \to u_{ε} strongly in L^{2} ((0, T); L^{2} (Ω_{ε}^{p})), \\ u_{ε δ} \to u_{ε} strongly in L^{2} ((0, T); H^{s, 2} (Ω_{ε}^{p})) \cap C ([0, T]; H^{- s, 2} (Ω_{ε}^{p})), \\ v_{ε δ} \to v_{ε} strongly in L^{2} ((0, T); L^{2} (Ω_{ε}^{p})) and \\ v_{ε δ} \to v_{ε} strongly in L^{2} ((0, T); H^{s, 2} (Ω_{ε}^{p})) \cap C ([0, T]; H^{- s, 2} (Ω_{ε}^{p})) . \end{array}$ Now by trace Theorem (cf. Satz 8.7 of [58]), we have $\begin{matrix} u_{ε δ} \to u_{ε} strongly in L^{2} {(Γ_{ε}^{})}^{t}, v_{ε δ} \to v_{ε} strongly in L^{2} {(Γ_{ε}^{})}^{t} . \end{matrix}$ Since R is Lipschitz, $R (u_{ε δ}, v_{ε δ}) \to R (u_{ε}, v_{ε})$ strongly in $L^{2} ({Γ_{ε}^{}}^{t})$ and pointwise a.e. in ${(Γ_{ε}^{})}^{t}$ . This and weak* convergence of $z_{ε δ}$ implies $u_{ε}$ , $v_{ε}$ , $w_{ε}$ satisfy (3.1c). Now we need to show that $z_{ε} \in ψ (w_{ε})$ a.e. in ${(Γ_{ε}^{})}^{t}$ . Let $\begin{matrix} \underline{w_{ε}} = \underset{δ \to 0}{lim inf} w_{ε δ} (t, x) ⩾ 0 a.e. in {(Γ_{ε}^{})}^{t} . \end{matrix}$ Now, we decompose ${(Γ_{ε}^{})}^{t} = K_{1} \cup K_{2}$ in almost everywhere sense, where $K_{1} = {\underline{w_{ε}} > 0}$ and $K_{2} = {\underline{w_{ε}} = 0}$ . We need to show that $w_{ε} > 0$ and $z_{ε} = 1$ in $K_{1}$ , while $w_{ε} = 0$ and $z_{ε} = R (u_{ε}, v_{ε}) \in [0, 1]$ in $K_{2}$ . Note that $\begin{matrix} \underline{w_{ε}} ⩽ w_{ε} ⟹ w_{ε} ⩾ \underline{w_{ε}} > 0 in K_{1} . \end{matrix}$ Let $(t, x) \in K_{1}$ such that $\underline{w_{ε}} (t, x) > 2 β > 0$ for β sufficiently small. We write $\underline{w_{ε}} = {lim}_{δ \to 0} A_{δ}$ , where $A_{δ} = inf w_{ε δ} (t, x)$ . Therefore, $A_{δ}$ is an increasing sequence and since $w_{ε δ}$ is bounded by Lemma 4.5, $A_{δ}$ is also bounded and converges to its least upper bound, i.e. $sup A_{δ} = \underline{w_{ε}}$ . So for any $β > 0$ there exists $w_{ε δ}$ such that $w_{ε δ} > \underline{w_{ε}} - β > β$ . Hence $z_{ε δ} (t, x) = 1$ for all δ small enough. So, $z_{ε} (t, x) = 1$ . Also, $\begin{matrix} (4.16) & \underset{δ \to 0}{lim inf} \int_{0}^{t} z_{ε δ} (s, x) d s ⩽ \int_{0}^{t} z_{ε} (s, x) d s in {(Γ_{ε}^{})}^{t} . \end{matrix}$ We choose $η = X_{(0, t)}$ and integrate w.r.t. t, then $\begin{array}{l} w_{ε δ} = w_{I ε δ} + k_{d} \int_{0}^{t} (R (u_{ε δ}, u_{ε δ}) - z_{ε δ}) d s \\ ⟹ w_{ε δ} = w_{I ε δ} + k_{d} \int_{0}^{t} (R (u_{ε δ}, u_{ε δ}) - R (u_{ε}, v_{ε})) d s - k_{d} \int_{0}^{t} (z_{ε δ} - z_{ε}) d s a.e. in {(Γ_{ε}^{})}^{t} . \end{array}$ Now, on $K_{2}$ taking ${lim inf}_{δ \to 0}$ we get $\begin{matrix} w_{ε} = k_{d} \underset{δ \to 0}{lim inf} \int_{0}^{t} (z_{ε δ} - z_{ε}) d s ⩽ 0, by (4.16) . \end{matrix}$ Hence, $w_{ε} (t, x) = 0$ in $K_{2}$ . As $\frac{\partial w_{ε}}{\partial t} \in L^{2} ({Γ_{ε}^{}}^{t})$ therefore we have, $\frac{\partial w_{ε}}{\partial t} = 0$ a.e. in $K_{2}$ . Consequently, $z_{ε} = R (u_{ε}, v_{ε})$ with $0 ⩽ z_{ε} ⩽ 1$ . □

5. An anticipated homogenized model of the problem $(P_{ε})$ via asymptotic expansion

Homogenization is a mathematically rigorous approach for averaging differential equations. It gives the macroscopic description of a medium that is microscopically heterogeneous, i.e. it aims to replace a complex, rapidly varying heterogeneous medium with a simple, slowly changing homogeneous medium. In this Section, we will obtain the upscaled version (macroscopic description) of the problem $(P_{ε})$ by using asymptotic expansion (see Chapter 1.3.1 of [21]). Although asymptotic expansion is a formal method and does not address the convergence of the micro problem to the macro problem, it still helps us to obtain (or at least gives an idea of) the upscaled equations. Besides asymptotic analysis, we use two-scale convergence (see [1,2,30]) and periodic unfolding method (see [12,13,16,17]) in Section 6 to perform the homogenization of $(P_{ε})$ . Now, let $\begin{matrix} (5.1) & \begin{matrix} u_{ε} (t, x) = \sum_{i = 0}^{\infty} ε^{i} u_{i} (t, x, \frac{x}{ε}), v_{ε} (t, x) = \sum_{i = 0}^{\infty} ε^{i} v_{i} (t, x, \frac{x}{ε}), \\ w_{ε} (t, x) = \sum_{i = 0}^{\infty} ε^{i} w_{i} (t, x, \frac{x}{ε}), \end{matrix} \end{matrix}$ where $u_{i} (t, x, y)$ , $v_{i} (t, x, y)$ and $w_{i} (t, x, y)$ are Y-periodic functions in $y (= x / ε)$ . Moreover, $\nabla = \nabla_{x} + \frac{1}{ε} \nabla_{y}$ . Now by (5.1) and (2.3a), $\begin{array}{l} [\frac{\partial u_{0}}{\partial t} + ε \frac{\partial u_{1}}{\partial t} + ε^{2} \frac{\partial u_{2}}{\partial t} + \dots] + [{(div)}_{x} + \frac{1}{ε} {(div)}_{y}] [- D_{1} {(\nabla_{x} u_{0} + \frac{1}{ε} \nabla_{y} u_{0}) \\ (5.2) & + ε (\nabla_{x} u_{1} + \frac{1}{ε} \nabla_{y} u_{1}) + ε^{2} (\nabla_{x} u_{2} + \frac{1}{ε} \nabla_{y} u_{2}) + \dots}] = 0 . \end{array}$ Now equating the coefficients of $ε^{- 2}$ on both sides of (5.2) yields $\begin{matrix} (5.3) & {(div)}_{y} (- D_{1} \nabla_{y} u_{0}) = 0, \forall y \in Y . \end{matrix}$ We multiply (5.3) by $u_{0} (t, x, y)$ and integrate w.r.t. y, then $\begin{array}{l} \int_{Y^{p}} {(div)}_{y} (- D_{1} \nabla_{y} u_{0}) u_{0} (t, x, y) d y = 0 \\ (5.4) & ⟹ - \int_{Γ} D_{1} \nabla_{y} u_{0} . \vec{n} u_{0} d σ_{y} - \int_{\partial Y} D_{1} \nabla_{y} u_{0} . \vec{n} u_{0} d σ_{y} + D_{1} \int_{Y^{p}} {| \nabla_{y} u_{0} (t, x, y) |}^{2} d y = 0 . \end{array}$ Note that, $\int_{\partial Y} D_{1} \nabla_{y} u_{0} . \vec{n} u_{0} d σ_{y} = 0$ since $u_{0}$ is $Y -$ periodic. From (2.3c) we see that, $\begin{array}{l} - D_{1} \nabla u_{ε} . \vec{n} = ε \frac{\partial w_{ε}}{\partial t} \\ (5.5) & ⟹ - D_{1} [(\nabla_{x} u_{0} + \frac{1}{ε} \nabla_{y} u_{0}) + ε (\nabla_{x} u_{1} + \frac{1}{ε} \nabla_{y} u_{1}) + \dots] . \vec{n} = ε [\frac{\partial w_{0}}{d t} + ε \frac{\partial w_{1}}{\partial t} + \dots] . \end{array}$ We compare the coefficient of $ε^{- 1}$ on both sides of (5.5), which gives $\begin{matrix} - D_{1} \nabla_{y} u_{0} . \vec{n} = 0 . \end{matrix}$ We put these in (5.4) to obtain, $\begin{matrix} D_{1} \int_{Y^{p}} {| \nabla_{y} u_{0} (t, x, y) |}^{2} d y = 0 ⟹ u_{0} (t, x, y) = u_{0} (t, x), \forall y \in Y as D_{1} \neq 0 . \end{matrix}$ Next, comparing the coefficients of $ε^{- 1}$ on both sides of (5.2) leads to $\begin{matrix} (5.6) & {(div)}_{y} (- D_{1} (\nabla_{x} u_{0} + \nabla_{y} u_{1})) = 0 ⟹ {(div)}_{y} (\nabla_{y} u_{1}) = - \sum_{j = 1}^{n} \frac{\partial}{\partial y_{j}} (\frac{\partial u_{0}}{\partial x_{j}}) . \end{matrix}$ From this it follows that $u_{1}$ is a linear combination of $\frac{\partial u_{0}}{\partial x_{j}}$ for all j, i.e. $u_{1} (t, x, y) = \sum_{j = 1}^{n} \frac{\partial u_{0} (t, x)}{\partial x_{j}} l_{j} (y) + p (x)$ , where, for each $j = 1, 2, \dots, n$ , $l_{j} (y)$ are the Y-periodic solution of the cell problems $\begin{matrix} (5.7) & \{\begin{matrix} \nabla_{y} . (\nabla_{y} l_{j} + e_{j}) = 0 for all y \in Y^{p}, \\ (\nabla_{y} l_{j} + e_{j}) . \vec{n} = 0 on Γ, \\ y \mapsto l_{j} (y) is Y -periodic . \end{matrix} \end{matrix}$ Again, we compare the coefficients of $ε^{0}$ on both sides of (5.2) to derive $\begin{matrix} (5.8) & \int_{Y^{p}} [\frac{\partial u_{0}}{\partial t} - D_{1} Δ_{x} u_{0} + {(div)}_{x} (- D_{1} \nabla_{y} u_{1}) + {(div)}_{y} (- D_{1} \nabla_{x} u_{1} - D_{1} \nabla_{y} u_{2})] d y = 0 . \end{matrix}$ Next, by integration by parts $\begin{matrix} (5.9) & \int_{Y^{p}} {(div)}_{y} (- D_{1} \nabla_{x} u_{1} - D_{1} \nabla_{y} u_{2}) d y = - D_{1} \int_{Γ} (\nabla_{x} u_{1} + \nabla_{y} u_{2}) . \vec{n} d σ_{y}, \end{matrix}$ since $\int_{\partial Y} (\nabla_{x} u_{1} + \nabla_{y} u_{2}) . \vec{n} d σ_{y} = 0$ . Now comparing the coefficient of ε on both sides of (5.5), we have $\begin{matrix} - D_{1} (\nabla_{x} u_{1} + \nabla_{y} u_{2}) . \vec{n} = \frac{\partial w_{0}}{\partial t} . \end{matrix}$ Now (5.9) takes the form $\begin{matrix} \int_{Y^{p}} {(div)}_{y} (- D_{1} \nabla_{x} u_{1} - D_{1} \nabla_{y} u_{2}) d y = \int_{Γ} \frac{\partial w_{0}}{\partial t} d σ_{y} . \end{matrix}$ From (5.8), we deduce that $\begin{array}{l} | Y^{p} | \frac{\partial u_{0}}{\partial t} - \sum_{i, j = 1}^{n} \int_{Y^{p}} D_{1} (δ_{i j} + \frac{\partial l_{j} (y)}{\partial y_{i}}) \frac{\partial^{2} u_{0}}{\partial x_{i} \partial x_{j}} d y + \int_{Γ} \frac{\partial w_{0}}{\partial t} d σ_{y} = 0, by the definition of u_{1} \\ (5.10) & ⟹ \frac{\partial u_{0}}{\partial t} - \nabla . (A \nabla u_{0}) + P (t, x) = 0 for x \in Ω, \end{array}$ where $\begin{matrix} A = {(a_{i j})}_{1 ⩽ i, j ⩽ n} = \int_{Y^{p}} \frac{D_{1}}{| Y^{p} |} (δ_{i j} + \sum_{i . j = 1}^{n} \frac{\partial l_{j} (y)}{\partial y_{i}}) d y and P (t, x) = \int_{Γ} \frac{1}{| Y^{p} |} \frac{\partial w_{0}}{\partial t} d σ_{y} . \end{matrix}$ For the boundary condition, we equate the coefficients of $ε^{0}$ on both sides of (5.5) and obtain $\begin{array}{l} - D_{1} (\nabla_{x} u_{0} + \nabla_{y} u_{1}) . \vec{n} = 0 \\ (5.11) & ⟹ - A \nabla u_{0} . \vec{n} = 0 for x \in \partial Ω, proceeding as above . \end{array}$ For the initial condition, inserting (5.1) in (2.3d) yields $\begin{matrix} u_{0} (0, x) + ε u_{1} (0, x, \frac{x}{ε}) + \dots = u_{I 0} (x) + ε u_{I 1} (x, \frac{x}{ε}) + \dots . \end{matrix}$ Equating the coefficient of $ε^{0}$ we get $\begin{matrix} u_{0} (0, x) = u_{I 0} (x) . \end{matrix}$ Similarly, the macromodel of the mobile species $M_{2}$ is given by $\begin{array}{l} \frac{\partial v_{0}}{\partial t} - \nabla . (B \nabla v_{0}) + P (t, x) = 0 in S \times Ω, \\ - B \nabla v_{0} . \vec{n} = 0 on S \times \partial Ω, \\ v_{0} (0, x) = v_{I 0} (x) in Ω, \end{array}$ where $\begin{matrix} B = (b_{i j}) = \int_{Y^{p}} \frac{D_{2}}{| Y^{p} |} (δ_{i j} + \sum_{i, j = 1}^{n} \frac{\partial l_{j}}{\partial y_{i}}) d y \end{matrix}$ and $l_{j}$ is the solution of the cell-problem (5.7). We now find out the effective equation for the immobile species and for that we need to get rid of the multivaluedness in (2.3i). In order to do this, we replace $ψ (w_{ε})$ by $ψ_{δ} (w_{ε})$ given by (4.8). Inserting (5.1) in (2.3i) and using series expansion we obtain $\begin{array}{l} [\frac{\partial w_{0}}{\partial t} + ε \frac{\partial w_{1}}{\partial t} + ε^{2} \frac{\partial w_{2}}{\partial t} + \dots] = & k_{d} [k \frac{k_{1} k_{2} (u_{0} v_{0} + ε (u_{0} v_{1} + u_{1} v_{0}) + \dots)}{{(1 + k_{1} u_{0} + k_{2} v_{0}) + ε (k_{1} u_{1} + k_{2} v_{1}) + \dots}^{2}} \\ - (ψ_{δ} (w_{0}) + ε (\dots))] . \end{array}$ We will first simplify the r.h.s. $\begin{array}{l} k \frac{k_{1} k_{2} (u_{0} v_{0} + ε (u_{1} v_{0} + u_{0} v_{1}) + \dots)}{{[(1 + k_{1} u_{0} + k_{2} v_{0}) + ε (k_{1} u_{1} + k_{2} v_{1}) + \dots]}^{2}} \\ = k \frac{k_{1} k_{2} (u_{0} v_{0} + ε (u_{1} v_{0} + u_{0} v_{1}) + \dots)}{{(1 + k_{1} u_{0} + k_{2} v_{0})}^{2}} {[1 + ε \frac{k_{1} u_{1} + k_{2} v_{1}}{1 + k_{1} u_{0} + k_{2} v_{0}} + \dots]}^{- 2} \\ = k \frac{k_{1} k_{2} (u_{0} v_{0} + ε (u_{1} v_{0} + u_{0} v_{1}) + \dots)}{{(1 + k_{1} u_{0} + k_{2} v_{0})}^{2}} [1 - 2 ε \frac{k_{1} u_{1} + k_{2} v_{1}}{1 + k_{1} u_{0} + k_{2} v_{0}} + \dots] \\ = k \frac{k_{1} k_{2} u_{0} v_{0}}{{(1 + k_{1} u_{0} + k_{2} v_{0})}^{2}} + k ε (\dots) . \end{array}$ Therefore, we have $\begin{matrix} [\frac{\partial w_{0}}{\partial t} + ε \frac{\partial w_{1}}{\partial t} + ε^{2} \frac{\partial w_{2}}{\partial t} + \dots] = k_{d} [(k \frac{k_{1} k_{2} u_{0} v_{0}}{{(1 + k_{1} u_{0} + k_{2} v_{0})}^{2}} + k ε (\dots)) - (ψ_{δ} (w_{0}) + ε (\dots))] . \end{matrix}$ Upon comparing the coefficients of $ε^{0}$ on both sides, we get $\begin{matrix} \frac{\partial w_{0}}{\partial t} = k_{d} [k \frac{k_{1} k_{2} u_{0} v_{0}}{{(1 + k_{1} u_{0} + k_{2} v_{0})}^{2}} - ψ_{δ} (w_{0})] . \end{matrix}$ Now taking $δ \to 0$ as shown in Section 4 (cf. [54,55]) we obtain $\begin{matrix} \frac{\partial w_{0}}{\partial t} = k_{d} [k \frac{k_{1} k_{2} u_{0} v_{0}}{{(1 + k_{1} u_{0} + k_{2} v_{0})}^{2}} - z_{0}], where z_{0} \in ψ (w_{0}) . \end{matrix}$ For the initial condition, comparing the coefficient of $ε^{0}$ on both sides of $\begin{matrix} w_{0} (0, x, \frac{x}{ε}) + ε w_{1} (0, x, \frac{x}{ε}) + \dots = w_{I 0} (x, \frac{x}{ε}) + ε w_{I 1} (x, \frac{x}{ε}) + \dots, \end{matrix}$ we can write $w_{0} (0, x, y) = w_{I 0} (x, y)$ . The required upscaled model can be summarized as For mobile species $M_{1}$ : $\begin{matrix} (5.12) & \begin{matrix} \frac{\partial u_{0}}{\partial t} - \nabla . (A \nabla u_{0}) + P (t, x) = 0 in S \times Ω, \\ - A \nabla u_{0} . \vec{n} = 0 on S \times \partial Ω, \\ u_{0} (0, x) = u_{I 0} (x) in Ω . \end{matrix} \end{matrix}$ For mobile species $M_{2}$ : $\begin{matrix} (5.13) & \begin{matrix} \frac{\partial v_{0}}{\partial t} - \nabla . (B \nabla v_{0}) + P (t, x) = 0 in S \times Ω, \\ - B \nabla v_{0} . \vec{n} = 0 on S \times \partial Ω, \\ v_{0} (0, x) = v_{I 0} (x) in Ω . \end{matrix} \end{matrix}$ For immobile species $M_{12}$ : $\begin{matrix} (5.14) & \begin{matrix} \frac{\partial w_{0}}{\partial t} = k_{d} (R (u_{0}, v_{0}) - z_{0}) in S \times Ω \times Γ, \\ where z_{0} \in ψ (w_{0}) in S \times Ω \times Γ, \\ w_{0} (0, x, y) = w_{I 0} (x, y) on Ω \times Γ . \end{matrix} \end{matrix}$

6. Proof of Theorem 3.2 (Homogenization of the problem $(P_{ε})$ via two-scale convergence)

Here we aim to transform the equations from the oscillating domain to a fixed domain by applying a rigorous approach of homogenization. As a first step toward achieving this aim, we need to extend the solutions $(u_{ε}, v_{ε})$ from $S \times Ω_{ε}^{p}$ to $S \times Ω$ . We also unfold the ODE to define the unfolded concentration on the fixed boundary Γ. We start with the following Lemma:

Lemma 6.1.
There exists a positive constant C independent of ε such that $\begin{array}{l} sup_{ε > 0} ({‖ u_{ε} (t) ‖}_{Ω} + D_{1} ‖ \nabla u_{ε} ‖_{{(Ω)}^{t}} + {‖ χ_{ε} \frac{\partial u_{ε}}{\partial t} ‖}_{L^{2} (S; H^{1, 2} {(Ω)}^{})} + {‖ v_{ε} (t) ‖}_{Ω} \\ (6.1) & + D_{2} ‖ \nabla v_{ε} ‖_{{(Ω)}^{t}} + {‖ χ_{ε} \frac{\partial v_{ε}}{\partial t} ‖}_{L^{2} (S; H^{1, 2} {(Ω)}^{})}) ⩽ C < \infty . \end{array}$
Proof.
The proof follows from the estimate (3.2c) and Lemma A.6. □
Lemma 6.2.
We can conclude the following convergence results upto a subsequence (still denoted by the same subscript) from the a-priori bounds ( 3.2c ) and ( 6.1 )
$u_{ε} ⇀ u_{0}$ in $L^{2} (S; H^{1, 2} (Ω))$ ,

$\frac{\partial u_{ε}}{\partial t} ⇀ \frac{\partial u_{0}}{\partial t}$ in $L^{2} (S; H^{1, 2} {(Ω)}^{})$ ,

$v_{ε} ⇀ v_{0}$ in $L^{2} (S; H^{1, 2} (Ω))$ ,

$\frac{\partial v_{ε}}{\partial t} ⇀ \frac{\partial v_{0}}{\partial t}$ in $L^{2} (S; H^{1, 2} {(Ω)}^{})$ ,

$u_{ε} (T, \cdot) \overset{2}{⇀} u_{0}^{}$ in $H^{1, 2} (Ω)$ ,

$v_{ε} (T, \cdot) \overset{2}{⇀} v_{0}^{}$ in $H^{1, 2} (Ω)$ ,

$u_{ε} \to u_{0}$ in $L^{2} (S \times Γ_{ε}^{})$ ,

$v_{ε} \to v_{0}$ in $L^{2} (S \times Γ_{ε}^{})$ ,

There exists $u_{0} \in L^{2} (S; H^{1, 2} (Ω))$ and $u_{1} \in L^{2} (S \times Ω; H_{per}^{1, 2} (Y) / R)$ such that $u_{ε} \overset{2}{⇀} u_{0}$ and $\nabla u_{ε} \overset{2}{⇀} \nabla_{x} u_{0} + \nabla_{y} u_{1}$ ,

There exists $v_{0} \in L^{2} (S; H^{1, 2} (Ω))$ and $v_{1} \in L^{2} (S \times Ω; H_{per}^{1, 2} (Y) / R)$ such that $v_{ε} \overset{2}{⇀} v_{0}$ and $\nabla v_{ε} \overset{2}{⇀} \nabla_{x} v_{0} + \nabla_{y} v_{1}$ ,

$w_{ε} \overset{2}{⇀} w_{0}$ in $L^{2} (S \times Ω \times Γ)$ ,

$\frac{\partial w_{ε}}{\partial t} \overset{2}{⇀} \frac{\partial w_{0}}{\partial t}$ in $L^{2} (S \times Ω \times Γ)$ ,

$w_{ε} (T, \cdot) \overset{2}{⇀} w_{0}^{} (x, y)$ in $L^{2} (Ω \times Γ)$ ,

$z_{ε} \overset{2}{⇀} z_{0}$ in $L^{2} (S \times Ω \times Γ)$ .

Proof.
The results (i)–(vi) follows from the boundedness of $u_{ε}$ , $v_{ε}$ in $L^{2} (S; H^{1, 2} (Ω))$ and $\frac{\partial u_{ε}}{\partial t}$ , $\frac{\partial v_{ε}}{\partial t}$ in $L^{2} (S; H^{1, 2} {(Ω)}^{})$ . We require some compact embedding results to establish (vii) and (viii). The sobolev space $H^{β^{'}, 2} (Ω)$ is compactly embedded in $H^{β, 2} (Ω)$ , that is $H^{β^{'}, 2} (Ω) ↪ ↪ H^{β, 2} (Ω)$ for $β \in (\frac{1}{2}, 1)$ and $0 < β < β^{'} ⩽ 1$ . So for a fixed ε, the space $U_{0} = {u_{0} \in L^{2} (S; H^{1, 2} (Ω)) and \frac{\partial u_{0}}{\partial t} \in L^{2} (S \times Ω)} ↪ ↪ L^{2} (S; H^{β, 2} (Ω))$ by the Lions–Aubin compactness Lemma. Then by the trace inequality (A.5) we get $\begin{array}{l} ‖ u_{ε} - u_{0} ‖_{L^{2} (S \times Γ_{ε}^{})} & ⩽ C_{0} ‖ u_{ε} - u_{0} ‖_{L^{2} (S; H^{β, 2} (Ω_{ε}^{p}))} \\ ⩽ C ‖ u_{ε} - u_{0} ‖_{L^{2} (S; H^{β, 2} (Ω))} \to 0 as ε \to 0 . \end{array}$

Let $V_{0} = {v_{0} \in L^{2} (S; H^{1, 2} (Ω)) and \frac{\partial v_{0}}{\partial t} \in L^{2} (S \times Ω)} ↪ ↪ L^{2} (S; H^{β, 2} (Ω))$ , then we proceed in the similar fashion to obtain for the second mobile species $\begin{matrix} ‖ v_{ε} - v_{0} ‖_{L^{2} (S \times Γ_{ε}^{})} ⩽ C ‖ v_{ε} - v_{0} ‖_{L^{2} (S; H^{β, 2} (Ω))} \to 0 as ε \to 0 . \end{matrix}$ The estimate (6.1) in combination with Lemma A.3 and Lemma A.5 gives (ix)–(xiii). The inequality $\begin{matrix} ‖ z_{ε} ‖_{L^{2} (S \times Γ_{ε}^{})}^{2} ⩽ T ‖ z_{ε} ‖_{L^{\infty} (S \times Γ_{ε}^{})} sup_{ε > 0} ε | Γ_{ε}^{} | ⩽ T ‖ z_{ε} ‖_{L^{\infty} (S \times Γ_{ε}^{})} \frac{| Γ | | Ω |}{| Y |} < \infty, \end{matrix}$ yields that $z_{ε}$ is bounded in $L^{2} (S \times Γ_{ε}^{})$ . So as an implication of Lemma A.5 we can conclude that $z_{ε} \overset{2}{⇀} z_{0}$ in $L^{2} (S \times Ω \times Γ)$ . □
Lemma 6.3.

The sequence* ${R (u_{ε}, v_{ε})}$ is strongly convergent to $R (u_{0}, v_{0})$ in $L^{2} (S \times Γ_{ε}^{})$ . We can deduce therefore that* $R (u_{ε}, v_{ε}) \overset{2}{⇀} R (u_{0}, v_{0})$ in $L^{2} (S \times Ω \times Γ)$ .

The unfolded sequence of the immobile species ${T_{ε}^{b} (w_{ε})}$ is strongly convergent to $w_{0}$ in $L^{2} (S \times Ω \times Γ)$ .

Proof.
(i) Since R is Lipschitz, we can write $\begin{matrix} | R (u_{ε}, v_{ε}) - R (u_{0}, v_{0}) | ⩽ L_{R} | u_{ε} - u_{0} | + L_{R} | v_{ε} - v_{0} | . \end{matrix}$ Now, we apply Minkowski’s inequality and Lemma 6.2 to obtain $\begin{array}{l} {‖ R (u_{ε}, v_{ε}) - R (u_{0}, v_{0}) ‖}_{L^{2} (S \times Γ_{ε}^{})} & ⩽ L_{R} {‖ u_{ε} - u_{0} ‖_{L^{2} (S \times Γ_{ε}^{})} + ‖ v_{ε} - v_{0} ‖_{L^{2} (S \times Γ_{ε}^{})}} \\ \to 0 as ε \to 0 . \end{array}$ Then we use Lemma A.10 together with Lemma A.8 to establish $\begin{array}{l} T_{ε}^{b} (R (u_{ε}, v_{ε})) \to R (u_{0}, v_{0}) strongly in L^{2} (S \times Ω \times Γ) \\ ⟹ R (u_{ε}, v_{ε}) \overset{2}{⇀} R (u_{0}, v_{0}) in L^{2} (S \times Ω \times Γ) . \end{array}$ (ii) We unfold the ODE (2.3i) by applying the boundary unfolding operator (A.7) and derive $\begin{array}{l} (6.2) & T_{ε}^{b} (\frac{\partial w_{ε}}{\partial t}) = \frac{\partial}{\partial t} (T_{ε}^{b} (w_{ε})) = k_{d} (T_{ε}^{b} (R (u_{ε}, v_{ε})) - T_{ε}^{b} (z_{ε})) \\ (6.3) & where T_{ε}^{b} (z_{ε}) \in ψ (T_{ε}^{b} (w_{ε})) = T_{ε}^{b} ψ (w_{ε}) = \{\begin{matrix} {0} & if T_{ε}^{b} w_{ε} < 0, \\ [0, 1] & if T_{ε}^{b} w_{ε} = 0, \\ {1} & if T_{ε}^{b} w_{ε} > 0 . \end{matrix} \end{array}$ For $λ > m$ , we subtract (6.2) for $ε = ε_{λ}$ from the choice of $ε = ε_{m}$ and get $\begin{matrix} \frac{\partial}{\partial t} (T_{ε_{λ}}^{b} w_{ε_{λ}} - T_{ε_{m}}^{b} w_{ε_{m}}) = k_{d} (T_{ε_{λ}}^{b} R (u_{ε_{λ}}, v_{ε_{λ}}) - T_{ε_{m}}^{b} R (u_{ε_{m}}, v_{ε_{m}})) - k_{d} (T_{ε_{λ}}^{b} z_{ε_{λ}} - T_{ε_{m}}^{b} z_{ε_{m}}) . \end{matrix}$ We now multiply $(T_{ε_{λ}}^{b} w_{ε_{λ}} - T_{ε_{m}}^{b} w_{ε_{m}})$ and integrate over $S \times Ω \times Γ$ to deduce $\begin{array}{l} \int_{0}^{t} \frac{\partial}{\partial t} {‖ T_{ε_{λ}}^{b} w_{ε_{λ}} (s) - T_{ε_{m}}^{b} w_{ε_{m}} (s) ‖}_{L^{2} (Ω \times Γ)}^{2} d s \\ = 2 k_{d} \int_{0}^{t} \int_{Ω \times Γ} (T_{ε_{λ}}^{b} R (u_{ε_{λ}}, v_{ε_{λ}}) - T_{ε_{m}}^{b} R (u_{ε_{m}}, v_{ε_{m}}) - T_{ε_{λ}}^{b} z_{ε_{λ}} + T_{ε_{m}}^{b} z_{ε_{m}}) (T_{ε_{λ}}^{b} w_{ε_{λ}} - T_{ε_{m}}^{b} w_{ε_{m}}) d x d σ_{y} d t . \end{array}$ As a consequence of the monotonicity of $T_{ε}^{b} (z_{ε})$ with respect to $T_{ε}^{b} (w_{ε})$ we have $\begin{matrix} (T_{ε_{λ}}^{b} z_{ε_{λ}} - T_{ε_{m}}^{b} z_{ε_{m})} (T_{ε_{λ}}^{b} w_{ε_{λ}} - T_{ε_{m}}^{b} w_{ε_{m}}) ⩾ 0 . \end{matrix}$ So we get the inequality $\begin{array}{l} {‖ T_{ε_{λ}}^{b} w_{ε_{λ}} (t) - T_{ε_{m}}^{b} w_{ε_{m}} (t) ‖}_{L^{2} (Ω \times Γ)}^{2} \\ ⩽ {‖ T_{ε_{λ}}^{b} w_{ε_{λ}} (0) - T_{ε_{m}}^{b} w_{ε_{m}} (0) ‖}_{L^{2} (Ω \times Γ)}^{2} + k_{d}^{2} \int_{0}^{t} {‖ T_{ε_{λ}}^{b} R (u_{ε_{λ}}, v_{ε_{λ}}) - T_{ε_{m}}^{b} R (u_{ε_{m}}, v_{ε_{m}}) ‖}_{L^{2} (Ω \times Γ)}^{2} d t \\ + \int_{0}^{t} {‖ T_{ε_{λ}}^{b} w_{ε_{λ}} (s) - T_{ε_{m}}^{b} w_{ε_{m}} (s) ‖}_{L^{2} (Ω \times Γ)}^{2} d s . \end{array}$ We then simplify each term of the r.h.s. We use Lemma A.9 to estimate the 1st term as $\begin{array}{l} {‖ T_{ε_{λ}}^{b} w_{ε_{λ}} (0) - T_{ε_{m}}^{b} w_{ε_{m}} (0) ‖}_{L^{2} (Ω \times Γ)}^{2} \\ ⩽ ε_{λ} | Y | \int_{Γ_{ε_{λ}}^{}} {| w_{ε_{λ}} (0, x) |}^{2} d σ_{x} + ε_{m} | Y | \int_{Γ_{ε_{m}}^{}} {| w_{ε_{m}} (0, x) |}^{2} d σ_{x} \\ ⩽ | Y | max (‖ w_{I ε_{λ}} ‖_{L^{\infty} (Γ_{ε_{λ}}^{})}, ‖ w_{I ε_{m}} ‖_{L^{\infty} (Γ_{ε_{m}}^{})}) (\sum_{k = 1}^{N_{ε}} \int_{Γ_{k}} d σ_{y}) ({ε_{λ}}^{n} + {ε_{m}}^{n}) \\ (6.4) & = C ({ε_{λ}}^{n} + {ε_{m}}^{n}) \end{array}$ and the second term of the r.h.s. can be estimated as $\begin{array}{l} \int_{Ω \times Γ} {| T_{ε_{λ}}^{b} R (u_{ε_{λ}}, v_{ε_{μ}}) - T_{ε_{m}}^{b} R (u_{ε_{m}}, v_{ε_{m}}) |}^{2} d x d σ_{y} \\ ⩽ \int_{Ω \times Γ} ({| T_{ε_{λ}}^{b} R (u_{ε_{λ}}, v_{ε_{λ}}) - T_{ε_{λ}}^{b} R (u_{0}, v_{0}) |}^{2} + {| T_{ε_{λ}}^{b} R (u_{0}, v_{0}) - R (u_{0}, v_{0}) |}^{2}) d x d σ_{y} \\ (6.5) & + \int_{Ω \times Γ} ({| T_{ε_{m}}^{b} R (u_{0}, v_{0}) - R (u_{0}, v_{0}) |}^{2} + {| T_{ε_{m}}^{b} R (u_{0}, v_{0}) - T_{ε_{m}}^{b} R (u_{ε_{m}}, v_{ε_{m}}) |}^{2}) d x d σ_{y} . \end{array}$ We see that $T_{ε_{λ}}^{b} R (u_{0}, v_{0}) \to R (u_{0}, v_{0})$ strongly in $L^{2} (S \times Ω \times Γ)$ since $R (u_{0}, v_{0})$ is independent of y. It follows from Lemma A.9 and the first part of Lemma 6.3 that $\begin{matrix} \int_{Ω \times Γ} {| T_{ε_{λ}}^{b} R (u_{ε_{λ}}, v_{ε_{λ}}) - T_{ε_{λ}}^{b} R (u_{0}, v_{0}) |}^{2} d x d σ_{y} = ε_{λ} | Y | \int_{Γ_{ε_{λ}}^{}} {| R (u_{ε_{λ}}, v_{ε_{λ}}) - R (u_{0}, v_{0}) |}^{2} d σ_{x} ⩽ C ε_{λ} . \end{matrix}$ Thus (6.5) takes the form $\begin{matrix} (6.6) & \int_{Ω \times Γ} {| T_{ε_{λ}}^{b} R (u_{ε_{λ}}, v_{ε_{λ}}) - T_{ε_{m}}^{b} R (u_{ε_{m}}, v_{ε_{m}}) |}^{2} d x d σ_{y} ⩽ C (ε_{λ} + ε_{m}) . \end{matrix}$ Putting together (6.4) and (6.6) we obtain $\begin{array}{l} {‖ T_{ε_{λ}}^{b} w_{ε_{λ}} (t) - T_{ε_{m}}^{b} w_{ε_{m}} (t) ‖}_{L^{2} (Ω \times Γ)}^{2} & ⩽ C ({ε_{λ}}^{n} + {ε_{m}}^{n}) + k_{d}^{2} T C (ε_{λ} + ε_{m}) \\ + \int_{0}^{t} {‖ T_{ε_{λ}}^{b} w_{ε_{λ}} (s) - T_{ε_{m}}^{b} w_{ε_{m}} (s) ‖}_{L^{2} (Ω \times Γ)}^{2} d s . \end{array}$ Gronwall’s inequality gives $\begin{array}{l} {‖ T_{ε_{λ}}^{b} w_{ε_{λ}} (t) - T_{ε_{m}}^{b} w_{ε_{m}} (t) ‖}_{L^{2} (Ω \times Γ)}^{2} & ⩽ {C ({ε_{λ}}^{n} + {ε_{m}}^{n}) + k_{d}^{2} T C (ε_{λ} + ε_{m})} e^{T} \\ \to 0 as λ, m \to \infty . \end{array}$ By the completeness of the space $L^{2} (S \times Ω \times Γ)$ , we can derive that $T_{ε}^{b} (w_{ε}) \to ζ$ in $L^{2} (S \times Ω \times Γ)$ . That imply, $w_{ε} \overset{2}{⇀} ζ$ in $L^{2} (S \times Ω \times Γ)$ (cf. Lemma A.8) but we already prove that $w_{ε} \overset{2}{⇀} w_{0}$ in $L^{2} (S \times Ω \times Γ)$ . Hence $ζ = w_{0}$ . □
Proof of Theorem 3.2.
We make use of the two-scale convergence technique to obtain the macroscopic equations. We test the PDEs by $Φ_{i} (t, x, \frac{x}{ε}) = ϕ_{i} (t, x) + ε ψ_{i} (t, x, \frac{x}{ε})$ where $ϕ_{i} \in L^{2} (S; H^{1, 2} (Ω))$ , $ψ_{i} (t, x, \frac{x}{ε}) \in L^{2} (S \times Ω; H_{per}^{1, 2} (Y))$ in such a way that $\frac{\partial ϕ_{i}}{\partial t} \in L^{2} (S; {(H^{1, 2} (Ω))}^{})$ , $\frac{\partial ψ_{i}}{\partial t} \in L^{2} (S \times Ω; {(H_{per}^{1, 2} (Y))}^{})$ , for $i = 1, 2$ . For the ODE we take $ψ (t, x, \frac{x}{ε}) \in L^{2} (S \times Ω; H_{per}^{1, 2} (Y))$ as a test function such that $\frac{\partial ψ}{\partial t} \in L^{2} (S \times Ω; {(H_{per}^{1, 2} (Y))}^{})$ . Now, multiplying the PDEs (2.3a) and (2.3e) by $Φ_{1}$ and $Φ_{2}$ and the ODE (2.3i) by ψ and adding up yields $\begin{array}{l} \int_{0}^{T} \int_{Ω_{ε}^{p}} \frac{\partial u_{ε}}{\partial t} [ϕ_{1} (t, x) + ε ψ_{1} (t, x, \frac{x}{ε})] d x d t \\ + \int_{0}^{T} \int_{Ω_{ε}^{p}} D_{1} \nabla u_{ε} [\nabla_{x} ϕ_{1} + ε \nabla_{x} ψ_{1} + \nabla_{y} ψ_{1}] d x d t \\ + ε \int_{0}^{T} \int_{Γ_{ε}^{}} \frac{\partial w_{ε}}{\partial t} [ϕ_{1} (t, x) + ε ψ_{1} (t, x, \frac{x}{ε})] d σ_{x} d t \\ + \int_{0}^{T} \int_{Ω_{ε}^{p}} \frac{\partial v_{ε}}{\partial t} [ϕ_{2} (t, x) + ε ψ_{2} (t, x, \frac{x}{ε})] d x d t \\ + \int_{0}^{T} \int_{Ω_{ε}^{p}} D_{2} \nabla v_{ε} [\nabla_{x} ϕ_{2} + ε \nabla_{x} ψ_{2} + \nabla_{y} ψ_{2}] d x d t \\ + ε \int_{0}^{T} \int_{Γ_{ε}^{}} \frac{\partial w_{ε}}{\partial t} [ϕ_{2} (t, x) + ε ψ_{2} (t, x, \frac{x}{ε})] d σ_{x} d t + \int_{0}^{T} \int_{Γ_{ε}^{}} \frac{\partial w_{ε}}{\partial t} ψ (t, x, \frac{x}{ε}) d σ_{x} d t \\ - k_{d} \int_{0}^{T} \int_{Γ_{ε}^{}} (R (u_{ε}, v_{ε}) - z_{ε}) ψ (t, x, \frac{x}{ε}) d σ_{x} d t = 0 . \end{array}$ Then integrate by parts in t to obtain $\begin{array}{l} - \int_{0}^{T} \int_{Ω_{ε}^{p}} u_{ε} (\frac{\partial ϕ_{1}}{\partial t} + ε \frac{\partial ψ_{1}}{\partial t}) d x d t + \int_{Ω_{ε}^{p}} u_{ε} (T, x) (ϕ_{1} (T, x) + ε ψ_{1} (T, x, \frac{x}{ε})) d x \\ - \int_{Ω_{ε}^{p}} u_{ε} (0, x) (ϕ_{1} (0, x) + ε ψ_{1} (0, x, \frac{x}{ε})) d x \\ + \int_{0}^{T} \int_{Ω_{ε}^{p}} D_{1} \nabla u_{ε} [\nabla_{x} ϕ_{1} + ε \nabla_{x} ψ_{1} + \nabla_{y} ψ_{1}] d x d t \\ + ε \int_{0}^{T} \int_{Γ_{ε}^{}} \frac{\partial w_{ε}}{\partial t} (ϕ_{1} (t, x) + ε ψ_{1} (t, x, \frac{x}{ε})) d σ_{x} d t - \int_{0}^{T} \int_{Ω_{ε}^{p}} v_{ε} (\frac{\partial ϕ_{2}}{\partial t} + ε \frac{\partial ψ_{2}}{\partial t}) d x d t \\ + \int_{Ω_{ε}^{p}} v_{ε} (T, x) (ϕ_{2} (T, x) + ε ψ_{2} (T, x, \frac{x}{ε})) d x \\ - \int_{Ω_{ε}^{p}} v_{ε} (0, x) (ϕ_{2} (0, x) + ε ψ_{2} (0, x, \frac{x}{ε})) d x \\ + \int_{0}^{T} \int_{Ω_{ε}^{p}} D_{2} \nabla v_{ε} [\nabla_{x} ϕ_{2} + ε \nabla_{x} ψ_{2} + \nabla_{y} ψ_{2}] d x d t \\ + ε \int_{0}^{T} \int_{Γ_{ε}^{}} \frac{\partial w_{ε}}{\partial t} (ϕ_{2} + ε ψ_{2}) d σ_{x} d t - \int_{0}^{T} \int_{Γ_{ε}^{}} w_{ε} (t, x) \frac{\partial ψ}{\partial t} d σ_{x} d t \\ + \int_{Γ_{ε}^{}} w_{ε} (T, x) ψ (T, x, \frac{x}{ε}) d σ_{x} - \int_{Γ_{ε}^{}} w_{ε} (0, x) ψ (0, x, \frac{x}{ε}) d σ_{x} \\ (6.7) & - k_{d} \int_{0}^{T} \int_{Γ_{ε}^{}} (R (u_{ε}, v_{ε}) - z_{ε}) ψ (t, x, \frac{x}{ε}) d σ_{x} d t = 0 . \end{array}$ Now we pass $ε \to 0$ in (6.7) and get $\begin{array}{l} - \int_{0}^{T} \int_{Ω} \int_{Y^{p}} u_{0} (t, x) \frac{\partial ϕ_{1}}{\partial t} d x d y d t + \int_{Ω \times Y^{p}} u_{0}^{} (x) ϕ_{1} (T, x) d x d y - \int_{Ω \times Y^{p}} u_{I 0} (x) ϕ_{1} (0, x) d x d y \\ + \int_{0}^{T} \int_{Ω} \int_{Y^{p}} D_{1} (\nabla u_{0} (t, x) + \nabla_{y} u_{1} (t, x, y)) (\nabla_{x} ϕ_{1} + \nabla_{y} ψ_{1}) d x d y d t \\ + \int_{0}^{T} \int_{Ω \times Γ} \frac{\partial w_{0}}{\partial t} ϕ_{1} d x d σ_{y} d t - \int_{0}^{T} \int_{Ω} \int_{Y^{p}} v_{0} (t, x) \frac{\partial ϕ_{2}}{\partial t} d x d y d t \\ + \int_{Ω \times Y^{p}} v_{0}^{} (x) ϕ_{2} (T, x) d x d y - \int_{Ω \times Y^{p}} v_{I 0} (x) ϕ_{2} (0, x) d x d y + \int_{0}^{T} \int_{Ω \times Γ} \frac{\partial w_{0}}{\partial t} ϕ_{2} d x d σ_{y} d t \\ + \int_{0}^{T} \int_{Ω} \int_{Y^{p}} D_{2} (\nabla v_{0} (t, x) + \nabla_{y} v_{1} (t, x, y)) (\nabla_{x} ϕ_{2} + \nabla_{y} ψ_{2}) d x d y d t \\ - \int_{0}^{T} \int_{Ω \times Γ} w_{0} (t, x, y) \frac{\partial ψ}{\partial t} d x d σ_{y} d t + \int_{Ω \times Γ} w_{0}^{} (x, y) ψ (T, x, y) d x d σ_{y} \\ - \int_{Ω \times Γ} w_{I 0} (x, y) ψ (0, x, y) d x d σ_{y} \\ (6.8) & - k_{d} \int_{0}^{T} \int_{Ω \times Γ} (R (u_{0}, v_{0}) - z_{0}) ψ (t, x, y) d x d σ_{y} d t = 0 . \end{array}$ We now derive the two-scale limit equations by making special choices of the test functions. First we choose $ϕ_{1}$ , $ϕ_{2}$ , $ψ_{1}$ , $ψ_{2}$ all equal to zero and ψ as above with the property $ψ (0, x, y) = 0 = ψ (T, x, y)$ then we have the macroscopic equation for the ODE as $\begin{matrix} (6.9) & \frac{\partial w_{0}}{\partial t} = k_{d} (R (u_{0}, v_{0}) - z_{0}) in S \times Ω \times Γ . \end{matrix}$ Next we choose $ϕ_{1}$ as above such that $ϕ_{1} (0, x) = 0 = ϕ_{1} (T, x)$ and the remaining test functions all zero in (6.8) and that leads to $\begin{array}{l} \int_{0}^{T} \int_{Ω} \frac{\partial u_{0} (t, x)}{\partial t} ϕ_{1} d x d t + \int_{0}^{T} \int_{Ω} \int_{Y^{p}} \frac{D_{1}}{| Y^{p} |} (\nabla u_{0} (t, x) + \nabla_{y} u_{1} (t, x, y)) \nabla_{x} ϕ_{1} d x d y d t \\ (6.10) & + \int_{0}^{T} \int_{Ω} \int_{Γ} \frac{1}{| Y^{p} |} \frac{\partial w_{0}}{\partial t} ϕ_{1} d x d σ_{y} d t = 0 . \end{array}$ We then set all other test functions is equal to zero except $ψ_{1}$ to derive the cell problem: $\begin{matrix} \int_{0}^{T} \int_{Ω} \int_{Y^{p}} D_{1} (\nabla u_{0} (t, x) + \nabla_{y} u_{1} (t, x, y)) \nabla_{y} ψ_{1} d x d y d t = 0 . \end{matrix}$ For the choice of $u_{1} (t, x, y) = \sum_{j = 1}^{n} \frac{\partial u_{0}}{\partial x_{j}} (t, x) l_{j} (y) + p (x)$ we have for $j = 1, 2, \dots, n$ , $l_{j} (y) \in H_{per}^{1, 2} (Y^{p})$ are the Y-periodic solution of the cell problems $\begin{array}{l} (6.11a) & \nabla_{y} . (\nabla_{y} l_{j} + e_{j}) = 0 \forall y \in Y^{p}, \\ (6.11b) & (\nabla_{y} l_{j} + e_{j}) . \vec{n} = 0 on Γ, \\ (6.11c) & y \mapsto l_{j} (y) is Y -periodic . \end{array}$ Now for the choice of $u_{1}$ as above equation (6.10) can be rewritten as $\begin{array}{l} \int_{0}^{T} \int_{Ω} \frac{\partial u_{0} (t, x)}{\partial t} ϕ_{1} d x d t + \int_{0}^{T} \int_{Ω} \int_{Y^{p}} \frac{D_{1}}{| Y^{p} |} (δ_{i j} + \sum_{i, j = 1}^{n} \frac{\partial l_{j}}{\partial y_{i}}) \frac{\partial u_{0} (t, x)}{\partial x_{j}} \nabla_{x} ϕ_{1} d x d y d t \\ + \int_{0}^{T} \int_{Ω} P (t, x) ϕ_{1} d x d t = 0, \end{array}$ which can be simplified as $\begin{matrix} (6.12) & \frac{\partial u_{0} (t, x)}{\partial t} + \nabla . (- A \nabla u_{0} (t, x)) + P (t, x) = 0 in S \times Ω, \end{matrix}$ where $\begin{matrix} A = {(a_{i j})}_{1 ⩽ i, j ⩽ n} = \int_{Y^{p}} \frac{D_{1}}{| Y^{p} |} (δ_{i j} + \sum_{i, j = 1}^{n} \frac{\partial l_{j}}{\partial y_{i}}) d y, P (t, x) = \int_{Γ} \frac{1}{| Y^{p} |} \frac{\partial w_{0}}{\partial t} d σ_{y} . \end{matrix}$ After that we take $ϕ_{2}$ as earlier and vanishing at $t = 0$ and $t = T$ together with $ϕ_{1}$ , $ψ_{1}$ , $ψ_{2}$ , ψ all zero and derive $\begin{array}{l} \int_{0}^{T} \int_{Ω} \frac{\partial v_{0} (t, x)}{\partial t} ϕ_{2} d x d t + \int_{0}^{T} \int_{Ω} \int_{Y^{p}} \frac{D_{2}}{| Y^{p} |} (\nabla v_{0} (t, x) + \nabla_{y} v_{1} (t, x, y)) \nabla_{x} ϕ_{2} d x d y d t \\ (6.13) & + \int_{0}^{T} \int_{Ω} \int_{Γ} \frac{1}{| Y^{p} |} \frac{\partial w_{0}}{\partial t} ϕ_{2} d x d σ_{y} d t = 0 . \end{array}$ We keep $ψ_{2}$ non-zero and all other test functions are equal to zero to compute the cell problem for $M_{2}$ : $\begin{matrix} \int_{0}^{T} \int_{Ω} \int_{Y^{p}} D_{2} (\nabla v_{0} (t, x) + \nabla_{y} v_{1} (t, x, y)) \nabla_{y} ψ_{2} d x d y d t = 0 . \end{matrix}$ So for $v_{1} (t, x, y) = \sum_{j = 1}^{n} \frac{\partial v_{0}}{\partial x_{j}} (t, x) l_{j} (y) + q (x)$ the cell problems are given by (6.11a)–(6.11c). Recalling (6.13), we get the upscaled model for $M_{2}$ as $\begin{matrix} (6.14) & \frac{\partial v_{0} (t, x)}{\partial t} + \nabla . (- B \nabla v_{0} (t, x)) + P (t, x) = 0 in S \times Ω, \end{matrix}$ where $\begin{matrix} B = {(b_{i j})}_{1 ⩽ i, j ⩽ n} = \int_{Y^{p}} \frac{D_{2}}{| Y^{p} |} (δ_{i j} + \sum_{i, j = 1}^{n} \frac{\partial l_{j}}{\partial y_{i}}) d y, P (t, x) = \int_{Γ} \frac{1}{| Y^{p} |} \frac{\partial w_{0}}{\partial t} d σ_{y} . \end{matrix}$ We now substitute (6.9)–(6.14) in (6.8) to identify the initial conditions. We define $ϕ_{1} (T, x) = 0$ together with $ϕ_{2} (T, x) = 0 = ϕ_{2} (0, x)$ and $ψ (0, x, y) = 0 = ψ (T, x, y)$ and deduce $\begin{matrix} \int_{Ω \times Y^{p}} (u_{0} (0, x) - u_{I 0} (x)) ϕ_{1} (0, x) d x d y ⟹ u_{0} (0, x) = u_{I 0} (x) . \end{matrix}$ Similarly, setting all the other test functions as zero except $ϕ_{2} (0, x)$ and $ψ (0, x, y)$ we have our initial conditions as $v_{0} (0, x) = v_{I 0} (x)$ and $w_{0} (0, x) = w_{I 0} (x, y)$ . We can calculate the final conditions as $u_{0} (T, x) = u_{0}^{} (x)$ , $v_{0} (T, x) = v_{0}^{} (x)$ and $w_{0} (T, x, y) = w_{0}^{} (x, y)$ . We finally derive the strong form of the homogenized system as (3.3a)–(3.5c) by fixing $ϕ_{1} (T, x) = ϕ_{2} (T, x) = ψ (T, x, y) = 0$ .

Now we need to characterize the two-scale limit of the multivalued dissolution rate term. According to Lemma 6.3, $T_{ε}^{b} (w_{ε}) \to w_{0}$ in $L^{2} (S \times Ω \times Γ)$ . Therefore by corollary on page 53 of [59], there exists a subsequence ${T_{ε}^{b} (w_{ε})}$ (still denoted by the same symbol) pointwise convergent to $w_{0}$ a.e. in $S \times Ω \times Γ$ . i.e. $\begin{matrix} lim_{ε \to 0} T_{ε}^{b} (w_{ε}) = w_{0} a.e. in S \times Ω \times Γ . \end{matrix}$ While $T_{ε}^{b} (w_{ε}) ⩾ 0$ , $w_{0} ⩾ 0$ so we have to consider two different cases.

Case 1: Let $w_{0} > 0$ .

As $w_{0}$ is the pointwise limit of $T_{ε}^{b} (w_{ε})$ therefore for any $η > 0$ there exists a $δ > 0$ such that $| ε | < δ ⟹ | T_{ε}^{b} (w_{ε}) - w_{0} | < η$ . Let us choose $η = \frac{w_{0}}{2}$ such that $T_{ε}^{b} (w_{ε}) > \frac{w_{0}}{2}$ . So by definition (6.3) we have $T_{ε}^{b} z_{ε} = 1 ⟹ z_{0} = 1$ as a consequence of weak convergence.

Case 2: Let $w_{0} = 0$ .

Since $\frac{\partial w_{ε}}{\partial t} \in L^{2} (S \times Γ_{ε}^{})$ thus for a test function $ϕ \in C_{0}^{\infty} (S \times Ω)$ we get $\begin{matrix} \int_{0}^{T} \int_{Γ_{ε}^{}} \frac{\partial w_{ε}}{\partial t} ϕ d σ_{x} d t = - \int_{0}^{T} \int_{Γ_{ε}^{}} w_{ε} \frac{\partial ϕ}{\partial t} d σ_{x} d t \to 0 as ε \to 0 because lim_{ε \to 0} w_{ε} = 0 . \end{matrix}$ since ϕ is arbitrary, it follows that $\frac{\partial w_{ε}}{\partial t} \overset{2}{⇀} 0$ that means, $\frac{\partial}{\partial t} (T_{ε}^{b} (w_{ε})) ⇀ 0$ in $L^{2} (S \times Ω \times Γ)$ . Then $\begin{matrix} \int_{0}^{T} \int_{Ω} \int_{Γ} \frac{\partial T_{ε}^{b} (w_{ε})}{\partial t} ϕ d x d σ_{y} d t = k_{d} \int_{0}^{T} \int_{Ω} \int_{Γ} (T_{ε}^{b} (R (u_{ε}, v_{ε})) - T_{ε}^{b} z_{ε}) ϕ d x d σ_{y} d t, \end{matrix}$ passing the limit $ε \to 0$ leads to $\begin{matrix} \int_{0}^{T} \int_{Ω} \int_{Γ} 0 ϕ d x d σ_{y} d t = k_{d} \int_{0}^{T} \int_{Ω} \int_{Γ} (R (u_{0}, v_{0}) - z_{0}) ϕ d x d σ_{y} d t, \end{matrix}$ i.e. $z_{0} = R (u_{0}, v_{0})$ and $0 ⩽ z_{0} ⩽ 1$ when $w_{0} = 0$ on Γ. □

We can show the existence, positivity and uniqueness of a solution of the macromodel as below.

Positivity: We multiply (3.3a) by $- {[u_{0}]}_{-}$ and integration over $S \times Ω$ yields $\begin{matrix} (6.15) & {‖ {[u_{0} (t)]}_{-} ‖}_{Ω}^{2} + 2 {⟨ A \nabla {[u_{0}]}_{-}, \nabla {[u_{0}]}_{-} ⟩}_{{(Ω)}^{t}} = {‖ {[u_{I 0}]}_{-} ‖}_{Ω}^{2} - 2 {⟨ P (t, x), - {[u_{0}]}_{-} ⟩}_{{(Ω)}^{t}} . \end{matrix}$ The initial condition term in the r.h.s. vanishes due to (A1). Utilizing the ellipticity of the matrix A and the assumption (A2) we can write $\begin{matrix} {⟨ P (t, x), - {[u_{0}]}_{-} ⟩}_{{(Ω)}^{t}} = - \frac{k_{d}}{| Y^{p} |} {⟨ z_{0}, - {[u_{0}]}_{-} ⟩}_{{(Ω \times Γ)}^{t}} ⩾ 0 . \end{matrix}$ Consequently, (6.15) takes the form $\begin{array}{l} {‖ {[u_{0} (t)]}_{-} ‖}_{Ω}^{2} + 2 α {‖ {[u_{0}]}_{-} ‖}_{{(Ω)}^{t}}^{2} ⩽ 0 ⟹ {[u_{0} (t)]}_{-} = 0 for a.e. t \in S . \end{array}$ The same procedure leads to ${[v_{0} (t)]}_{-} = 0$ for a.e. $t \in S$ . Next we use $- {[w_{0}]}_{-}$ as a multiplier in (3.5a) and integrate over $S \times Ω \times Γ$ to get $\begin{array}{l} {‖ {[w_{0} (t)]}_{-} ‖}_{Ω \times Γ}^{2} - {‖ {[w_{I 0}]}_{-} ‖}_{Ω \times Γ}^{2} & = k_{d} {⟨ R (u_{0}, v_{0}) - z_{0}, - {[w_{0}]}_{-} ⟩}_{{(Ω \times Γ)}^{t}} \\ = k_{d} {⟨ R (u_{0}, v_{0}), - {[w_{0}]}_{-} ⟩}_{{(Ω \times Γ)}^{t}} . \end{array}$ Since $z_{0} = 0$ for $w_{0} ⩽ 0$ . Now as $u_{0}, v_{0} ⩾ 0$ so $R (u_{0}, v_{0}) = k \frac{k_{1} u_{0} k_{2} v_{0}}{{(1 + k_{1} u_{0} + k_{2} v_{0})}^{2}} ⩾ 0$ and $R (u_{0}, v_{0}) (- {[w_{0}]}_{-}) ⩽ 0$ . Thus we have, ${[w_{0} (t)]}_{-} = 0$ for a.e. $t \in S$ .

Existence: We can calculate the estimate (3.6) by proceeding as in Section 4. The proof of existence is based on Galerkin method since the source term $P (t, x) \in L^{2} (S \times Ω)$ .

Uniqueness: Suppose there are two weak solutions to the macroscopic equations (3.4a)–(3.5c). We denote $V_{0} = v_{01} - v_{02}$ , $W_{0} = w_{01} - w_{02}$ and $Z_{0} = z_{01} - z_{02}$ . Now $V_{0}$ and $W_{0}$ satisfy the weak formulations $\begin{array}{l} (6.16) & {⟨ \frac{\partial V_{0}}{\partial t}, θ ⟩}_{{(Ω)}^{t}} + {⟨ B \nabla V_{0}, \nabla θ ⟩}_{{(Ω)}^{t}} + \frac{1}{| Y^{p} |} \int_{0}^{t} \int_{Ω \times Γ} \frac{\partial W_{0}}{\partial t} θ d x d σ_{y} d t = 0, \\ (6.17) & {⟨ \frac{\partial W_{0}}{\partial t}, η ⟩}_{{(Ω \times Γ)}^{t}} = k_{d} {⟨ R (u_{0}, v_{01}) - R (u_{0}, v_{02}) - Z_{0}, η ⟩}_{{(Ω \times Γ)}^{t}} . \end{array}$ With the choice of $η = W_{0}$ and the monotonicity of $Z_{0}$ , (6.17) takes the form $\begin{matrix} {‖ W_{0} (t) ‖}_{Ω \times Γ}^{2} ⩽ k_{d}^{2} L_{R}^{2} ‖ V_{0} ‖_{{(Ω \times Γ)}^{t}}^{2} + \int_{0}^{t} {‖ W_{0} (s) ‖}_{Ω \times Γ}^{2} d s . \end{matrix}$ Gronwall’s Inequality implies $\begin{matrix} (6.18) & {‖ W_{0} (t) ‖}_{Ω \times Γ}^{2} ⩽ k_{d}^{2} L_{R}^{2} ‖ V_{0} ‖_{{(Ω \times Γ)}^{t}}^{2} e^{t} . \end{matrix}$ Take $θ = V_{0}$ in (6.16) to obtain $\begin{matrix} \frac{1}{2} {‖ V_{0} (t) ‖}_{Ω}^{2} + {⟨ B \nabla V_{0}, \nabla θ ⟩}_{{(Ω)}^{t}} ⩽ \frac{1}{| Y^{p} |} \int_{0}^{t} \int_{Ω \times Γ} | \frac{\partial W_{0}}{\partial t} | | V_{0} | d x d σ_{y} d t . \end{matrix}$ Now using the ellipticity of B and the fact that $V_{0}$ is independent of y we get $\begin{matrix} {‖ V_{0} (t) ‖}_{Ω}^{2} + 2 α ‖ \nabla V_{0} ‖_{{(Ω)}^{t}}^{2} ⩽ \frac{2 k_{d} L_{R} | Γ |}{| Y^{p} |} \int_{0}^{t} {‖ V_{0} (s) ‖}^{2} d s . \end{matrix}$ So application of Gronwall’s Inequality gives $\begin{matrix} {‖ V_{0} (t) ‖}_{Ω}^{2} ⟹ v_{01} (t) = v_{02} (t) . \end{matrix}$ Consequently, We have from (6.18) that $\begin{matrix} {‖ W_{0} (t) ‖}_{Ω \times Γ}^{2} ⟹ w_{01} (t) = w_{02} (t) . \end{matrix}$
7. Conclusion

We studied a system of diffusion–reaction equations coupled with a precipitation-dissolution model. We started with a micro model in a heterogeneous porous medium. We also assumed that the two mobile species have different diffusion coefficients. We first proved the existence of a unique global in time weak solution and further performed the homogenization of the micro model in this paper. The upscaled model has a similar form as the micro problem and it can be verified via numerical simulation that this method approximates the micro problem. The numerical validation for similar type of model can be found in [32,50]. This paper is the first part of the series of papers where we will explore the possibility of different diffusion coefficients of mobile species together with non-linear reaction rate terms. As mentioned earlier that having different diffusion coefficients raises serious questions on the existence of global in time solutions. To our knowledge for a general model, this question is still unknown and we will address this question elsewhere.

Footnotes

Acknowledgements

The authors are grateful to the anonymous referee(s) for his/her/their constructive inputs that helped us to considerably improve the initial manuscript. The authors are also thankful to the editor for a smooth and prompt communication regarding the manuscript.

Appendix

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Diffusion–reaction–dissolution–precipitation model in a heterogeneous porous medium with nonidentical diffusion coefficients: Analysis and homogenization

Abstract

Keywords

1. Introduction

3. Mathematical preliminaries

6. Proof of Theorem 3.2 (Homogenization of the problem ( P ε ) via two-scale convergence)

Footnotes

Acknowledgements

Appendix

References

6. Proof of Theorem 3.2 (Homogenization of the problem $(P_{ε})$ via two-scale convergence)