{{\mathbb T}}_{\iota'}$, $\iota,\iota'\in\{-,+\}$, can be extended to bounded and continuous functions on the respective closures $\bar{{\mathbb R}}_+\times\bar{{\mathbb R}}_{\iota}\times \bar{{\mathbb T}}_{\iota'}$,

-   for each $(t,y,k)\in {{\mathbb R}}_+\times{{\mathbb R}}_*\times  {{\mathbb T}}_*$ fixed, the function $W(t+s,y+\bar{\omega}'(k) s,k)$ is of the $C^1$ class in the $s$-variable in a neighborhood of $s=0$, and the directional derivative $$\label{Dt}
    D_tW(t,y,k)=\left(\partial_t+\bar{\omega}'(k)\partial_y\right)W(t,y,k):=\frac{d}{ds}_{|s=0} W(t+s,y+\bar{\omega}'(k) s,k)$$ is bounded in ${{\mathbb R}}_+\times{{\mathbb R}}_*\times {{\mathbb T}}_*$ and satisfies $$\label{eq:8a}
     \begin{array}{ll}
     D_tW(t,y,k) = {\gamma}_0 L_k W(t,y,k), &
    \quad (t,y,k)\in{{\mathbb R}}_+\times {{\mathbb R}}_*\times{{\mathbb T}}_*,
    \end{array}$$

-   $W(t)$ satisfies , and $$\label{010102-19}
    \lim_{t\to0+}W(t,y,k)=W_0(y,k),\quad (y,k)\in{{\mathbb R}}_*\times{{\mathbb T}}_*.$$

The following result has been shown in [@koran], see Proposition 2.2.

\[prop013001-19\] Suppose that $W_0\in{\cal C}_T$. Then, under the above hypotheses on the scattering kernel $R(k,k')$ and the dispersion relation ${\omega}(k)$, there exists a unique classical solution to equation with the interface conditions and in the sense of Definition $\ref{df013001-19}$.

### $L^2$ solution {#sec2.6.4}

We assume that $T=0$. Define ${\frak W}_t(W_0):=W(t)$. Thanks to Proposition \[prop013001-19\] the family $\left({\frak W}_t\right)_{t\ge0}$ forms a semigroup on the linear space ${\cal C}_0$. Furthermore, we let $$\label{cR}
{\cal R}F(k):=  \int_{{{\mathbb T}}}R(k,k')
F\left(k'\right) dk',\quad k\in{{\mathbb T}},\quad F\in L^1({{\mathbb T}}).$$ Let $
 {\cal C}_0':=  {\cal C}_0\cap L^2({{\mathbb R}}\times {{\mathbb T}}).
$ The following result holds.

\[prop011406-19\] We 