A reaction diffusion model for understanding phyllotactic formation

Abstract

Phyllotactic patterns in plants are well known to be related to the golden ratio. Actually, many mathematical models using the theoretical inhibitory effect were proposed to reproduce these phyllotactic patterns. In 1996, Douady and Couder introduced a model using magnetic repulsion and succeeded in reproducing phyllotactic patterns numerically. On the other hand, it was recently revealed in biological experiments that a plant hormone, auxin, regulates the phyllotactic formation as an activator (Reinhardt et al., Nature 426:255–260, 2003). Then, there arises a natural question as to how the inhibitory effect can be related to the auxin. In this paper, a reaction diffusion model is proposed by taking account of auxin behavior in plant tips. The relationship between Douady and Couder’s model and our model is shown by singular limit analysis. It also provides us with the potential function corresponding to the inhibitory effect, and the bifurcation diagram.

Appendix

Here we will provide an outline of the proof of the theorem and explain the key proposition to overcome the difficulty in the proof. We divide $S_n^{\varepsilon }(x,t)$ defined in (6) into the following two parts:

$$\begin{aligned} P_n^{\varepsilon }(x,t) : =&\int _{{\mathbb {R}}^2} K \left( x - y, \frac{ 1 }{ \varepsilon }( t - ( n - 1 )T ) \right) e^{ - \frac{ b }{ \varepsilon }( t - ( n - 1 )T ) }S^{\varepsilon }_{n-1} (y,(n-1)T)dy, \\ N_n^{\varepsilon }(x,t) : =&\int _{ 0}^{\frac{1}{\varepsilon }(t-(n-1)T)} ae^{ -b \tau }\sum \limits _{j=0}^{n-1} K\left( x - X^{\varepsilon }_j( t - \varepsilon \tau ), \tau \right) d\tau . \end{aligned}$$

To show that $S^{ \varepsilon }_n(x,t)$ is the weak solution of $(M_n^{\varepsilon })$, we introduce the following auxiliary system:

$$\begin{aligned} (M_n^{\varepsilon ,\eta })\left\{ \begin{array}{l@{\quad }l} X^{\varepsilon ,\eta }_j(t) = ( R_0 + (t-jT)V )\left( \cos \theta ^{\varepsilon ,\eta }_j , \sin \theta ^{\varepsilon ,\eta }_j \right) , &{} t \in ((n-1)T, nT], \\ \varepsilon \dfrac{\partial S_n}{\partial t} = D\varDelta S_n - bS_n + a\sum \limits _{j=0}^{n-1} \rho _{\eta } \left( x - X^{\varepsilon ,\eta }_j(t)\right) , &{} (x,t) \in {{\mathbb {R}}}^2 \times ((n-1)T, nT], \\ S_n(x,(n-1)T)=S_{n-1}(x,(n-1)T), &{} x \in {\mathbb {R}}^2, \end{array} \right. \end{aligned}$$

where $j=0, \ldots , n-1$, $n \ge 1$, and $\rho _{\eta }(x)$ is the Friedrichs mollifier with a small parameter $0<\eta \ll 1$ defined by

$$\begin{aligned} \rho _{\eta }(x)=\frac{1}{\eta }\rho (\frac{1}{\eta }x), \quad \rho (x) := \left\{ \begin{array}{l@{\quad }l} C \exp \left( -\frac{1}{1- \left| x \right| ^2} \right) , &{} \left| x \right| < 1,\\ 0, &{} \left| x \right| \ge 1. \end{array}\right. \end{aligned}$$

The initial conditions for $(S_n, \{X_j^{\varepsilon ,\eta } \}_{j=0}^{2})$, are set similarly to those of $(M_n^{\varepsilon })$. Using the heat kernel, we have the classical solution of $(M_n^{\varepsilon ,\eta })$,

$$\begin{aligned} S_n^{\varepsilon , \eta }(x,t) =&\int _{{\mathbb {R}}^2} K\left( x-y, \frac{1}{\varepsilon } ( t-(n-1)T) \right) e^ { -\frac{b}{\varepsilon }(t- (n-1)T) } S^{\varepsilon ,\eta }_{n-1} (y,(n-1)T)dy \\&+\int ^{ \frac{1}{\varepsilon }(t-(n-1)T) }_ { 0 } \int _{{\mathbb {R}}^2} a e ^ { - b\tau } K\left( x-y, \tau \right) \sum \limits _{j=0}^{n-1}\rho _{\eta } \left( y - X^{\varepsilon ,\eta }_j(t -\varepsilon \tau )\right) dyd\tau . \end{aligned}$$

Theorem A.1

Assume that

$(a1)_n$ :

$S^{\varepsilon , \eta }_{n-1}(\cdot , (n-1)T), \ S^{\varepsilon }_{n-1}(\cdot , (n-1)T) \in L^{1}({{\mathbb {R}}}^2)$,

$(a2)_n$ :

$\sup _{\varepsilon ,\eta >0} \Vert S^{\varepsilon ,\eta }_{n-1}(\cdot , (n-1)T) \Vert _{L^1({{\mathbb {R}}}^2)} <C_1$,

$(a3)_n$ :

$\sup _{\varepsilon >0} \Vert S^{\varepsilon }_{n-1}(\cdot , (n-1)T) \Vert _{L^1({{\mathbb {R}}}^2)} <C_2$,

$(a4)_n$ :

$\left\| S^{\varepsilon , \eta }_{n-1}(\cdot , (n-1)T) - S^{\varepsilon }_{n-1}(\cdot , (n-1)T) \right\| _{L^1({{\mathbb {R}}}^2)} \rightarrow 0 \ as \ \eta \rightarrow +0$,

$(a5)_n$ :

$\theta _{j }^{\varepsilon , \eta } \rightarrow \theta _{j }^{\varepsilon } \ as \ \eta \rightarrow +0 \ (j=2, \ldots , n-1)$,

$(a6)_n$ :

$\theta _{j }^{\varepsilon } \rightarrow \theta _{j }^{0} \ as \ \varepsilon \rightarrow +0 \ (j=2, \ldots , n-1)$.

Then $S_n^{\varepsilon }(x,t)$ is the unique weak solution of $(M_n^{\varepsilon })$ for $t \in ((n-1)T, nT]$. Moreover,

$$\begin{aligned} \left\| S^{\varepsilon }_n(\cdot , t) - g_n(\cdot , t) \right\| _{L^1({\mathbb {R}}^2)} \rightarrow 0 \end{aligned}$$

for $t \in ((n-1)T, nT]$ and

$$\begin{aligned} \min { \tilde{S}}_n^{\varepsilon } \rightarrow \min { \tilde{S}}_n^{0} \end{aligned}$$

(A.1)

as $\varepsilon \rightarrow +0$. Additionally, if $\theta \in [0,2\pi )$ attaining the minimum of ${ \tilde{S}}^{0}_n(\theta )$ on $C_{R_0}(\theta )$ is unique,

$$\begin{aligned} \theta _n^{\varepsilon } \rightarrow \theta _n^{0} \end{aligned}$$

as $\varepsilon \rightarrow +0$.

Proof of Theorem 4.1

Theorem 4.1 follows from Theorem A.1 via the induction. Indeed, $S_1^{\varepsilon }(x,t)$ and $S_1^{0}(x,t)$ satisfy the all assumptions $(a1)_1$–$(a6)_1$ for (0, T]. Hence, we obtain that $S_1^{\varepsilon }(x,t)$ is the unique weak solution of $(M_1^{\varepsilon })$, $\left\| S^{\varepsilon }_1(\cdot , t) - g_1(\cdot , t) \right\| _{L^1({\mathbb {R}}^2)} \rightarrow 0$ for (0, T] as $\varepsilon \rightarrow +0$, $\left\| { \tilde{S}}_{1}^{\varepsilon }( \cdot ) - { \tilde{g}_1}(\cdot ) \right\| _{C^1( [0, 2\pi ) )} \rightarrow 0$ and $\theta ^{\varepsilon }_1 \rightarrow \theta ^{0}_1$ as $\varepsilon \rightarrow +0$ since $\theta _1^0$ is unique at $t=T$. Thus, the first step of the induction is established. Next suppose that the theorem for $( 0, (n-1)T]$ holds. All assumptions $(a1)_n$–$(a6)_n$ can be confirmed by Theorem A.1. Therefore, by induction, the proof of Theorem 4.1 is completed. $\square$

Proof of Theorem A.1

We mainly explain the outline of the proof. From the assumptions $(a1)_n$–$(a5)_n$, we can prove $\Vert S_n^{\varepsilon ,\eta }(\cdot , t)- S_n^{\varepsilon }(\cdot , t) \Vert _{L^1({{\mathbb {R}}}^2)} \rightarrow 0$ as $\eta \rightarrow +0$ by the continuity of $\varPsi _h$ in $L^1({{\mathbb {R}}}^2)$ where $\varPsi _h$ is the shift operator defined by $\varPsi _hf(\cdot ):=f(x-h)$ for $x,h \in {{\mathbb {R}}}^2$, namely, $\Vert \varPsi _h f(\cdot ) -f(\cdot ) \Vert _{L^1({{\mathbb {R}}}^2)} \rightarrow 0$ as $h \rightarrow 0$. It also follows from $(a1)_n$, $(a4)_n$, $(a5)_n$ and the dominated convergence theorem that $\Vert {\tilde{S}}_n^{\varepsilon ,\eta }(\cdot )- {\tilde{S}}_n^{\varepsilon }(\cdot ) \Vert _{C^1([0, 2\pi ))} \rightarrow 0$ as $\eta \rightarrow +0$. Thus, we can determine that $\theta _n^{\varepsilon , \eta } \rightarrow \theta ^{\varepsilon }$ as $\eta \rightarrow +0$. Using $\Vert S_n^{\varepsilon ,\eta }(\cdot , t)- S_n^{\varepsilon }(\cdot , t) \Vert _{L^1({{\mathbb {R}}}^2)} \rightarrow 0$ for $t \in [(n-1)T, nT]$ and $(a5)_n$, we can show that $S_n^{\varepsilon }(x, t)$ is the unique weak solution of $(M_n^{\varepsilon })$ for $t \in ((n-1)T, nT]$ by the Hölder inequality. From $(a3)_n$ and $(a6)_n$ we can show $\Vert {\tilde{S}}_n^{\varepsilon }(\cdot )- {\tilde{S}}_n^{0}(\cdot ) \Vert _{C^1([0, 2\pi ))} \rightarrow 0$ as $\varepsilon \rightarrow +0$ which implies (A.1). Since $S_n^{\varepsilon }$ has the singularity points moving along $X^{\varepsilon }_j( t - \varepsilon \tau ) (j=0,\ldots ,n-1)$ for $\tau \in [0, (t-(n-1)T)/\varepsilon ]$, we cannot apply the dominated convergence theorem to show (7). We prepare the following proposition. $\square$

Proposition A.1

Under the assumptions $(a3)_n$ and $(a6)_n$,

$$\begin{aligned} \left\| S^{\varepsilon }_n(\cdot ,t) - S^{0}_n(\cdot ,t) \right\| _{L^1({\mathbb {R}}^2)} \rightarrow 0 \end{aligned}$$

as $\varepsilon \rightarrow +0$ for $t \in ((n-1)T,nT]$.

Proof

It is easily shown that $S_n^{\varepsilon }( \cdot ,t),S_n^{0}( \cdot ,t) \in L^1({\mathbb {R}}^2)$. Therefore, we omit it. Next, we show $\Vert S^{\varepsilon }_n(\cdot ,t) - S^{0}_n(\cdot ,t) \Vert _{L^1({\mathbb {R}}^2)} \rightarrow 0$ as $\varepsilon \rightarrow +0$. Note that $S_n^{\varepsilon }( x,t) - S_n^{0}( x,t) = P_n^{\varepsilon }( x,t) + N_n^{\varepsilon }( x,t) - S_n^{0}( x,t)$. Since

$$\begin{aligned} \left\| P_n^{\varepsilon }( \cdot ,t) \right\| _{L^1({\mathbb {R}}^2)}= & {} e^{ -\frac{b}{\varepsilon }(t-(n-1)T)} \int _{{\mathbb {R}}^2}S^{\varepsilon }_{n-1}(y,(n-1)T) dy\\\le & {} C e^{ -\frac{b}{\varepsilon }(t-(n-1)T)} \rightarrow 0 \end{aligned}$$

as $\varepsilon \rightarrow +0$, we only need to estimate $\Vert N_n^{\varepsilon }( \cdot ,t) - S^{0}_n( \cdot ,t) \Vert _{L^1({\mathbb {R}}^2)}$. We have that

$$\begin{aligned} \left| N^{\varepsilon }_n - S^{0}_n \right|&\le \int ^{ \frac{1}{\varepsilon } (t-(n-1)T) } _0 ae^{-b\tau } \sum \limits _{j=0}^{n-1} \left| K( x-X^{\varepsilon }_j( t - \varepsilon \tau ), \tau ) - K( x-X^0_j( t ), \tau ) \right| d\tau \\&\ \ \ \ \ \ + \int ^{\infty }_{\frac{1}{\varepsilon } (t-(n-1)T)} ae^{ -b \tau } \sum \limits _{j=0}^{n-1} K( x-X^0_j( t ), \tau ) d\tau . \end{aligned}$$

We denote the first and second terms on the right hand side by $Q_n^{\varepsilon }(x,t)$ and $R_n^{\varepsilon }(x,t)$, respectively. We calculate that

$$\begin{aligned} \left\| R_n^{\varepsilon } ( \cdot , t ) \right\| _{L^1({{\mathbb {R}}}^2)}&= \int _{{{\mathbb {R}}}^2} \int ^{ \infty } _{\frac{1}{\varepsilon }(t-(n-1)T)} ae^{ -b \tau } \sum \limits _{j=0}^{n-1} K( x-X^0_j( t ), \tau ) d\tau dx\\&= \int ^{ \infty } _{\frac{1}{\varepsilon }(t-(n-1)T)} ane^{-b\tau }d\tau \\&=\frac{an}{b}e^{-\frac{b}{\varepsilon }(t-(n-1)T)} \rightarrow 0 \end{aligned}$$

as $\varepsilon \rightarrow +0$. Because $N^{\varepsilon }_n(x,t)$ and $S^{0}_n(x,t)$ have singularities at $X_j^{\varepsilon }(t-\varepsilon \tau )$ and $X_j^{0}(t)$, respectively, we divide $\Vert Q_n^{\varepsilon } \Vert _{L^1({{\mathbb {R}}}^2)}$ into three parts.

$$\begin{aligned} H_n^{\varepsilon }( t):= & {} \int _{{{\mathbb {R}}}^2} \int ^{ \sigma } _0 ae^{-b\tau } \sum \limits _{j=0}^{n-1} \left| K( x-X^{\varepsilon }_j( t - \varepsilon \tau ), \tau ) - K( x-X^0_j( t ), \tau ) \right| d\tau dx, \\ I_n^{\varepsilon }( t):= & {} \int _{\varOmega _\varepsilon } \int ^{ \frac{1}{\varepsilon } (t-(n-1)T) } _{ \sigma } ae^{-b\tau }\\&\quad \times \sum \limits _{j=0}^{n-1} \left| K( x-X^{\varepsilon }_j( t - \varepsilon \tau ), \tau ) - K( x-X^0_j( t ), \tau ) \right| d\tau dx, \\ J_n^{\varepsilon }( t ):= & {} \int _{{{\mathbb {R}}}^2 \backslash \varOmega _\varepsilon } \int ^{ \frac{1}{\varepsilon } (t-(n-1)T) } _{ \sigma } ae^{-b\tau }\\&\quad \times \sum \limits _{j=0}^{n-1} \left| K( x-X^{\varepsilon }_j( t - \varepsilon \tau ), \tau ) - K( x-X^0_j( t ), \tau ) \right| d\tau dx, \end{aligned}$$

where $0<\sigma \ll 1$ and

$$\begin{aligned} \varOmega _{\varepsilon }:= \bigcup _{ \tau \in [ \sigma , \frac{1}{\varepsilon }(t-(n-1)T)] } \left\{ x\in {\mathbb {R}}^2 \bigg | \ \ \left| x - X^{\varepsilon }_j(t - \varepsilon \tau ) \right| \bigg . < \frac{1}{2} \left| x - X^0_j(t) \right| \right\} . \end{aligned}$$

Firstly, we estimate $H_n^{\varepsilon }( t)$. As $C_0( {{\mathbb {R}}}^2 \times [0, \sigma ])$ is dense in $L^1( {{\mathbb {R}}}^2 \times [0, \sigma ])$, for any ${\tilde{\varepsilon }} > 0$ there exists a function $\xi (\cdot ,\cdot ) \in C_0( {{\mathbb {R}}}^2 \times [0, \sigma ])$ such that $\Vert K(\cdot ,\cdot ) - \xi (\cdot ,\cdot ) \Vert _{L^1({{\mathbb {R}}}^2 \times [0,\sigma ))} < {\tilde{\varepsilon }}$. Moreover, for any ${ \tilde{\varepsilon }}>0$, there exists ${ \tilde{\delta }}>0$, such that whenever $\left| X^0_j(t )-X^{\varepsilon }_j(t -\varepsilon \tau ) \right| < { \tilde{\delta }}$ for $\tau \in [0, \sigma ]$, we have $\left| \varPsi _{X^0_j(t )-X^{\varepsilon }_j(t -\varepsilon \tau ) }\xi (\cdot , \tau ) - \xi (x, \tau ) \right| < { \tilde{\varepsilon }}$ since $\theta ^{\varepsilon } _j \rightarrow \theta ^{0}_j$ as $\varepsilon \rightarrow +0 \ (j=1, \ldots , n-1)$. Thus we have

$$\begin{aligned} H_n^{\varepsilon } ( t )= & {} \int ^{ \sigma } _0 a e^{-b\tau } \sum \limits _{j=0}^{n-1} \int _{{{\mathbb {R}}}^2} \left| \varPsi _{X^0_j(t )-X^{\varepsilon }_j(t -\varepsilon \tau ) }K(\cdot , \tau )- K(x , \tau ) \right| dxd\tau \\\le & {} \int ^{ \sigma } _0 a e^{-b\tau } \sum \limits _{j=0}^{n-1} \int _{{{\mathbb {R}}}^2} \left| \varPsi _{X^0_j(t )-X^{\varepsilon }_j(t -\varepsilon \tau ) }K(\cdot , \tau )- \varPsi _{X^0_j(t )-X^{\varepsilon }_j(t -\varepsilon \tau ) }\xi (\cdot ,\tau ) \right| dxd\tau \\&+ \int ^{ \sigma } _0 a e^{-b\tau } \sum \limits _{j=0}^{n-1} \int _{{{\mathbb {R}}}^2} \left| \varPsi _{X^0_j(t )-X^{\varepsilon }_j(t -\varepsilon \tau ) }\xi (\cdot , \tau )- \xi (x , \tau ) \right| dxd\tau \\&+ \int ^{ \sigma } _0 a e^{-b\tau } \sum \limits _{j=0}^{n-1} \int _{{{\mathbb {R}}}^2} \left| \xi (x, \tau )- K(x , \tau ) \right| dxd\tau \\\le & {} { \tilde{\varepsilon }} + 2\int ^{ \sigma } _0 a e^{-b\tau } \sum \limits _{j=0}^{n-1} \int _{{{\mathbb {R}}}^2} \left| \xi (x, \tau )- K(x , \tau ) \right| dxd\tau \\< & {} 3{ \tilde{\varepsilon }}. \end{aligned}$$

Secondly, we consider $I^{\varepsilon }_n(t)$. If $x \in \varOmega _{\varepsilon }$,

$$\begin{aligned}&\left| \mid x - X^{\varepsilon }_j(t-\varepsilon \tau ) \mid ^2 -\mid x - X^0_j(t) \mid ^2 \right| \\&\quad \le \left| \mid x - X^{\varepsilon }_j(t-\varepsilon \tau ) \mid + \mid x - X^0_j(t) \mid \right| \left| X^{\varepsilon }_j(t-\varepsilon \tau ) - X^0_j(t) \right| \\&\quad \le M\left( (V \varepsilon \tau )^2 +C_n( \varepsilon + ( 1 - \cos ( \theta _j - \theta ^{\varepsilon }_j) ) ) \right) ^{1/2},\\ \end{aligned}$$

where

$$\begin{aligned} M := \sup _{x \in \varOmega _{\varepsilon }} \left\{ \mid x - X^{\varepsilon }_j(t-\varepsilon \tau ) \mid + \mid x - X^0_j(t) \mid \right\} , \quad C_n = 2 (R_0 + nVT )^2. \end{aligned}$$

Thus, using $(a6)_n$, we compute

$$\begin{aligned}&I^{\varepsilon }_n(t)\\= & {} \int _{\varOmega _\varepsilon } \int ^{ \frac{1}{\varepsilon } (t-(n-1)T) } _{ \sigma } \sum \limits _{j=0}^{n-1} \frac{ae^{-b\tau }}{4\pi D \tau } e^{- \frac{ \left| x-X^0_j( t ) \right| ^2 }{4D \tau }} \left| e^{- \frac{ \left| x-X^{\varepsilon }_j( t - \varepsilon \tau ) \right| ^ 2 -\left| x-X^0_j( t ) \right| ^2 }{4D \tau }} - 1 \right| d\tau dx\\\le & {} \sum \limits _{j=0}^{n-1} \left| e^{ \frac{ M \left( (V\varepsilon \sigma )^2 + C_n (\varepsilon + ( 1 - \cos ( \theta ^{0}_j - \theta ^{\varepsilon }_j) ) ) \right) ^{\frac{1}{2}} }{4D \sigma }} - 1 \right| \int _{\varOmega _\varepsilon } \int ^{ \frac{1}{\varepsilon } (t-(n-1)T) } _{ \sigma } \frac{ ae^{-b\tau }}{4\pi D \tau } e^{- \frac{ \left| x-X^0_j( t ) \right| ^2 }{4D \tau }} d\tau dx\\&\rightarrow 0 \end{aligned}$$

as $\varepsilon \rightarrow 0$. Finally,we consider $J_n^{\varepsilon }(t)$. Set

$$\begin{aligned} f_n^{\varepsilon }(x,\tau )=ae^{-b\tau } \sum \limits _{j=0}^{n-1} \left| K( x-X^{\varepsilon }_j( t - \varepsilon \tau ), \tau ) - K( x-X^0_j( t ), \tau ) \right| . \end{aligned}$$

Then

$$\begin{aligned} J_n^{\varepsilon }(t)&= \int _{{{\mathbb {R}}}^2 \backslash \varOmega _\varepsilon } \int ^{ \frac{1}{\varepsilon } (t-(n-1)T) } _{ \sigma } f_n^{\varepsilon }(x,\tau ) d\tau dx = \int _{{{\mathbb {R}}}^2 } \int ^{ \infty } _{ \sigma } f_n^{\varepsilon }(x,\tau )\chi _{\varTheta _n}(x,\tau )dx, \end{aligned}$$

where $\chi _{\varTheta _n}(x,\tau )$ is a characteristic function and

$$\begin{aligned} \varTheta _n:= \left\{ {{\mathbb {R}}}^2 \backslash \varOmega _{\varepsilon } \right\} \times \left[ \sigma , \frac{1}{\varepsilon }(t-(n-1)T) \right) . \end{aligned}$$

As a result of the characteristic function $\chi _{\varTheta _n}(x,\tau )$, the domain of integration $J_n^{\varepsilon }(t)$ is independent of $\varepsilon$. Thus, we have that $f_n^{\varepsilon }(x,\tau )\chi _{\varTheta _n}(x,\tau ) \rightarrow 0$ pointwise in ${{\mathbb {R}}}^2 \times [\sigma , \infty ]$ as $\varepsilon \rightarrow +0$. Since $\left| x - X^{\varepsilon }_j(t - \varepsilon \tau ) \right| > \left| x - X^0_j(t) \right| /2$ for $x \notin \varOmega _{\varepsilon }, \ 0 \le \tau < (t-(n-1)T)/\varepsilon$, we have

$$\begin{aligned} \left| f_n^{\varepsilon }(x,\tau )\psi _n^{\varepsilon }(x,\tau ) \right|\le & {} ae^{-b\tau } \sum \limits _{j=0}^{n-1} \left\{ K\left( \frac{1}{2}(x-X^0_j( t )), \tau \right) + K( x-X^0_j( t ), \tau ) \right\} \\&\quad \times \psi _n^{\varepsilon }(x,\tau ) \\\le & {} ae^{-b\tau } \sum \limits _{j=0}^{n-1} \left\{ K\left( \frac{1}{2}(x-X^0_j( t )), \tau \right) + K( x-X^0_j( t ), \tau ) \right\} \end{aligned}$$

which is measurable in ${{\mathbb {R}}}^2 \times [\sigma ,\infty )$. Thus from the dominated convergence theorem, we compute that

$$\begin{aligned} \lim _{\varepsilon \rightarrow +0} J_n^{\varepsilon }(t)= & {} \lim _{\varepsilon \rightarrow +0} \int _{{{\mathbb {R}}}^2 \backslash \varOmega _\varepsilon } \int ^{ \frac{1}{\varepsilon } (t-(n-1)T) } _{ \sigma } f_n^{\varepsilon }(x,\tau ) d\tau dx \\= & {} \int _{{{\mathbb {R}}}^2 } \int ^{ \infty } _{ \sigma } \lim _{\varepsilon \rightarrow +0} f_n^{\varepsilon }(x,\tau )\psi _n^{\varepsilon }(x,\tau ) d\tau dx \\= & {} 0. \end{aligned}$$

$\square$

This proposition immediately implies (7). Thus, we have completed the proof of Theorem A.1. $\square$