content_image/318028.jpg content_image/318026.jpg content_image/318027.jpg
s > 0. From the assumption on R , we have 

<img src='content_image/318028.jpg'>

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 

<img src='content_image/318026.jpg'>

Here and henceforth we set 

<img src='content_image/318027.jpg'>

