have ${\frak W}_t\left( {\cal C}_0'\right)\subset  {\cal C}_0'$ for all $t\ge0$. The semigroup $\left({\frak W}_t\right)_{t\ge0}$ extends by the $L^2$ closure from ${\cal C}_0'$ to a $C_0$-continuous semigroup of contractions on $L^2({{\mathbb R}}\times{{\mathbb T}})$. Moreover, it is the unique solution in $L^2({{\mathbb R}}\times{{\mathbb T}})$ of the integral equation $$\label{integral}
{\frak W}_t={\frak W}_t^{\rm un}+2{\gamma}_0\int_0^t {\frak W}_{t-s}^{\rm
  un}{\cal R} {\frak W}_sds,\quad t\ge0.$$

The proof of this result is contained in Appendix \[appa\]. We shall refer to the semigroup solution described in Proposition \[prop011406-19\] as the $L^2$-solution of quation with the interface conditions and for $T=0$. To extend the definition of such a solution to the case of an arbitrary $T\ge0$ we proceed as follows. Suppose that $W_0\in L^2({{\mathbb R}}\times{{\mathbb T}})$. Let $\chi\in C^\infty_c({{\mathbb R}})$ be an arbitrary real valued, even function that satisfies $$\label{011406-19}
\chi(y)=\left\{\begin{array}{ll}
1,&\mbox{ for }|y|\le 1/2, \\
0,&\mbox{ for }|y|\ge 1,\\
\mbox{belongs to }[0,1],& \mbox{ if otherwise.}
\end{array}\right.$$

\[df011406-19\] We say that $W(t,y,k)$ is the $L^2$-solution of quation with the interface conditions and for a given $T\ge0$ and an initial condition $W_0\in L^2({{\mathbb R}}\times{{\mathbb T}})$, if it is of the form $$\label{021406-19}
W(t,y,k):={\frak W}_t(\widetilde W_0)(y,k)+\int_0^t {\frak
  W}_s(F)(y,k)ds+T\chi(y),\quad (t,y,k)\in \bar{{\mathbb R}}_+\times{{\mathbb R}}\times{{\mathbb T}}.$$ Here $$\label{eqF}
F(y,k):=-T\bar{\omega}'(k)\chi'(y),\quad \widetilde W_0(y,k):= W_0(y,k)-T\chi(y).$$

\[rm011406-19\] [*Note that 