s > 0. From the assumption on R , we have By (4.12), we see that ∂$_{t}$χ = 0 for | t | ≤ T − 2 δ or T − δ ≤ | t | ≤ T and |∇ $_{y}$χ | = Δ$_{y}$ χ = 0 for | y − y$_{0}$ | ≤ L − 2 δ or L − δ ≤ | y − y$_{0}$ | ≤ L. Therefore, if | t | ∈ [0 , T − 2 δ ] ∪ [ T − δ, T ] and | y − y$_{0}$ | ∈ [0 , L − 2 δ ] ∪ [ L − δ, L ], then ∂$_{t}$χ = |∇ $_{y}$χ | = Δ$_{y}$ χ = 0. Hence Here and henceforth we set S$_{2}$ = C ( ∫ { T − 2 δ ≤| t |≤ T − δ }∩ Ω$_{1}$ 2 ∑ k =1 ∣ 2 ( ∇ $_{y}$∂ k t u · ∇ $_{y}$χ ) + ∂ k t u (Δ$_{y}$ χ + i∂$_{t}$χ ) ∣ 2 e 2 $^{sϕ}$dxdydt ) + C ( ∫ { L − 2 δ ≤| y − y$_{0}$ |≤ L − δ }∩ Ω$_{1}$ 2 ∑ k =1 ∣ 2 ( ∇ $_{y}$∂ k t u · ∇ $_{y}$χ ) + ∂ k t u (Δ$_{y}$ χ + i∂$_{t}$χ ) ∣ 2 e 2 $^{sϕ}$dxdydt ) = C ( ∫ { T − 2 δ ≤| t |≤ T − δ }∩ Ω$_{1}$ 2 ∑ k =1 ∣ 2 ( ∇ $_{y}$∂ k t u · ∇ $_{y}$χ ) + ∂ k t u (Δ$_{y}$ χ + i∂$_{t}$χ ) ∣ 2 ( exp(2 se − $^{γϵ}$) ) ) dxdydt + C ( ∫ { L − 2 δ ≤| y − y$_{0}$ |≤ L − δ }∩ Ω$_{1}$ 2 ∑ k =1 ∣ 2 ( ∇ $_{y}$∂ k t u · ∇ $_{y}$χ ) + ∂ k t u (Δ$_{y}$ χ + i∂$_{t}$χ ) ∣ 2 ( exp(2 se − $^{γϵ}$) ) ) dxdydt ) ≤ C ∫ Ω$_{1}$ 2 ∑ k =1 ( ∣ ∂ k t u ∣ 2 + ∇ $_{y}$∂ k t u ∣ 2 ) e 2 $^{sκ$_{1}$}$dxdydt ≤ CM $^{2}$e2 $^{sκ$_{1}$}$. (4.18) κ$_{1}$ = e − $^{γϵ}$, κ$_{2}$ = e $^{γϵ}$. S$_{1}$ = C ∫ Ω$_{1}$ 2 ∑ k =1 χ $^{2}$f 2 ∣ ∂ k t R ∣ 2 e 2 $^{sϕ}$dxdydt ≤ C Ω$_{1}$ χ $^{2}$f $^{2}$e2 $^{sϕ}$dxdydt. 27