We set where a$_{i}$, b$_{j}$ ∈ L $^{∞}$(Ω) , 0 ≤ i ≤ n, 1 ≤ j ≤ m. Proposition 1 Let us assume that 0 < α, β < 1 be small and γ > 0 be sufficiently large, and let with some δ$_{0}$ > 0 . Then there exist constants C > 0 and s$_{0}$ > 0 such that for all s ≥ s$_{0}$, provided that Proof Let us set and Lu := i∂$_{t}$u ( x, y, t ) + Δ$_{y}$ u ( x, y, t ) − Δ$_{x}$ u ( x, y, t ) + n ∑ i =1 a$_{i}$ ( x, y, t ) u$_{x}$$_{i}$ + m ∑ j =1 b$_{j}$ ( x, y, t ) u$_{y}$$_{j}$ + a$_{0}$ ( x, y, t ) u, (3.4) | x − x$_{0}$ | 2 − α $^{2}$| y | 2 − β 2 | t | 2 > δ 2 $_{0}$, ( x, y, t ) ∈ Ω (3.5) ∫ Ω ( s |∇ $_{y}$u | 2 + s |∇ $_{x}$u | 2 + s 3 | u | $^{2}$) e 2 $^{sϕ}$dxdydt ≤ C ∫ Ω | Lu | $^{2}$e 2 $^{sϕ}$dxdydt + C ∫ ∂D$_{+}$ × G × ( − T,T ) s | ∂$_{ν}$u | $^{2}$e 2 $^{sϕ}$dS$_{x}$ dydt Lu ∈ L $^{2}$(Ω) , u ∈ H $^{2}$(Ω) , u ( x, y, t ) = 0 , ( x, y, t ) ∈ Γ$_{x}$ , u ( x, y, t ) = |∇ $_{y}$u ( x, y, t ) | = 0 , ( x, y, t ) ∈ Γ$_{y}$ , u ( x, y, T ) = u ( x, y, − T ) = 0 , ( x, y ) ∈ D × G. (3.6) L$_{0}$u := i∂$_{t}$u + Δ$_{y}$ u − Δ$_{x}$ u = F, (3.7) z ( x, y, t ) = e $^{sϕ}$u ( x, y, t ) , P$_{s}$z ( x, y, t ) = e $^{sϕ}$L$_{0}$ u. (3.8) 6