We set Then and From (4.6) we note that By (4.14)-(4.15), we can apply the Carleman estimate (see Proposition 1) to w$_{1}$, w$_{2}$ : Here and henceforth, C > 0 denotes a generic constant which is independent of 26 w$_{k}$ = ( ∂ k t u ) χ, k = 1 , 2 . Aw$_{k}$ = f∂ k t Rχ + 2 ( ∇ $_{y}$∂ k t u · ∇ $_{y}$χ ) + ∂ k t u (Δ$_{y}$ χ + i∂$_{t}$χ ) , x ∈ D, y ∈ G$_{1}$, k = 1 , 2 (4.14) w$_{k}$ ( x, y, t ) = |∇ $_{y}$w$_{k}$ ( x, y, t ) | = 0 , ( x, y, t ) ∈ D × ∂G$_{1}$ × ( − T, T ) , w$_{k}$ ( x, y, t ) = 0 , ( x, y, t ) ∈ ∂D × G$_{1}$ × ( − T, T ) , w$_{k}$ ( x, y, T ) = w$_{k}$ ( x, y, − T ) = 0 , ( x, y, t ) ∈ D × G$_{1}$. (4.15) G$_{1}$ ⊂ G$_{2}$. (4.16) ∫ Ω$_{1}$ 2 ∑ k =1 ( s |∇ $_{x}$w$_{k}$ | 2 + s |∇ $_{y}$w$_{k}$ | 2 + s | w$_{k}$ | 2 ) e 2 $^{sϕ}$dxdydt ≤ C ∫ Ω$_{1}$ 2 k =1 χ $^{2}$f 2 ∣ ∂ k t R ∣ 2 e 2 $^{sϕ}$dxdydt + C ∫ 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 ∂D$_{+}$ × G$_{1}$ × ( − T,T ) 2 ∑ k =1 s | ∂$_{ν}$w$_{k}$ | 2 e 2 $^{sϕ}$dS$_{x}$ dydt = S$_{1}$ + S$_{2}$ + S$_{3}$. (4.17)