Existence of the Global Solution of the Problem (4.2.36)-(4.2.37) 64

Theorem 4.2.1.2.1. Let (ˆu_εδ,ˆv_εδ)∈ F_ε^u× G_ε^v. Then there exists a positive global solution w_εδ∈ H^w_ε of the problem (4.2.36)-(4.2.37).

Proof. (i) Positivity: This is shown in the lemma 4.2.1.1.2.

(ii)Existence of the local solution: Letx∈Ω^p_εbe fixed but chosen arbitrarily. The function ψ_δm(.) is continuous w.r.t. w_εδm for every fixed t. Since

|ψδ(w_εδ)−ψ_δ( ¯w_εδ)|²_I=

I2

m=1

(((ψ_δ(w_εδm)−ψ_δ( ¯w_εδm)⁽⁽(²,

the vector functionψ_δ(.) is also continuous w.r.t. w_εδ for every fixed t. Moreover,ψ_δ(w_εδ) is measurable w.r.t. tand

|ψ_δ(w_εδ)|_I=

⎡

⎣^I2

m=1

(((ψ_δ(w_εδm)⁽⁽(²

⎤

⎦

12

≤

⎡

⎣^I2

m=1

1

⎤

⎦

12

=I2

12 =:m(t),

i.e.,ψ_δ(.) is bounded by a measurable functionm(.). Thus the application of theCaratheodory’s theorem yields the existence of an absolutely continuous function w_εδ(x) on [0, T1) which solves (4.2.36)-(4.2.37) (cf. theorem 2.1.1 in [CL55]), i.e., w_εδ(x)∈[H^1,1(0, T1)]^I2, where T1≤T, i.e, the solution is local. Sincexis arbitrary, for a.e. x∈Ω^p_ε,w_εδ(x)∈[H^1,1(0, T1)]^I2. For allφ∈[C₀^∞((0, T1))]^I2, the weak formulation of (4.2.36) is given by

" _T1

0

/∂w_εδ(t)

∂t , φ(t) 0

I2

dt=−k_d^{" T1}

0 ψ_δ(w_εδ(t)), φ(t)_I2dt, i.e.,

" _T1

0

/

w_εδ(t),∂φ(t)

∂t 0

I2

dt=k_d

" _T1

0 ψ_δ(w_εδ(t)), φ(t)_I2 dt. (4.2.44) Since w_εδ is a function of both x and t, we shall show that the weak derivative of w_εδ depends on bothx and tand belongs toL^p((0, T);L^p(Γ_ε))^I2 which accomplishes the claim thatw_εδ∈ H^w_ε. Let us choose another functionζ∈C₀^∞(Γ_ε). Multiplying (4.2.44) byζ and integrating over Γ_ε, we obtain

" _T1

0

"

Γ_ε

/

w_εδ(t, x),∂φ(t)

∂t ζ(x) 0

I2

ds dt=k_d

" _T1

0

"

Γ_εψδ(w_εδ(t, x)), φ(t)ζ(x)_I2 ds dt, (4.2.45)

4.2. Model M2 65 for all φ∈[C₀^∞((0, T1))]^I2 and ζ∈C₀^∞(Γ_ε). As φ∈[C₀^∞((0, T1))]^I2 and ζ∈C₀^∞(Γ_ε), φζ∈ L^q((0, T1);L^q(Γ_ε))^I2 such that the weak time derivative is inL^q((0, T1);L^q(Γ_ε))^I2, i.e.,

" _T1

0

/

w_εδ,∂φ(t)

∂t ψ 0

L^p(Γ_ε)^I2×L^q(Γ_ε)^I2 dt=k_d

" _T1

0 ψ_δ(w_εδ(t)), φ(t)ψ_Lp(Γ_ε)^I2×L^q(Γ_ε)^I2 dt.

Therefore for anyη∈L^q((0, T1);L^q(Γ_ε))^I2 such that ^∂η_∂t ∈L^q((0, T1);L^q(Γ_ε))^I2, we have

" _T1

0

/

w_εδ(t, x),∂η(t, x)

∂t 0

L^p(Γ_ε)^I2×L^q(Γ_ε)^I2

dt=k_d

" _T1

0 ψ_δ(w_εδ(t, x)), η(t, x)_Lp(Γ_ε)^I2×L^q(Γ_ε)^I2 dt.

This leads to the fact that the weak derivative of w_εδ, ^∂w_∂t^εδ ∈L^p((0, T1);L^p(Γ_ε))^I2, i.e., w_εδ∈H^1,p((0, T1);L^p(Γ_ε))^I2.

(iii) Extension of the solution: Clearly,

|ψ_δ(w_εδ)|_I=

⎡

⎣^I2

k=1

(((ψ_δ(w_εδi)⁽⁽(²

⎤

⎦

12

≤

⎡

⎣^I2

i=1

1²

⎤

⎦

12

=I2

12,

i.e., the r.h.s. of (4.2.36)-(4.2.37) is bounded. Therefore from corollary II.3.4 of [MM02], there exists a global solution of the problem (4.2.36)-(4.2.37) on [0, T) for any T >0.

4.2.1.3 Existence of the Global Solution of the Problem (4.2.31)-(4.2.37)

Lemma 4.2.1.3.1. Suppose that p > n+ 2 is fixed and κ∈L^∞(∂Ω_in). If we define the map Q_∂Ω_in: [H^1,p(Ω^p_ε)]^I2 →[H^1,q(Ω^p_ε)]^I2∗ by

Q_∂Ω_in(φ), ξ:=

I2

k=1

Q_∂Ω_in(φ_k), ξ_k:=

I2

k=1

"

∂Ω_inκφ_kξ_kds for ξ∈[H^1,q(Ω^p_ε)]^I2, (4.2.46) then Q_∂Ω_in is well defined and continuous.

Proof. Forφ∈H^1,p(Ω^p_ε)^I2, the map is given by Q_∂Ω_in(φ), ξ=

I2

k=1

"

∂Ω_inκφ_kξ_kds

≤ ||κ||_L∞(∂Ω_in) I2

k=1

"

∂Ω_in|φ_k||ξ_k|ds

≤ ||κ||_L∞(∂Ω_in) I2

k=1

"

∂Ω|φ_k||ξ_k|ds

≤ ||κ||_L∞(∂Ω_in) I2

k=1

||φk||_Lp(∂Ω)||ξk||_Lq(∂Ω). (4.2.47) From theorem 3.4.2.2, we know that for 1≤p <∞ and φ_k ∈H^1,p(Ω^p_ε), there exists an extension ˜φ_k of φ_k (for the sake of notation we still denote the extension byφ_k) such that

||φ_k||_H1,p(Ω)≤C||φ_k||_H1,p(Ω^p_ε), (4.2.48) where C is independent of ε. Also from theorem B.5, for a domain Ω with sufficiently smooth boundary and for 1≤p <∞, there exists a bounded linear operatorT:H^1,p(Ω)→ L^p(∂Ω) such that forφ_k∈H^1,p(Ω^p_ε), T φ_k:=φ_k|_∂Ω and

||φk||_Lp(∂Ω)≤C||φk||_H1,p(Ω), (4.2.49)

whereC depends onpand Ω only. Combining (4.2.48) and (4.2.49), we obtain

||φ_k||_Lp(∂Ω)≤C||φ_k||_H1,p(Ω)≤C||φ_k||_H1,p(Ω^p_ε). (4.2.50) An inequality similar to (4.2.50) holds forξ_k too, i.e.,

||ξ_k||_Lq(∂Ω)≤C||ξ_k||_H1,q(Ω)≤C||ξ_k||_H1,q(Ω^p_ε). (4.2.51) Using (4.2.50) and (4.2.51) in (4.2.47), we get

|Q_∂Ω_in(φ), ξ|

≤C

I2

k=1

||φ_k||_H1,p(Ω^p_ε)||ξ_k||_H1,q(Ω^p_ε)

≤C

⎡

⎣^I2

k=1

||φ_k||^p_H1,p(Ω^p_ε)

⎤

⎦

1 p⎡

⎣^I2

k=1

||ξ_k||^q_H1,q(Ω^p_ε)

⎤

⎦

1 q

, by discrete H¨older’s inequality

=C|||φ|||_[H1,p(Ω^p_ε)]^I2|||ξ|||_[H1,q(Ω^p_ε)]^I2

=⇒ sup

|||ξ|||_[H1,q(Ωp ε)]I2=1

|Q_∂Ω_in(φ), ξ| ≤C|||φ|||_[H1,p(Ω^p_ε)]^I2

⎛

⎝ sup

|||ξ|||_[H1,q(Ωp ε)]I2=1

|||ξ|||_[H1,q(Ω^p_ε)]^I2

⎞

⎠

=⇒ |||Q_∂Ω_in(φ)|||_[H1,q(Ω^p_ε)^∗]^I2 ≤C|||φ|||_[H1,p(Ω^p_ε)]^I2

=⇒ ||Q_∂Ω_in||||_L([H1,p(Ω^p_ε)]^I2,[H^1,q(Ω^p_ε)^∗]^I2)≤C.

This shows that the map Q_∂Ω_in: [H^1,p(Ω^p_ε)]^I2 →[H^1,q(Ω^p_ε)]^I2∗ is well-defined and bounded,

hence continuous.

Lemma 4.2.1.3.2. Let p > n+ 2 be fixed. Then the map RΓ_ε : [L^p(Γ_ε)]^I2 →[H^1,q(Ω^p_ε)^∗]^I2 given by

RΓ_ε(υ), η:=

I2

k=1

RΓ_ε(υ_k), η_k:=

I2

k=1

ε

"

Γ_ευ_k(x)η_k(x)dσ_x, for η∈[H^1,q(Ω^p_ε)]^I2, (4.2.52) is well defined and continuous.

Proof. We proceed like previous lemma. Here the map is given as²⁹ RΓ_ε(υ), η=

I2

k=1

ε

"

Γ_ευ_k(x)η_k(x)dσ_x η∈[H^1,q(Ω^p_ε)]^I2

≤^I2

k=1

ε

"

Γ_ε|υ_k(x)|^pdσ_x

¹

p"

Γ_ε|η_k(x)|^qdσ_x

¹

q

≤^I2

k=1

ε

"

Γ_ε|υ_k(x)|^pdσ_x

¹

p ε

"

Γ_ε|η_k(x)|^qdσ_x

¹

q

≤C

I2

k=1

||υ_k||_Lp(Γ_ε)

!"

Ω^p_ε|η_k(x)|^q+ε^q|∇η_k(x)|^q

#¹

q

≤C max (1, ε^q)^1q

I2

k=1

||υ_k||_Lp(Γ_ε)||η_k||_H1,q(Ω^p_ε) 29Note that we have used the theorem3.4.1.3. Also see thatε=ε^1p+1q.

4.2. Model M2 67

≤C max (1, ε^q)^1q

⎡

⎣^I2

k=1

||υ_k||^p_Lp(Γ_ε)

⎤

⎦

1 p⎡

⎣^I2

k=1

||η_k||^q_H1,q(Ω^p_ε)

⎤

⎦

1 q

≤C max (1, ε^q)^1q |||υ|||_Lp(Γ_ε)^I2|||η|||_H1,q(Ω^p_ε)^I2.

The proof follows.

Theorem 4.2.1.3.3. Let the assumptions (4.2.1)-(4.2.10) hold true and uˆ_εδ ∈ F_ε^u. Then there exists a global weak solution (v_εδ, w_εδ)∈ G_ε^v× H^w_ε of the problem

∂v_εδ

∂t − ∇ ·(D∇v_εδ−q_εv_εδ) =S2R(ˆ¯ u_εδ, v_εδ) in (0, T)×Ω^p_ε,

−(D∇v_εδ−q_εv_εδ)·n= 0 on (0, T)×∂Ω_in,

−D∇v_εδ·n= 0 on (0, T)×∂Ω_out,

−D∇vεδ·n=ε∂w_εδ

∂t on (0, T)×Γ_ε,

v_εδ(0, x) =v0(x) in Ω^p_ε,

∂w_εδ

∂t =−kdψ_δ(w_εδ) on (0, T)×Γ_ε, w_εδ(0, x) =w0(x) on Γ_ε.

Let us denote this problem by ( ¯P_ε²_δ⁺). We follow the approach shown in section4.1.1 but here we pay special attention to the boundary terms due to the presence ofinflow-outflow boundaryconditions. We define the Lyapunov functional in the following way: Letμ⁰∈R^I2 be a solution of the linear system

S₂^Tμ⁰=−logK, (4.2.53)

where K∈R^J is the vector of equilibrium constants K_j= ^k

f j

k^b_j. Due to (4.2.4), the system (4.2.53) has a solution. Letg_k:R⁺0 →Rbe defined as

g_k(v_εδk) :=μ⁰_k−1 + logv_εδkv_εδk+e^(1−μ0k) for each k= 1,2, ..., I2. (4.2.54) We define g:R⁺0^I2 →R,f_r:R⁺0^I2 →RandF_r:L^∞₊(Ω^p_ε)^I2 →Rin a similar way as we did in section4.1.1.2. We also note that all the properties ofg_k,g,f_r andF_r from section4.1.1.2 (see propositions4.1.1.2.1 and 4.1.1.2.2) hold good. For technical reason, we add an extra term on both sides of the first PDE in the problem ( ¯P_ε²_δ⁺), i.e., for anyκ >0, we have

∂v_εδ

∂t − ∇ ·(D∇v_εδ−q_εv_εδ) +κv_εδ=S2R(ˆ¯ u_εδ, v_εδ) +κv_εδ in (0, T)×Ω^p_ε, (4.2.55)

−(D∇v_εδ−q_εv_εδ)·n= 0 on (0, T)×∂Ω_in, (4.2.56)

−D∇v_εδ·n= 0 on (0, T)×∂Ω_out, (4.2.57)

−D∇vε_δ·n=ε∂w_εδ

∂t on (0, T)×Γ_ε, (4.2.58)

v_εδ(0, x) =v0(x) in Ω^p_ε, (4.2.59)

∂w_εδ

∂t =−k_dψ_δ(w_εδ) on (0, T)×Γ_ε, (4.2.60)

w_εδ(0, x) =w0(x) on Γ_ε. (4.2.61)

We denote the problem (4.2.55)-(4.2.61) by ( ¯P_ε²⁺

δM). Since a solution of ( ¯P_ε²⁺

δM) is also a solution of ( ¯P_ε²_δ⁺), we prove the global existence of the weak solution of ( ¯P_ε²⁺

δM). Let us

define the fixed point operatorZ1:G_ε^v→ G_ε^v via v_εδ:=Z1(ˆv_εδ), where v_εδ is the solution of the linear problem given by

∂v_εδ

∂t − ∇ ·(D∇v_εδ−q_εv_εδ) +κv_εδ=S2R(ˆ¯ u_εδ,vˆ_εδ) +κˆv_εδ in (0, T)×Ω^p_ε, (4.2.62)

−(D∇vεδ−q_εv_εδ)·n= 0 on (0, T)×∂Ω_in, (4.2.63)

−D∇v_εδ·n= 0 on (0, T)×∂Ω_out, (4.2.64)

−D∇vε_δ·n=ε∂w_εδ

∂t on (0, T)×Γ_ε, (4.2.65)

v_εδ(0, x) =v0(x) in Ω^p_ε, (4.2.66)

∂w_εδ

∂t =−k_dψ_δ(w_εδ) on (0, T)×Γ_ε, (4.2.67)

w_εδ(0, x) =w0(x) on Γ_ε. (4.2.68)

Remark 4.2.1.3.4. For fixed ˆu_εδ and ˆv_εδ, the subproblem (4.2.67)-(4.2.68) has a unique positive global solution w_εδ in H^w_ε. The reformulation of the problem (4.2.62)-(4.2.66) is given by

∂v_εδ

∂t +Av_εδ =f_bound(v_εδ) +f(ˆu_εδ,vˆ_εδ), v_εδ(0, x) =v0(x),

(AP)

where A is defined as in the remark 4.1.1.1.1 and satisfies the maximal regularity on [H^1,q(Ω^p_ε)^∗]^I2,f_bound(v_εδ) :=Q_∂Ω_in(v_εδ) +RΓ_ε

−^∂w_∂t^εδ−q_ε· ∇vεδ, and f(ˆu_εδ,vˆ_εδ) :=κˆv_εδ+ S2R(ˆ¯ u_εδ,ˆv_εδ), where κ >0. Note that the theorem 3.4.3.4 implies ˆu_εδ ∈L^∞((0.T)×Ω^p_ε)^I1. Similar arguments as in remark4.1.1.1.1 leads to the fact that f∈L^p((0, T);H^1,q(Ω^p_ε)^∗)^I2. Using lemmas 4.2.1.3.1, 4.2.1.3.2 and the assumption (4.2.7), the boundary term f_bound∈ L^p((0, T);H^1,q(Ω^p_ε)^∗)^I2 . The condition v0∈^!6H^1,q(Ω^p_ε)^∗, H^1,p(Ω^p_ε)⁷₁₋1

p,p

#_I2

is fulfilled by (4.2.3). Then theorem3.3.1assures the existence of a unique solution of the problem (AP).

Therefore the operatorZ1 is well-defined.

The application of Schaefer’s fixed point theorem resides on the verification of the following two conditions:

(i) The operatorZ1 is continuous and compact.

(ii) The set{v_εδ∈ G_ε^v|∃λ∈[0,1] :v_εδ=λZ1(v_εδ)}is bounded, i.e., there exists a constant C >0 independent of v_εδ and λ such that any arbitrary solution v_εδ ∈ G_ε^v of the equation

v_εδ=λZ1(v_εδ) (4.2.69)

satisfies

|v_εδ|G_ε^v≤C. (4.2.70)

4.2. Model M2 69 Equations (4.2.62)-(4.2.68) and (4.1.69) imply

∂v_εδ

∂t − ∇ ·(D∇vεδ−q_εv_εδ) +κv_εδ =λS2R(ˆ¯ u_εδ, v_εδ) +λκv_εδ in (0, T)×Ω^p_ε, (4.2.71) v_εδ(0, x) =λv0(x) in Ω^p_ε, (4.2.72)

−(D∇v_εδ−q_εv_εδ)·n= 0 on (0, T)×∂Ω_in, (4.2.73)

−D∇vε_δ·n= 0 on (0, T)×∂Ω_out, (4.2.74)

−D∇v_εδ·n=λε∂w_εδ

∂t on (0, T)×Γ_ε, (4.2.75)

∂w_εδ

∂t =−k_dψ_δ(w_εδ) on (0, T)×Γ_ε, (4.2.76)

w_εδ(0, x) =w0(x) on Γ_ε. (4.2.77)

Note thatw_εδ is the solution of the ODE problem (4.2.76)-(4.2.77). Let us call the problem (4.2.71)-(4.2.77) as ( ¯P_ε²⁺

δλM). To show the inequality (4.2.70), we aim to prove a theorem like4.1.1.2.3 which is the following:

Theorem 4.2.1.3.5. Let r∈N(r ≥2), 0≤t≤T and 0≤λ≤1. Suppose that uˆ_εδ ∈ F_ε^u. Further assume that v_εδ ∈ G_ε^v is a solution of ( ¯P_ε²⁺

δλM). Then the following inequality holds good:

F_r(v_εδ(t))≤e^C34tF_r(v_εδ(0)) for a.e. t, (4.2.78) where C34 is independent of ε, δ, λand t.

Our starting point is the following lemma which is similar to the lemma 4.1.1.2.7.

Lemma 4.2.1.3.6. Let p > n+ 2, uˆ_εδ∈ F_ε^u and r∈N (r≥2). Assume that v_εδ ∈ G_ε^v is a solution of ( ¯P_ε²⁺

δλM) and for τ >0,

v_εδ,τ :=v_εδ+τ. (4.2.79)

Then the following inequality holds:

" _t

0

/∂v_εδ,τ

∂θ , ∂f_r(v_εδ,τ) 0

[H^1,q(Ω^p_ε)^∗]^I2×H^1,q(Ω^p_ε)^I2

dθ

≤h(t, τ, v_εδ,τ) +l(t, τ, v_εδ,τ) +C34

" _t

0

F_r(v_εδ,τ)dθ for a.e. t, (4.2.80) where h(t, τ, v_εδ,τ) and l(t, τ, v_εδ,τ) tend to zero as τ→0 for a.e. t, andC34 is independent of ε, δ, λand t.

Proof. Obviously v_εδ,τ ∈ G_ε^v. For p > n+ 2, v_εδ,τ ∈L^∞((0, T)×Ω^p_ε)^I2 (cf. theorem 3.4.3.4) and ∂f_r(v_εδ,τ)∈L^q((0, T);H^1,q(Ω^p_ε))^I2. Using ∂f_r(v_εδ,τ) in the weak formualtion of the PDE (4.2.71), we get

" _t

0 ∂_θv_εδ, ∂f_r(v_εδ,τ)_[H1,q(Ω^p_ε)^∗]^I2×[H^1,q(Ω^p_ε)]^I2 dθ+κ

I2

k=1

" _t

0

"

Ω^p_ε

v_εδk∂f_r(v_εδ,τ)_kdx dθ

−^I2

k=1

!" _t

0

"

∂Ω_inq_ε·nvε_δk(∂f_r(v_εδ,τ))_kds dθ+λεk_d

" _t

0

"

Γ_εψ_δ(w_εδk) (∂f_r(v_εδ,τ))_kdσ_xdθ

#

+

n

l=1

" _t

0

"

Ω^p_ε

/ D ∂

∂x_lv_εδ, ∂

∂x_l(∂f_r(v_εδ,τ)) 0

I2

dx dθ+

" _t

0

"

Ω^p_εq_ε· ∇v_εδ, ∂f_r(v_εδ,τ)_I2dx dθ

=

" _t

0

4

S2R(ˆ¯ u_εδ, v_εδ), ∂f_r(v_εδ,τ)⁵

[H^1,q(Ω^p_ε)^∗]^I2×[H^1,q(Ω^p_ε)]^I2 dθ+λκ

I2

k=1

" _t

0

"

Ω^p_ε

v_εδk∂f_r(v_εδ,τ)_kdx dθ,

i.e.,

" _t

0 ∂_θv_εδ, ∂f_r(v_εδ,τ)_[H1,q(Ω^p_ε)^∗]^I2×[H^1,q(Ω^p_ε)]^I2 dθ

=−ⁿ

l=1

" _t

0

"

Ω^p_ε

/ D ∂

∂x_lv_εδ, ∂

∂x_l(∂f_r(v_εδ,τ)) 0

I2

dx dθ

+

I2

k=1

!" _t

0

"

∂Ω_inq_ε·nvε_δk(∂f_r(v_εδ,τ))_kds dθ+λεk_d

" _t

0

"

Γ_εψ_δ(w_εδk) (∂f_r(v_εδ,τ))_kdσ_xdθ

#

−

" _t

0

"

Ω^p_εq_ε· ∇v_εδ, ∂f_r(v_εδ,τ)_I2 dx dθ+

" _t

0

4

S2R(ˆ¯ u_εδ, v_εδ), ∂f_r(v_εδ,τ)⁵

[H^1,q(Ω^p_ε)^∗]^I2×[H^1,q(Ω^p_ε)]^I2 dθ

−(1−λ)κ

I2

k=1

" _t

0

"

Ω^p_εv_εδk∂f_r(v_εδ,τ)_kdx dθ

=:I_{dif f}^(t) +I_bound^(t) +I_advec^(t) +I_reac^(t) +I_Ex^(t) for a.e. t, (4.2.81) where

I_{dif f}^(t) :=−ⁿ

l=1

" _t

0

"

Ω^p_ε

/ D ∂

∂x_lv_εδ, ∂

∂x_l(∂f_r(v_εδ,τ)) 0

I2

dx dθ,

I_bound^(t) :=

I2

k=1

!" _t

0

"

∂Ω_inq_ε·nv_εδk(∂f_r(v_εδ,τ))_kds dθ+λεk_d

" _t

0

"

Γ_εψ_δ(w_εδk) (∂f_r(v_εδ,τ))_kdσ_xdθ

# , I_advec^(t) :=−

" _t

0

"

Ω^p_εq_ε· ∇v_εδ, ∂f_r(v_εδ,τ)_I2 dx dθ, I_reac^(t) :=

" _t

0

4

S2R(ˆ¯ u_εδ, v_εδ), ∂f_r(v_εδ,τ)⁵

[H^1,q(Ω^p_ε)^∗]^I2×[H^1,q(Ω^p_ε)]^I2 dθ, I_Ex^(t) :=−(1−λ)κ

I2

k=1

" _t

0

"

Ω^p_ε

v_εδk∂f_r(v_εδ,τ)_kdx dθ.

Now we simplify and estimate the terms I_{dif f}^(t) , I_bound^(t) , I_advec^(t) , I_reac^(t) and I_Ex^(t) one by one.

With the help of (4.2.9), the termI_reac^(t) can be estimated in the same way as we did in the lemma 4.1.1.2.7and this will give

I_reac^(t) ≤λ r C

I2

k=1

" _t

0

"

Ω^p_ετ^$(((μ⁰_k⁽⁽(+T|Ω||logτ|+v_εδk,τ^%dx dθ=:h(t, τ, v_εδ,τ) for a.e. t, where C is independent of λ, ˆu_εδ and v_εδ,τ and all the other terms of h(t, τ, v_εδ,τ) are bounded and tend to zero as τ→0 for a.e. t, i.e.,

I_reac^(t) ≤h(t, τ, v_εδ,τ)→0 as τ→0 for a.e. t. (4.2.82) I_Ex^(t) =−κ(1−λ)

I2

i=1

" _t

0

"

Ω^p_εv_εδk∂f_r(v_εδ, τ)_kdx dθ

=κ(1−λ)

I2

i=1

" _t

0

"

Ω^p_ε

r(τ−v_εδk,τ)f_r−1(v_εδ, τ)(μ⁰_k+ logv_εδk,τ)dx dθ sincev_εδk,τ =v_εδk+τ

=τ κ(1−λ)

I2

i=1

" _t

0

"

Ω^p_ε

r(μ⁰_k+ logv_εδk,τ)f_r−1(v_εδ,τ)dx dθ +rκ(1−λ)

I2

i=1

" _t

0

"

Ω^p_ε−v_εδk,τ(μ⁰_k+ logv_εδk,τ)f_r−1(v_εδ,τ)dx dθ. (4.2.83)

4.2. Model M2 71 It can be shown that

−v_εδk,τ(μ⁰_k+ logv_εδk,τ)≤e^−(1+μ0k) ∀i. (4.2.84) We have logv_εδk,τ ≤v_εδk,τ ≤g_k(v_εδk,τ) and g_k(v_εδk,τ)≥(e−1)e^−μ0k. Choosing a constant C= max

1≤k≤I2

1 +⁽⁽μ⁰_k⁽⁽e^−μ0k(e−1)

, we obtain

μ⁰_k+ logv_εδk,τ ≤μ⁰_k+g_k(v_εδk,τ)≤⁽⁽(μ⁰_k⁽⁽(+g_k(v_εδk,τ)≤C g_k(v_εδk,τ). (4.2.85) Combining (4.2.83), (4.2.84) and (4.2.85), we get

I_Ex^(t) ≤(1−λ)

⎡

⎣rτ κ

I2

i=1

" _t

0

"

Ω^p_εCg_k(v_εδk,τ)f_r−1(v_εδ,τ)dx dθ+κ

I2

i=1

" _t

0

"

Ω^p_εre^−(1+μ0k)f_r−1(v_εδ,τ)dx dθ

⎤

⎦

≤rτ κ(1−λ)

I2

i=1

" _t

0

"

Ω^p_εCg_k(v_εδk,τ)f_r−1(v_εδ,τ)dx dθ +κ(1−λ)

I2

i=1

" _t

0

"

Ω^p_ε

r(e(e−1))⁻¹g_k(v_εδk,τ)f_r−1(v_εδ,τ)dx dθ

≤rτ κ(1−λ)C

I2

i=1

" _t

0

"

Ω^p_ε

g(v_εδ,τ)f_r−1(v_εδ,τ)dx dθ +κ(1−λ)

I2

i=1

" _t

0

"

Ω^p_εr(e(e−1))⁻¹g(v_εδ,τ)f_r−1(v_εδ,τ)dx dθ sinceg_k(v_εδk,τ)≤g(v_εδ,τ)

≤I2rκτ C

" _t

0

"

Ω^p_εf_r(v_εδ,τ)dx dθ+I2rκ(e(e−1))⁻¹

" _t

0

"

Ω^p_εf_r(v_εδ,τ)dx dθ since 0≤λ≤1 and f_r=f_r−1g for a.e. t.

Asτ →0,f_r(v_εδ,τ) is bounded inL¹((0, T)×Ω^p_ε). Therefore for a.e. t the first term in the r.h.s. of the above inequality tends to zero as τ →0. Denote the first term byl(t, τ, v_εδ,τ), then

I_Ex^(t) ≤l(t, τ, v_εδ,τ) +I2rk(e(e−1))⁻¹

" _t

0

"

Ω^p_εf_r(v_εδ,τ)dx dθ for a.e. t. (4.2.86) Again,

I_advec^(t) =−

" _t

0

"

Ω^p_εq_ε· ∇v_εδ, ∂f_r(v_εδ,τ)_I2 dx dθ

=−^I2

k=1

" _t

0

"

Ω^p_εq_ε· ∇vε_δk(∂f_r(v_εδ,τ))_kdx dθ

=−^I2

k=1

n

l=1

" _t

0

"

Ω^p_εq_l∂v_εδk

∂x_l (∂f_r(v_εδ,τ))_kdx dθ

=−ⁿ

l=1

" _t

0

"

Ω^p_ε

∂f_r(v_εδ,τ)

∂x_l q_ldx dθ

=−^{" t}

0

"

Ω^p_ε∇_xf_r(v_εδ,τ)·q_εdx dθ

=

" _t

0

"

Ω^p_εf_r(v_εδ,τ)∇ ·q_εdx dθ−

" _T

0

"

∂Ω^p_εf_r(v_εδ,τ)q_ε·n ds dθ

=−^{" t}

0

"

∂Ω^p_ε

f_r(v_εδ,τ)q_ε·n ds dθ, since∇ ·q_ε= 0 in Ω^p_ε

=−^{" t}

0

"

∂Ωf_r(v_εδ,τ)q_ε·n ds dθ−^{" t}

0

"

Γ_εf_r(v_εδ,τ)q_ε·n dσxdθ

=−^{" t}

0

"

∂Ωf_r(v_εδ,τ)q_ε·n ds dθ, sinceq_ε= 0 on Γ_ε

=−

" _t

0

"

∂Ω_in

f_r(v_εδ,τ)q_ε·n ds dθ−

" _t

0

"

∂Ω_out

f_r(v_εδ,τ)q_ε·n ds dθ

≤ −^{" t}

0

"

∂Ω_inf_r(v_εδ,τ)q_ε·n ds dθ, sinceq_ε·n≥0 on∂Ω_out and f_r(v_εδ,τ)≥0

≤ ||q_ε·n||_L∞((0,T)×∂Ω_in)

" _t

0

"

∂Ω_in|f_r(v_εδ,τ)|ds dθ

=C26

" _t

0

"

∂Ω

f_r(v_εδ,τ)ds dθ for a.e. t, (4.2.87) where C26:=||q_ε·n||_L∞((0,T)×∂Ω_in) and f_r≥0. Note thatC26 is independent of ε, δ,λ,τ and t. Again,

I_bound^(t) =

I2

k=1

!" _t

0

"

∂Ω_inq_ε·nv_εδk(∂f_r(v_εδ,τ))_kds dθ+λεk_d

" _t

0

"

Γ_εψ_δ(w_εδk) (∂f_r(v_εδ,τ))_kdσ_xdθ

#

=

I2

k=1

!" _t

0

"

∂Ω_inq_ε·n

v_εδk,τ−τ

(∂f_r(v_εδ,τ))_kds dθ +λεk_d

" _t

0

"

Γ_ε

ψ_δ(w_εδk) (∂f_r(v_εδ,τ))_kdσ_xdθ

#

=

I2

k=1

!" _t

0

"

∂Ω_inq_ε·nv_εδk,τ(∂f_r(v_εδ,τ))_kds dθ−^{" t}

0

"

∂Ω_inq_ε·nτ(∂f_r(v_εδ,τ))_kds dθ

#

+

I2

k=1

! λεk_d

" _t

0

"

Γ_ε

ψ_δ(w_εδk) (∂f_r(v_εδ,τ))_kdσ_xdθ

#

=:

I2

k=1

$

Boundary_1,k+ Boundary_2,k+ Boundary_3,k^%, (4.2.88)

where

Boundary_1,k:=

" _t

0

"

∂Ω_in

q_ε·nv_εδk,τ(∂f_r(v_εδ,τ))_kds dθ, (4.2.89) Boundary_2,k:=−^{" t}

0

"

∂Ω_inq_ε·nτ(∂f_r(v_εδ,τ))_kds dθ and (4.2.90) Boundary_3,k:=λεk_d

" _t

0

"

Γ_εψ_δ(w_εδk) (∂f_r(v_εδ,τ))_kdσ_xdθ. (4.2.91) Now,

Boundary_1,k=

" _t

0

"

∂Ω_inq_ε·nvε_δk,τ(∂f_r(v_εδ,τ))_kds dθ

=

" _t

0

"

∂Ω_in−|q_ε·n|v_εδk,τ(∂f_r(v_εδ,τ))_kds dθ

=

" _t

0

"

∂Ω_in−rf_r−1(v_εδ,τ)|q_ε·n|v_εδk,τμ⁰_k+ logv_εδk,τds dθ.

4.2. Model M2 73 It can be shown that−v_εδk,τμ⁰_k+ logv_εδk,τ≤ _e(e−¹₁₎g_k(v_εδk,τ). This gives³⁰

Boundary_1,k≤

" _t

0

"

∂Ω_in

rf_r−1

1

e(e−1)g_k(v_εδk,τ)|q_ε·n|ds dθ

≤r||q_ε·n||_L∞((0,T)×∂Ω_in)

1 e(e−1)

" _t

0

"

∂Ω_in

f_r−1(v_εδ,τ)g_k(v_εδ,τ)ds dθ

≤r||q_ε·n||_L∞((0,T)×∂Ω_in)

1 e(e−1)

" _t

0

"

∂Ω_inf_r−1(v_εδ,τ)g(v_εδ,τ)ds dθ, sinceg_k≤g

=C27

" _t

0

"

∂Ω_in

f_r(v_εδ,τ(t, x))ds dθ, (4.2.92)

whereC27 (:=r||q_ε·n||_L∞((0,T)×∂Ω_in) 1

e(e−1)) is independent of ε,δ,λ,τ and t.

Boundary_2,k:=−

" _t

0

"

∂Ω_inq_ε·nτ(∂f_r(v_εδ,τ))_kds dθ

=

" _t

0

"

∂Ω_in|q_ε·n|τ rf_r−1(v_εδ,τ)μ⁰_k+ logv_εδk,τds dθ.

Let∂Ω⁺_in:=^-x∈∂Ω_in:μ⁰_k+ logv_εδk,τ ≥0^.and∂Ω⁻_in:=^-x∈∂Ω_in:μ⁰_k+ logv_εδk,τ ≤0^.. On the boundary∂Ω⁻_in, the integrand is nonpositive and it can be estimated by zero. This gives

Boundary_2,k≤

" _t

0

"

∂Ω⁺_in|q_ε·n|τ rf_r−1(v_εδ,τ)μ⁰_k+ logv_εδk,τds dθ.

From the definition of g_k, (e−1)e^−μ0k ≤g_k(v_εδk,τ) and it can be shown that logv_εδk,τ ≤ v_εδk,τ ≤g_k(v_εδk,τ). Choosing C28:= max

1≤k≤I2

1 +⁽⁽₍μ^k₀⁽⁽₍e^μ0k(e−1)⁻¹, we have μ⁰_k+ logv_εδk,τ ≤C28g_k(v_εδk,τ)≤C28g(v_εδ,τ).

Boundary_2,k≤

" _t

0

"

∂Ω⁺_in|q_ε·n|τ rf_r−1(v_εδ,τ)C28g(v_εδ,τ)ds dθ

≤C28rτ||q_ε·n||_L∞((0,T)×∂Ω_in)

" _t

0

"

∂Ω_in

f_r(v_εδ,τ)ds dθ

=C29

" _T

0

"

∂Ω_in

f_r(v_εδ,τ)ds dθ, (4.2.93) whereC29 (:=C28rτ||q_ε·n||_L∞((0,T)×∂Ω_in)) is independent of ε,δ,λand t.

Boundary_3,k=λεk_d

" _t

0

"

Γ_εψ_δ(w_εδk) (∂f_r(v_εδ,τ))_kdσ_xdθ

≤k_dε

" _t

0

"

Γ_ε(∂f_r(v_εδ,τ))_kdσ_xdθ,³¹

=k_dε

" _t

0

"

Γ_εrf_r−1(v_εδk,τ)μ⁰_k+ logv_εδk,τdσ_xdθ

30The simplification of the terms Boundary_1,k and Boundary_2,kare imitated from [Kr¨a08].

31Note that 0≤λ≤1,ψ_δ(vε_δk)≤1 andε1.

≤εk_dC28r

" _t

0

"

Γ_εf_r−1(v_εδ,τ)g(v_εδ,τ)dσ_xdθ,³²

=C30ε

" _t

0

"

Γ_εf_r(v_εδ,τ)dσ_xdθ, (4.2.94) where C30 (:=k_dC28r) is independent of ε, δ, λ and t. Substituting the estimates for Boundary_p,k for 1≤p≤3 in (4.2.88), we obtain³³

I_bound^(t) =

I2

k=1

$

Boundary_1,k+ Boundary_2,k+ Boundary_3,k

%

≤^I2

k=1

C31

!" _t

0

"

∂Ω_inf_r(v_εδ,τ)ds dθ+

" _t

0

"

∂Ω_inf_r(v_εδ,τ)ds dθ+ε

" _t

0

"

Γ_εf_r(v_εδ,τ)dσ_xdθ

#

= 2C31I2

" _t

0

"

∂Ω_in

f_r(v_εδ,τ)ds dθ+εC31I2

" _t

0

"

Γ_ε

f_r(v_εδ,τ)dσ_xdθ

≤C32

+" _t

0

"

∂Ωf_r(v_εδ,τ)ds dθ+ε

" _T

0

"

Γ_εf_r(v_εδ,τ)dσ_xdθ ,

for a.e. t. (4.2.95) The term I_{dif f} can be estimated as in lemma 4.1.1.2.7.

I_{dif f}^(t) =−D^I2

k=1

n

l=1

" _t

0

"

Ω^p_ε

r(r−1)f_r−2 I2

υ=1

μ⁰_υ+ logv_εδυ,τμ⁰_k+ logv_εδk,τ∂v_εδυ,τ

∂x_l

∂v_εδk

∂x_l dx dθ

−D

I2

k=1

n

l=1

" _t

0

"

Ω^p_ε

rf_r−1

1 v_εδk

∂v_εδk

∂x_l

∂v_εδk

∂x_l dx dθ

≤0

≤ −Dⁿ

l=1

" _t

0

"

Ω^p_εr(r−1)f_r−2

⎛

⎝^I2

k=1

μ⁰_k+ logv_εδk,τ ∂v_εδk

∂x_l

⎞

⎠

2

dx dθ for a.e. t.

(4.2.96) Combining (4.2.81), (4.2.82), (4.2.86), (4.2.87), (4.2.95) and (4.2.96), we get³⁴

" _t

0 ∂_θv_εδ, ∂f_r(v_εδ,τ)_[H1,q(Ωp

ε)]^∗I2×[H^1,q(Ω^p_ε)]^I2 dθ

=I_diif^(t) +I_bound^(t) +I_advec^(t) +I_reac^(t) +I_Ex^(t)

≤ −Dⁿ

l=1

" _t

0

"

Ω^p_ε

r(r−1)f_r−2

⎛

⎝^I2

k=1

μ⁰_k+ logv_εδk,τ∂v_εδk

∂x_l

⎞

⎠

2

dx dθ +C32

!" _t

0

"

∂Ω

f_r(v_εδ,τ)ds dθ+ε

" _t

0

"

Γ_ε

f_r(v_εδ,τ)dσ_xdθ

# +C26

" _t

0

"

∂Ω

f_r(v_εδ,τ)ds dθ +h(t, τ, v_εδ,τ) +l(t, τ, v_εδ,τ) +I2rk(e(e−1))⁻¹

" _t

0

"

Ω^p_ε

f_r(v_εδ,τ)dx dθ

32See the above calculation for Boundary_2,k.

33WhereC₃₁= max (C₂₇, C₂₉, C₃₀) andC₃₂= I₂max (2C₃₁, C₃₁).

34The idea to further estimate the termI_{dif f}+I_bound+I_advec+Ireac+I_Ex is borrowed from [Kr¨a08].

We also note thatfr(u) = [g(u)]^r= [g^r2(u)]²=f²r 2(u).

4.2. Model M2 75

=−D

n

l=1

" _t

0

"

Ω^p_ε

r(r−1)f_r−2

⎛

⎝^I2

k=1

μ⁰_k+ logv_εδk,τ∂v_εδk

∂x_l

⎞

⎠

2

dx dθ + (C32+C26)

" _t

0

"

∂Ω

f_r(v_εδ,τ)ds dθ+C32ε

" _t

0

"

Γ_ε

f_r(v_εδ,τ)dσ_xdθ +h(t, τ, v_εδ,τ) +l(t, τ, v_εδ,τ) +I2rk(e(e−1))⁻¹

" _t

0

"

Ω^p_ε

f_r(v_εδ,τ)dx dθ

=−Dⁿ

l=1

" _t

0

"

Ω^p_εr(r−1)f_r−2

⎛

⎝^I2

k=1

μ⁰_k+ logv_εδk,τ ∂v_εδk

∂x_l

⎞

⎠

2

dx dθ + (C32+C26)

" _t

0

"

∂Ωf²r

2(v_εδ,τ)ds dθ+C32ε

" _t

0

"

Γ_εf_r(v_εδ,τ)dσ_xdθ +h(t, τ, v_εδ,τ) +l(t, τ, v_εδ,τ) +I2rk(e(e−1))⁻¹

" _t

0

"

Ω^p_εf_r(v_εδ,τ)dx dθ (4.2.97) for a.e. t, where C26 and C32 are independent of ε, δ, λ and t. We further simplify the terms in (4.2.97).

C32ε

" _t

0

"

Γ_εf_r(v_εδ,τ)dσ_xdθ

= C32ε

" _t

0

"

∪_k∈ZnεΓ_kf_r(v_εδ,τ)dσ_xdθ

= C32ε

" _t

0

"

∪k∈ZnΓ_k

f_r(v_εδ,τ)εⁿ⁻¹dσ_ydθ

= C32εⁿ

k∈Zⁿ

" _t

0

"

Γ_kf_r(v_εδ,τ)dσ_ydθ

= C32εⁿ

k∈Zⁿ

" _t

0

"

Γ_k

f_r(v_εδ,τ)dσ_ydθ× 1 ((Y_k^p((

"

Y_k^p

dy

= C32εⁿ

k∈Zⁿ

((Y1_k^p((

" _t

0

"

Y_k^p

"

Γ_k

f_r(v_εδ,τ)dσ_ydθ dy

= C32εⁿ

k∈Zⁿ

((Y1_k^p((

" _t

0

"

Y_k^p

f_r(v_εδ,τ)dθ dy×

"

Γ_k

dσ_y

= C32εⁿ

k∈Zⁿ

((Y1_k^p((

" _t

0

"

Y_k^pf_r(v_εδ,τ)dθ dy× |Γ_k|

= C32εⁿ

k∈Zⁿ

|Γ_k| ((Y_k^p((

" _t

0

"

Y_k^pf_r(v_εδ,τ)dθ dy

= C32 |Γ|

|Y^p|εⁿ

k∈Zⁿ

" _t

0

"

Y_k^p

f_r(v_εδ,τ)dθ dy since ⁽⁽Y_k^p((=|Y^p| and |Γ_k|=|Γ|

= C32 |Γ|

|Y^p|εⁿ

" _t

0

"

∪k∈ZnY_k^p

f_r(v_εδ,τ)dθ dy

= C32 |Γ|

|Y^p|

" _t

0

"

∪k∈ZnεY_k^p

f_r(v_εδ,τ)dx dθ

= C32 |Γ|

|Y^p|

" _t

0

"

Ω^p_ε

f_r(v_εδ,τ)dx dθ. (4.2.98)

Also using theorem3.4.3.2 we have

" _t

0

"

∂Ω

f²r

2(v_εδ,τ(t, x))ds dθ

=

" _t

0

((((((f^r

2(v_εδ,τ(t))⁽⁽((((²

L²(∂Ω)dθ

≤C8

" _t

0

((((((∇f^r₂(v_εδ,τ(t))⁽⁽((((

[L²(Ω^p_ε)]ⁿ

((((((f^r

2(v_εδ,τ(t))⁽⁽((((

L²(Ω^p_ε)+⁽⁽((((f^r

2(v_εδ,τ(t)⁽⁽((((²

L²(Ω^p_ε)

≤C8

" _t

0

⎛

⎜⎜

⎜⎝ς⁽⁽((((∇f^r

2(v_εδ,τ)⁽⁽((((²

[L²(Ω^p_ε)]ⁿ+ ˆΛ_ς⁽⁽((((f^r

2(v_εδ,τ(t))⁽⁽((((²

L²(Ω^p_ε)

due to Young’s inequality

+⁽⁽((((f^r

2(v_εδ,τ(t))⁽⁽((((²

L²(Ω^p_ε)

⎞

⎟⎟

⎟⎠dθ

=C8

" _t

0

ς⁽⁽((((∇f^r₂(v_εδ,τ)⁽⁽((((²

[L²(Ω^p_ε)]ⁿ+ Λ_ς⁽⁽((((f^r

2(v_εδ,τ(t))⁽⁽((((²

L²(Ω^p_ε)

dθ,Λ_ς= ˆΛ_ς+ 1, (4.2.99) whereC8 (independent ofεandς) is a constant in Young’s inequality which will be chosen later. Further note that

((((((∇f^r₂(v_εδ,τ)⁽⁽((((²

[L²(Ω^p_ε)]ⁿ=

n

l=1

((((

( ((((

(

∂f^r

2(v_εδ,τ)

∂x_l ((((

( ((((

(

2

L²(Ω^p_ε)

=r² 4

"

Ω^p_εf_r−2(v_εδ,τ)

n

l=1

⎛

⎝^I2

k=1

μ⁰_k+ logv_εδk∂v_εδk

∂x_l

⎞

⎠

2

dx. (4.2.100) Combining (4.2.97), (4.2.98), (4.2.99) and (4.2.100), we obtain

" _t

0 ∂_θv_εδ, ∂f_r(v_εδ,τ)_[H1,q(Ωp

ε)]^∗I2×[H^1,q(Ω^p_ε)]^I2 dθ

≤ −Dⁿ

l=1

" _t

0

"

Ω^p_ε

r(r−1)f_r−2

⎛

⎝^I2

k=1

μ⁰_k+ logv_εδk,τ∂v_εδk

∂x_l

⎞

⎠

2

dx dθ

+ (C32+C26)C8

" _t

0

"

Ω^p_ε

r² 4f_r−2ς

n

l=1

⎛

⎝^I2

k=1

μ⁰_k+ logv_εδk∂v_εδk

∂x_l

⎞

⎠

2

dx dθ + (C32+C26)C8

" _t

0

"

Ω^p_εΛ_ς⁽⁽(f^r

2(v_εδ,τ(t))⁽⁽(²

L²(Ω^p_ε)dx dθ+C32 |Γ|

|Y^p|

" _t

0

"

Ω^p_εf_r(v_εδ,τ)dx dθ +h(t, τ, v_εδ,τ) +l(t, τ, v_εδ,τ) +I2rk(e(e−1))⁻¹

" _t

0

"

Ω^p_εf_r(v_εδ,τ)dx dθ

≤ +

−Dr(r−1) +ςC33

r² 4

," _t

0

"

Ω^p_εf_r−2

n

l=1

⎛

⎝^I2

k=1

μ⁰_k+ logv_εδk∂v_εδk

∂x_l

⎞

⎠

2

dx dθ+h(t, τ, v_εδ,τ) +l(t, τ, v_εδ,τ) +

C33Λ_ς+C32 |Γ|

|Y^p|+I2rκ(e(e−1))⁻¹

" _t

0

"

Ω^p_εf_r(v_εδ,τ)dx dθ for a.e. t.³⁵ (4.2.101) Choosingς≤^4D_C⁽₃₃^r−_r¹⁾, this shows that Λ_ς+ 1 is independent of ε,λand δ. This gives

" _t

0 ∂_θv_εδ, ∂f_r(v_εδ,τ)_[H1,q(Ωp

ε)]^∗I2×[H^1,q(Ω^p_ε)]^I2 dθ

≤h(t, τ, v_εδ,τ) +l(t, τ, v_εδ,τ) +

C33Λ_ς+C32 |Γ|

|Y^p|+I2rκ(e(e−1))⁻¹

" _t

0

"

Ω^p_εf_r(v_εδ,τ)dx dθ

35WhereC₃₃= C₈(C₂₆+C₃₂).

4.2. Model M2 77

≤h(t, τ, v_εδ,τ) +l(t, τ, v_εδ,τ) +C34

" _t

0

"

Ω^p_εf_r(v_εδ,τ)dx dθ

≤h(t, τ, v_εδ,τ) +l(t, τ, v_εδ,τ) +C34

" _t

0

F_r(v_εδ,τ)dθ for a.e. t. (4.2.102) whereC34

:=C33Λ_ς+C32 |Γ|

|Y^p|+I2rκ(e(e−1))⁻¹, andh(t, τ, v_εδ,τ) andl(t, τ, v_εδ,τ) tend

to zero as τ→0 for a.e. t.

Proof of theorem 4.2.1.3.5. Letv_εδ be a solution of the problem ( ¯P_ε²⁺

δλM). Since we only the know the nonnegativity of v_εδ, let v_εδ,τ :=v_εδ+τ for τ >0. Clearly v_εδ,τ ∈ G_ε^v. Here also we introduce the regularization ofv_εδ,τ and replicating the steps of theorem4.1.1.2.3, we obtain an inequality similar to (4.1.34), i.e.,

|F_r(v_εδ,τ(t))−F_r(v_εδ,τ(0))| ≤h(t, τ, v_εδ,τ) +l(t, τ, v_εδ,τ) +C34

" _t

0

F_r(v_εδ,τ)dθ

=⇒F_r(v_εδ,τ(t))−F_r(v_εδ,τ(0))≤h(t, τ, v_εδ,τ) +l(t, τ, v_εδ,τ) +C34

" _t

0 F_r(v_εδ,τ)dθ for a.e. t.

(4.2.103) Since v_εδ,τ →v_εδ as τ →0. h(t, τ, v_εδ,τ)→0 and l(t, τ, v_εδ,τ) as τ →0 for a.e. t. F_r(v_εδ,τ) is continuous (cf. lemma 4.1.1.2.4), i.e., F_r(v_εδ,τ)→F_r(v_εδ) asτ →0. Taking the limit on both sides of (4.2.103) as τ →0, we get

F_r(v_εδ(t))−F_r(v_εδ(0))≤C34

" _t

0 F_r(v_εδ)dθ for a.e. t, i.e.,

F_r(v_εδ(t))≤F_r(v_εδ(0)) +C34

" _t

0

F_r(v_εδ)dθ for a.e. t.

Gronwall’s inequality gives

F_r(v_εδ(t))≤e^C34tF_r(v_εδ(0)) for a.e. t. (4.2.104) whereC34 is independent of ε,δ,λand t. This establishes the inequality (4.2.78).

Now we use the theorem 4.2.1.3.5 to obtain L^r - and L^∞ - estimates of the solution v_εδ. Let

C36(r) :=C36:=

+ I2 sup

k,ε,δ >0

ess sup

0≤t≤T |Ω|C13e^C34t )

1 +

I₂¹²|||v0|||_L∞(Ω^p_ε)^I2

_r(1+α)*,¹

r

(4.2.105) and

C37:= sup

ε,δ >0

+ 1 +

I₂¹²|||v0|||_L∞(Ω^p_ε)^I2

1+α,

. (4.2.106)

Corollary 4.2.1.3.7. Let uˆ_εδ∈ F_ε^u be fixed. For any arbitrary solution v_εδ ∈ G_ε^v of ( ¯P_ε²⁺

δλM) the following estimates hold true:

sup

ε,δ >0|||v_εδ(t)|||_Lr(Ω^p_ε)^I2 ≤C36<∞ for all r and for a.e. t, (4.2.107) and

sup

ε,δ >0|||vε_δ(t)|||_L∞(Ω^p_ε)^I2 ≤C37<∞ for a.e. t. (4.2.108)

Proof. Givenr∈N(r≥2) and for the problem ( ¯P_ε²⁺

δλM),v_εδ(0, x) =λv0(x). From inequality (4.2.104), we have

F_r(v_εδ(t))≤e^C34tF_r(v_εδ(0)) for a.e. t.

A straightforward application of Gronwall’s inequality and arguments similar to the proof

of corollary 4.1.1.2.8yield the desired results.

Corollary 4.2.1.3.8. Let the assumptions (4.2.1)-(4.2.10), uˆ_εδ ∈ F_ε^u, 0≤λ≤1 and r∈N be satisfied. Then there exists a constant C independent of uˆ_εδ, v_εδ, ε, λ and t such that any arbitrary solution v_εδ ∈ G_ε^v of the problem ( ¯P_ε²⁺

δλM) satisfies

|||v_εδ|||_Gv

ε ≤C. (4.2.109)

Proof. Forp > n+2, ˆu_εδ∈L^∞((0, T)×Ω^p_ε)^I1. Note thatv_εδk satisfies the estimates (4.2.107) and (4.2.108). The abstract formulation of the problem (4.2.71)-(4.2.75) is given by

∂v_εδ

∂t +Av_εδ =f_bound(v_εδ) +f(v_εδ), (4.2.110)

v_εδ(0, x) =v0(x), (4.2.111)

where the operator Ais defined as in remark 4.1.1.1.1 with maximal regularity on [H^1,q(Ω^p_ε)^∗]^I2,f(v_εδ) =λS2R(ˆ¯ u_εδ, v_εδ) +λ κ v_εδ and f_bound(v_εδ) =−q_ε· ∇v_εδ+Q_∂Ω_in(v_εδ)+

RΓ_ε(−λ^∂w_∂t^εδ). Choosing r sufficiently large in (4.2.107) and application of H¨older’s in-equality imply that f ∈L^p((0, T);H^1,q(Ω^p_ε)^∗)^I2. Since from lemma 4.2.1.3.1 Q_∂Ω_in(v_εδ)∈ L^p((0, T);H^1,q(Ω^p_ε)^∗)^I2, by lemma4.2.1.3.2RΓ_ε(−λ^∂w_∂t^εδ)∈L^p((0, T);H^1,q(Ω^p_ε)^∗)^I2 and−q_ε·

∇vεδ∈L^p((0, T);H^1,q(Ω^p_ε)^∗)^I2, the termf_bound is inL^p((0, T);H^1,q(Ω^p_ε)^∗)^I2. Moreover from (4.2.3), we have v0 ∈[(H^1,q(Ω^p_ε)^∗, H^1,p(Ω^p_ε))₁₋1

p,p]^I2. Therefore from theorem 3.3.1 there exists a uniquev_εδ∈ G_ε^v such that

|||v_εδ|||_Gv

ε ≤C, (4.2.112)

whereC is independent of λandv_εδ.

Lemma 4.2.1.3.9. The operator Z1 is compact and continuous.

Proof. We will only show the compactness ofZ1as the continuity follows analogously. Let ˆ

u_εδ ∈ F_ε^u be fixed. Let {vˆ_εδn}^∞_n=1 be a bounded sequence in G_ε^v. For p > n+ 2, G_ε^v →→ L^∞((0, T)×Ω^p_ε)^I2. Then up to a subsequence (still denoted by same symbol) {ˆv_εδn}^∞_n=1 is strongly convergent inL^∞((0, T)×Ω^p_ε)^I2. Therefore the r.h.s of the PDE

∂v_εδn

∂t − ∇(D∇vε_δn−q_εv_εδn) +κv_εδn=S2R(ˆ¯ u_εδ,vˆ_εδn) +κvˆ_εδn

is strongly convergent inL^p((0, T);L^p(Ω^p_ε))^I2, i.e., inL^p((0, T);H^1,q(Ω^p_ε)^∗)^I2. Thus by the-orem 3.3.1, the sequence{vε_δn}^∞_n=1 is strongly convergent in G^v_ε. Proof of theorem 4.2.1.3.3. The compactness and continuity of the operatorZ1 is shown in the lemma4.2.1.3.9and the corollary4.2.1.3.8gives the estimate (4.2.70). By Schaefer’s fixed point theorem the operator Z1 has a fixed point, i.e., the problem ( ¯P_ε²⁺

δM) has a solution. This solution is also a solution of ( ¯P_ε²⁺

δ ).

4.2. Model M2 79 4.2.1.4 Existence of the Global Solution of the Complete Problem (P_ε²⁺_δ )

Theorem 4.2.1.4.1. There exists a positive weak solution (u_εδ, v_εδ, w_εδ)∈ F_ε^u× G_ε^v× H^w_ε of the following problem:

∂u_εδ

∂t − ∇ ·(D∇u_εδ−q_εu_εδ) =S1R(u¯ _εδ, v_εδ) in (0, T)×Ω^p_ε, u_εδ(0, x) =u0(x) in Ω^p_ε,

−(D∇u_εδ−q_εu_εδ)·n=d on (0, T)×∂Ω_in,

−D∇u_εδ·n= 0 on (0, T)×∂Ω_out,

−D∇uε_δ·n= 0 on (0, T)×Γ_ε,

∂v_εδ

∂t − ∇ ·(D∇v_εδ−q_εv_εδ) =S2R(u¯ _εδ, v_εδ) in (0, T)×Ω^p_ε, v_εδ(0, x) =v0(x) in Ω^p_ε,

−(D∇v_εδ−q_εv_εδ)·n= 0 on (0, T)×∂Ω_in,

−D∇v_εδ·n= 0 on (0, T)×∂Ω_out,

−D∇vε_δ·n=ε∂w_ε

∂t on (0, T)×Γ_ε,

∂w_εδ

∂t =−k_dψ_δ(w_εδ) on (0, T)×Γ_ε, w_εδ(0, x) =w0(x) on Γ_ε.

The positivity of the solution has already been shown in lemma 4.2.1.1.2. To prove the existence of the global solution of the problem (P_ε²⁺

δ ), we employ the similar techniques which we used to solve ( ¯P_ε²⁺

δ ). Here also the basic ingredients are Schaefer’s fixed point theorem, a Lyapunov functional and theorem3.3.1. The Lyapunov functional is defined in the following way: Let ¯μ⁰∈R^I1 be the solution of

S₁^Tμ¯⁰=−logK, (4.2.113)

whereK∈R^J is the vector of equilibrium constantsK_j=^k

f j

k^b_j. Due to (4.2.4), (4.2.113) has a solution. The functiong_i:R⁺0 →Ris defined byg_i(u_εδi) :=

¯

μ⁰_i −1 + logu_εδi

u_εδi+e^(1−μ¯0i). We define the functionsg,f_r andF_rin the similar way as we did in section4.1.1.2and their relevant properties hold good (see propositions 4.1.1.2.1 and 4.1.1.2.2). Now for technical reasons, we modify the right hand side of the first PDE in (P_ε²⁺_δ ), i.e., for anyκ >0, we get

∂u_εδ

∂t − ∇ ·(D∇uεδ−q_εu_εδ) +κu_εδ=S1R(u¯ _εδ, v_εδ) +κu_εδ in (0, T)×Ω^p_ε, (4.2.114) u_εδ(0, x) =u0(x) in Ω^p_ε, (4.2.115)

−(D∇uε_δ−q_εu_εδ)·n=d on (0, T)×∂Ω_in, (4.2.116)

−D∇uεδ·n= 0 on (0, T)×∂Ω_out, (4.2.117)

−D∇u_εδ·n= 0 on (0, T)×Γ_ε, (4.2.118)

∂v_εδ

∂t − ∇ ·(D∇v_εδ−q_εv_εδ) =S2R(u¯ _εδ, v_εδ) in (0, T)×Ω^p_ε, (4.2.119) v_εδ(0, x) =v0(x) in Ω^p_ε, (4.2.120)

−(D∇v_εδ−q_εv_εδ)·n= 0 on (0, T)×∂Ω_in, (4.2.121)

−D∇v_εδ·n= 0 on (0, T)×∂Ω_out, (4.2.122)

−D∇v_εδ·n=ε∂w_ε

∂t on (0, T)×Γ_ε, (4.2.123)

∂w_εδ

∂t =−k_dψ_δ(w_εδ) on (0, T)×Γ_ε, (4.2.124)

w_εδ(0, x) =w0(x) on Γ_ε. (4.2.125)

Let us denote this problem by (P_ε²⁺

δM). We define the fixed point operator Z2:F_ε^u→ F_ε^u via Z2(ˆu_εδ) :=u_εδ, whereu_εδ is the solution of the following linear problem

∂u_εδ

∂t − ∇ ·(D∇u_εδ−q_εu_εδ) +κu_εδ=S1R(ˆ¯ u_εδ, v_εδ) +κˆu_εδ in (0, T)×Ω^p_ε, (4.2.126) u_εδ(0, x) =u0(x) in Ω^p_ε, (4.2.127)

−(D∇u_εδ−q_εu_εδ)·n=d on (0, T)×∂Ω_in, (4.2.128)

−D∇u_εδ·n= 0 on (0, T)×∂Ω_out, (4.2.129)

−D∇u_εδ·n= 0 on (0, T)×Γ_ε, (4.2.130) wherev_εδ ∈ G_ε^v is the solution of the problem

∂v_εδ

∂t − ∇ ·(D∇v_εδ−q_εv_εδ) =S2R(ˆ¯ u_εδ, v_εδ) in (0, T)×Ω^p_ε, (4.2.131) v_εδ(0, x) =v0(x) in Ω^p_ε, (4.2.132)

−(D∇v_εδ−q_εv_εδ)·n= 0 on (0, T)×∂Ω_in, (4.2.133)

−D∇v_εδ·n= 0 on (0, T)×∂Ω_out, (4.2.134)

−D∇vεδ·n=ε∂w_εδ

∂t on (0, T)×Γ_ε, (4.2.135)

and w_εδ∈ H_ε^w is the solution of the problem

∂w_εδ

∂t =−k_dψ_δ(w_εδ) on (0, T)×Γ_ε, (4.2.136)

w_εδ(0, x) =w0(x) on Γ_ε. (4.2.137)

Note that for fixed ˆu_εδ the problem (4.2.131)-(4.2.137) has a solution and satisfies the estimates (4.2.107)-(4.2.108). The operatorZ2 is well-defined (can be verified as in remark 4.2.1.3.4). Now in order to apply the Schaefer’s fixed point theorem, we show the following two condition:

(i) The operatorZ2 is continuous and compact.

(ii) The set{u_εδ∈ F_ε^u|∃λ∈[0,1] :u_εδ=λZ2(u_εδ)}is bounded, i.e., there exists a constant C >0 independent of u_εδ and λ such that any arbitrary solution u_εδ ∈ F_ε^u of the equation

u_εδ=λZ2(u_εδ) (4.2.138)

satisfies

|u_εδ|F_ε^u≤C. (4.2.139)

4.2. Model M2 81 Combining (4.2.126)-(4.2.137) and (4.2.138), we obtain

∂u_εδ

∂t − ∇ ·(D∇uεδ−q_εu_εδ) +κu_εδ =λS1R(u¯ _εδ, v_εδ) +λκu_εδ in (0, T)×Ω^p_ε, (4.2.140) u_εδ(0, x) =λu0(x) in Ω^p_ε, (4.2.141)

−(D∇uε_δ−q_εu_εδ)·n=λd on (0, T)×∂Ω_in,

(4.2.142)

−D∇u_εδ·n= 0 on (0, T)×∂Ω_out,

(4.2.143)

−D∇u_εδ·n= 0 on (0, T)×Γ_ε, (4.2.144) wherev_εδ ∈ G_ε^v is the solution of the problem

∂v_εδ

∂t − ∇ ·(D∇v_εδ−q_εv_εδ) =S2R(u¯ _εδ, v_εδ) in (0, T)×Ω^p_ε, (4.2.145) v_εδ(0, x) =v0(x) in Ω^p_ε, (4.2.146)

−(D∇vεδ−q_εv_εδ)·n= 0 on (0, T)×∂Ω_in, (4.2.147)

−D∇v_εδ·n= 0 on (0, T)×∂Ω_out, (4.2.148)

−D∇v_εδ·n=ε∂w_εδ

∂t on (0, T)×Γ_ε, (4.2.149)

and w_εδ∈ H_ε^w is the solution of the problem

∂w_εδ

∂t =−k_dψ_δ(w_εδ) on (0, T)×Γ_ε, (4.2.150)

w_εδ(0, x) =w0(x) on Γ_ε. (4.2.151)

Let us call the problem (4.2.140)-(4.2.151) as (P_ε²⁺

δλM). The inequality (4.2.139) is the consequence of the following three results:

Lemma 4.2.1.4.2. Let p > n+ 2, 0≤λ≤1 and r∈N (r≥2). Assume that u_εδ ∈ F_ε^u is a solution of (P_ε²⁺

δλM) and for τ >0,

u_εδ,τ :=u_εδ+τ.

Then the following inequality holds:

" _t

0

/∂u_εδ,τ

∂θ , ∂f_r(u_εδ,τ) 0

[H^1,q(Ω^p_ε)^∗]^I1×H^1,q(Ω^p_ε)^I1

dθ

≤h(t, τ, u_εδ,τ) +l(t, τ, u_εδ,τ) +C38

" _t

0

F_r(u_εδ,τ)dθ for a.e. t,

where h(t, τ, u_εδ,τ) and l(t, τ, u_εδ,τ) tend to zero asτ →0 for a.e. t, andC38 is independent of ε, δ, λand t.

Proof. Note that foru_εδ∈ F_ε^u, the problem (4.2.145)-(4.2.151) has a solution (v_εδ, w_εδ)∈ G_ε^v×H^w_ε with estimates (4.2.107)-(4.2.108). We use∂f_r(u_εδ)∈L^q((0, T);H^1,q(Ω^p_ε))^I1 as the test function in the weak formulation of (4.2.140). Replicating the steps of lemma4.2.1.3.6

and use of (4.2.10) will finish the proof.

Theorem 4.2.1.4.3. Let r∈N(r≥2), 0≤t≤T and 0≤λ≤1. Suppose that u_εδ ∈ F_ε^u is a solution of (P_ε²⁺

δλM). Then the following inequality holds good:

F_r(u_εδ(t))≤e^C38tF_r(u_εδ(0)) for a.e. t, (4.2.152) where C38 is independent of ε, δ, λand t.

Proof. The proof follows from lemma 4.2.1.4.2by using arguments similar to the proof of

theorem 4.2.1.3.5.

Corollary 4.2.1.4.4. For any arbitrary solutionu_εδ∈ F_ε^u of(P_ε²⁺

δλM) the following estimates hold true:

|||u_εδ(t)|||_Lr(Ω^p_ε)^I1 ≤C39<∞ for all r and for a.e. t, (4.2.153) and

|||uεδ(t)|||_L∞(Ω^p_ε)^I1 ≤C40<∞ for a.e. t, (4.2.154) where C39 and C40 are independent of ε, δ, λand t.

Proof. By using arguments from the proof of corollary 4.2.1.3.7 in (4.2.152) yield the

proof.

Corollary 4.2.1.4.5. Let the assumptions (4.2.1)-(4.2.10),0≤λ≤1 andr∈Nbe satisfied.

Then there exists a constant C independent of u_εδ, ε, δ, λ and t such that any arbitrary solution u_εδ ∈ F_ε^u of the problem(P_ε²⁺

δλM) satisfies

|||uεδ|||_Fu

ε ≤C. (4.2.155)

Proof. The proof is analogous to the proof of corollary 4.2.1.3.8.

Lemma 4.2.1.4.5. The operator Z2 is compact and continuous.

Proof. Here we shall prove only the compactness of Z2. Let {uˆ_εδn}^∞_n=1 be a bounded sequence in F_ε^u. The proof will be done if we can show that up to a subsequence the r.h.s of the PDE

∂u_εδn

∂t − ∇ ·(∇uε_δn−q_εu_εδn) +κu_εδn =S1R(ˆ¯ u_εδn, v_εδn) +κˆu_εδn (4.2.156) is strongly convergent in L^p((0, T);H^1,q(Ω^p_ε)^∗)^I1, where v_εδn ∈ G_ε^v is the solution of the problem

∂v_εδn

∂t − ∇ ·(D∇v_εδn−q_εv_εδn) =S2R(ˆ¯ u_εδn, v_εδn) in (0, T)×Ω^p_ε, (4.2.157) v_εδn(0, x) =v0(x) in Ω^p_ε, (4.2.158)

−(D∇v_εδn−q_εv_εδn)·n= 0 on (0, T)×∂Ω_in, (4.2.159)

−D∇v_εδn·n= 0 on (0, T)×∂Ω_out, (4.2.160)

−D∇vε_δn·n=ε∂w_εδn

∂t on (0, T)×Γ_ε, (4.2.161)

with estimate (4.2.109) and w_εδn ∈ H^w_ε is the solution of the problem

∂w_εδn

∂t =−k_dψ_δ(w_εδn) on (0, T)×Γ_ε, (4.2.162)

w_εδn(0, x) =w0(x) on Γ_ε. (4.2.163)

Thus the sequence {vε_δn}^∞_n=1 is bounded in G^v_ε. Since F_ε^u, G_ε^v →→L^∞((0, T)×Ω^p_ε), up to a subsequence (still denoted by same symbol), {uˆ_εδn}^∞_n=1 and {v_εδn}^∞_n=1 are strongly convergent in L^∞((0, T)×Ω^p_ε) and this yields the strong convergence of the r.h.s of the

PDE (4.2.156) in L^q((0, T);H^1,q(Ω^p_ε)^∗)^I1.

4.2. Model M2 83 Proof of theorem 4.2.1.4.1. The corollary 4.2.1.4.5 and lemma 4.2.1.4.5 show that the conditions of Schaefer’s fixed point theorem are satisfied. Hence there exists at least one fixed point of Z2, i.e., the problem (P_ε²⁺

δM) has a solution (u_εδ, v_εδ, w_εδ)∈ F_ε^u× G_ε^v× H^w_ε. The solution of (P_ε²⁺

δM) is also a solution of (P_ε²_δ⁺).

Proof of theorem 4.2.1.1.1. Since in lemma 4.2.1.1.2we have shown that the solution of (P_ε²_δ⁺) is nonnegative, the solution also solves the problem (P_ε²_δ). In the next section, we

prove the uniqueness of the solution of (P_ε²_δ).

4.2.1.5 Uniqueness of the Solution of the Problem (P_ε²_δ)

Theorem 4.2.1.5.1. There exists a unique positive global solution(u_εδ, v_εδ, w_εδ)∈ F_ε^u×G_ε^v× H^w_ε of the problem (P_ε²

δ).

Proof. On the contrary, let us assume that (u_εδ,1, v_εδ,1, w_εδ,1) and (u_εδ,2, v_εδ,2, w_εδ,2) be the solutions of the problem (P_ε²_δ). Set ¯u_εδ:=u_εδ,1−u_εδ,2, ¯v_εδ:=v_εδ,1−v_εδ,2 and ¯w_εδ :=w_εδ,1− w_εδ,2. Let the systems satisfied by (u_εδ,1, v_εδ,1, w_εδ,1) and (u_εδ,2, v_εδ,2, w_εδ,2) be denoted by (P_ε²_δ

1) and (P_ε²_δ

2) respectively. Substracting the systems of equations of (P_ε²_δ

1) and (P_ε²_δ

2), we get

∂u¯_εδ

∂t − ∇ ·(D∇¯u_εδ−q_εu¯_εδ) =S1R(u_εδ,1, v_εδ,1)−S1R(u_εδ,2, v_εδ,2) in (0, T)×Ω^p_ε, (4.2.164)

¯

u_εδ(0) =0 in Ω^p_ε, (4.2.165)

−(D∇u¯_εδ−q_εu¯_εδ)·n =0 on (0, T)×∂Ω_in, (4.2.166)

−D∇¯u_εδ·n=0 on (0, T)×∂Ω_out, (4.2.167)

−D∇¯u_εδ·n=0 on (0, T)×Γ_ε, (4.2.168)

∂¯v_εδ

∂t − ∇ ·(D∇¯v_εδ−q_εv¯_εδ) =S2R(u_εδ,1, v_εδ,1)−S2R(u_εδ,2, v_εδ,2) in (0, T)×Ω^p_ε, (4.2.169)

¯

v_εδ(0) =0 in Ω^p_ε, (4.2.170)

−(D∇¯v_εδ−q_ε¯v_εδ)·n=0 on (0, T)×∂Ω_in, (4.2.171)

−D∇v¯_εδ·n=0 on (0, T)×∂Ω_out, (4.2.172)

−D∇¯v_εδ·n=ε∂w¯_εδ

∂t on (0, T)×Γ_ε, (4.2.173)

∂w¯_εδ

∂t =−k_d(ψ_δ(w_εδ,1)−ψ_δ(w_εδ,2)) on (0, T)×Γ_ε, (4.2.174)

¯

w_εδ(0) =0 on Γ_ε. (4.2.175)

(i) Uniqueness of the ODE (4.2.42)-(4.2.43): Here ¯w_εδ :=w_εδ,1−w_εδ,2∈H^1,p((0, T);

L^p(Γ_ε))^I2. We multiply the ODE (4.2.174) with ¯w_εδ and integrate over (0, t)×Γ_ε. Em-ployement of theLipschitz continuityofψ_δand a straightforward application of Gronwall’s inequality yield the desired result.

(ii)Uniqueness of the PDE (4.2.32)-(4.2.36) and (4.2.37)-(4.2.41): Testing the equation

(4.2.169) with ¯v_εδ, we obtain 1

2

I2

k=1

" _t

0

d dθ

((((((¯v_εδk(t)⁽⁽((((²

L²(Ω^p_ε)dθ+D

I2

k=1

" _t

0

"

Ω^p_ε

(((∇v¯_εδk⁽⁽(²dx dθ

+

I2

k=1

" _t

0

"

Ω^p_εq_ε∇¯v_εδkv¯_εδkdx dθ

−^I2

k=1

" _t

0

"

∂Ω_inq_ε·n⁽⁽(¯v_εδk⁽⁽(²ds dθ+

I2

k=1

" _t

0

"

Γ_εε∂w¯_εδk

∂t v¯_εδkdσ_xdθ

=

I2

k=1

" _t

0

4

S2R(u_εδ,1, v_εδ,1)_k−S2R(u_εδ,2, v_εδ,2)_k, v_εδk,1−v_εδk,2

5 dθ, i.e.,

1 2

I2

k=1

" _t

0

d dθ

(((¯v_εδk(t)⁽⁽(²

L²(Ω^p_ε)dθ+D

I2

k=1

" _t

0

"

Ω^p_ε

(((∇¯v_εδk⁽⁽(²dx dθ

=

I2

k=1

" _t

0

"

∂Ω_inq_ε·n⁽⁽(v¯_εδk⁽⁽(²ds dθ−^I2

k=1

" _t

0

"

Γ_εε∂w¯_εδk

∂t ¯v_εδkdσ_xdθ

=:Ibound

−^I2

k=1

" _t

0

"

Ω^p_ε

q_ε∇v¯_εδkv¯_εδkdx dθ

=:Iadvec

+

I2

k=1

" _t

0

4

S2R(u_εδ,1, v_εδk,1)_k−S2R(u_εδ,2, v_εδk,2)_k, v_εδk,1−v_εδk,2

5 dθ

=:Ireac

. (4.2.176)

We simplify the boundary, advective and reaction terms separately. We start with the advective term.

I_advec = −^I2

k=1

" _t

0

"

Ω^p_εq_ε∇¯v_εδkv¯_εδkdx dθ

≤ ^I2

k=1

" _t

0

"

Ω^p_ε|q_ε|⁽⁽(∇¯v_εδk⁽⁽((((v¯_εδk⁽⁽(dx dθ

≤ Q

I2

k=1

" _t

0

"

Ω^p_ε

(((∇¯v_εδk⁽⁽((((v¯_εδk⁽⁽(dx dθ, where Q=||q_ε||_L∞((0,T)×Ω^p_ε)

≤

Young’s

2Q² D

I2

k=1

" _t

0

"

Ω^p_ε

(((¯v_εδk⁽⁽(²dx dθ+D 8

I2

k=1

" _t

0

"

Ω^p_ε

(((∇¯v_εδk⁽⁽(²dx dθ

inequality

≤ C

I2

k=1

" _t

0

"

Ω^p_ε

(((¯v_εδk⁽⁽(²dx dθ+D 8

I2

k=1

" _t

0

"

Ω^p_ε

(((∇¯v_εδk⁽⁽(²dx dθ. (4.2.177)

Next we simplify the boundary term.

I_bound=

I2

k=1

" _t

0

"

∂Ω_in

q_ε·n⁽⁽(¯v_εδk⁽⁽(²ds dθ−^I2

k=1

" _t

0

"

Γ_ε

ε∂w¯_εδk

∂t v¯_εδkdσ_xdθ.

4.2. Model M2 85 By part (i), w_εδ,1(t, x) =w_εδ,2(t, x) for a.e. t and x. This implies that the boundary term on Γ_ε vanishes. On ∂Ω_in,q_ε·n≤0. Thus the integrand on ∂Ω_in is nonpositive. Therefore

I_bound≤0. (4.2.178)

Finally, we simplify the reaction term.

I_reac=

I2

k=1

" _t

0

4

S2R(u_εδ,1, v_εδ,1)_k−S2R(u_εδ,2, v_εδ,2)_k, v_εδk,1−v_εδk,2

5 dθ

≤1 2

I2

k=1

" _t

0

!

||S2R(u_εδ,1, v_εδ,1)_k−S2R(u_εδ,2, v_εδ,2)_k||²_L2(Ω^p_ε)+⁽⁽₍⁽⁽₍v_εδk,1−v_εδk,2((((((²

L²(Ω^p_ε)

# dθ.

(4.2.179) Note that

||S2R(u_εδ,1, v_εδ,1)−S2R(u_εδ,2, v_εδ,2)||²_L2(Ω^p_ε)

≤

"

Ω^p_ε

⎛

⎝^J

j=1

|νkj||Rj(u_εδ,1, v_εδ,1)−R_j(u_εδ,2, v_εδ,2)|

⎞

⎠

2

dx.

Expanding the termR_j(u_εδ,1, v_εδ,1)−R_j(u_εδ,2, v_εδ,2), we will obtain two terms in which each term contains a factor of the type u_εδl,1−u_εδl,2 and v_εδm,1−v_εδm,2 whereas all the other factors are in L^∞((0, T)×Ω^p_ε). Therefore we obtain

||S2R(u_εδ,1, v_εδ,1)−S2R(u_εδ,2, v_εδ,2)||²_L2(Ω^p_ε)

≤Cˆ

⎡

⎣^I1

i=1

"

Ω^p_ε

(((u_εδi,1−u_εδi,2(((²dx+

I2

k=1

"

Ω^p_ε

(((v_εδk,1−v_εδk,2(((²dx

⎤

⎦. (4.2.180)

Combining (4.2.176), (4.2.177), (4.2.178), (4.2.179) and (4.2.180), we obtain 1

2

I2

k=1

" _t

0

d dθ

((((((v¯_εδk(t)⁽⁽((((²

L²(Ω^p_ε)dθ+D

I2

k=1

" _t

0

"

Ω^p_ε

(((∇¯v_εδk⁽⁽(²dx dθ

=I_bound+I_advec+I_reac

≤0 +C

I2

k=1

" _t

0

"

Ω^p_ε

(((v¯_εδk⁽⁽(²dx dθ+D 8

I2

k=1

" _t

0

"

Ω^p_ε

(((∇v¯_εδk⁽⁽(²dx dθ

+C

⎡

⎣" _t

0

"

Ω^p_ε I1

i=1

(((u¯_εδi⁽⁽(²dx dθ+

" _t

0

"

Ω^p_ε I2

k=1

(((v¯_εδk⁽⁽(²dx dθ

⎤

⎦

=⇒ 1 2

I2

k=1

" _t

0

d dθ

((((((v¯_εδk(t)⁽⁽((((²

L²(Ω^p_ε)dθ≤C¯1

⎛

⎝^I2

k=1

" _t

0

"

Ω^p_ε

(((¯v_εδk⁽⁽(²dx dθ+

I1

i=1

" _t

0

"

Ω^p_ε

(((u¯_εδi⁽⁽(²dx dθ

⎞

⎠. (4.2.181) Now we test the equation (4.2.164) by ¯u_εδ and proceed in the similar fashion as above, we obtain an inequality like (4.2.181) as

1 2

I1

i=1

" _t

0

d dθ

((((((u¯_εδi(t)⁽⁽((((²

L²(Ω^p_ε)dθ≤C¯2

⎛

⎝^I2

k=1

" _t

0

"

Ω^p_ε

(((v¯_εδk⁽⁽(²dx dθ+

I1

i=1

" _t

0

"

Ω^p_ε

(((u¯_εδi⁽⁽(²dx dθ

⎞

⎠. (4.2.182)

Adding (4.2.181) and (4.2.182), we get 1

2

" _t

0

d dθ

||¯u_εδ||²_[L2(Ω^p_ε)]^I1+||¯v_εδ||²_[L2(Ω^p_ε)]^I2

dθ≤C¯3

" _t

0

||¯u_εδ||²_[L2(Ω^p_ε)]^I1+||¯v_εδ||²_[L2(Ω^p_ε)]^I2

dθ, i.e.,

||u¯_εδ(t)||²_[L2(Ω^p_ε)]^I1+||¯v_εδ(t)||²_[L2(Ω^p_ε)]^I2 ≤2 ¯C3

" _t

0

||¯u_εδ(θ)||²_[L2(Ω^p_ε)]^I1+||¯v_εδ(θ)||²_[L2(Ω^p_ε)]^I2

dθ.

Since u_εδi,1(0) =u_εδi,2(0) and v_εδk,1(0) =v_εδk,2(0) for all i and k, therefore Gronwall’s in-equality gives

||¯u_εδ(t)||²_[L2(Ω^p_ε)]^I1+||¯v_εδ(t)||²_[L2(Ω^p_ε)]^I2 = 0 for a.e. t

=⇒u_εδ,1=u_εδ,2 andv_εδ,1=v_εδ,2.

Hence the problem (P_ε²_δ) has a unique positive global weak solution inF_ε^u× G_ε^v× H_ε^w.