T}}_*)$ that consists of continuous functions $F$ that can be continuously extended to $\bar{{\mathbb R}}_\pm\times{{\mathbb T}}_*$ and satisfy the interface conditions $$\label{feb1408}
F(0^+,k)=p_-(k)F(0^+, -k)+p_+(k)F(0^-,k)+{{\fgeeszett}}(k)T, \quad\hbox{ for $0< k\le 1/2$},\\$$ and $$\label{feb1410}
F(0^-, k)=p_-(k)F(0^-,-k) + p_+(k)F(0^+, k) +{{\fgeeszett}}(k)T,\quad \hbox{ for $-1/2< k< 0$.}$$

Note that $F\in{\cal C}_T$ if and only if $F-T'\in{\cal C}_{T-T'}$ for any $T,T'\in{{\mathbb R}}$, because of (\[012304\]).

Let us fix $T\ge0$. We consider the kinetic interface problem given by equation $$\label{eq:8}
\begin{aligned} 
 &\partial_tW(t,y,k) + \bar{\omega}'(k) \partial_y W(t,y,k) = {\gamma}_0 L_k W(t,y,k), 
  \quad (t,y,k)\in{{\mathbb R}}_+\times {{\mathbb R}}_*\times{{\mathbb T}}_*,
 \\
&
W(0,y,k)=W_0(y,k),
\end{aligned}$$ with the interface condition $$\label{feb1408aa}
W(t)\in {\cal C}_T,\quad t\ge0.$$ Here $ L_k $ denotes the operator $L$ acting on the $k$ variable. We shall omit writing the subscript if there is no danger of confusion.

### Simplified case. Explicit solution {#sec2.6.2}

We consider first the situation when equation is replaced by $$\label{eq:8p}
\begin{aligned} 
 &\partial_tW^{\rm un}(t,y,k) + \bar{\omega}'(k) \partial_y W^{\rm un}(t,y,k) = -2{\gamma}_0R(k)  W^{\rm un}(t,y,k), 
  \quad (t,y,k)\in{{\mathbb R}}_+\times {{\mathbb R}}_*\times{{\mathbb T}}_*,
 \\
&
W^{\rm un}(0,y,k)=W_0(y,k),
\end{aligned}$$ with the interface conditions – , with $T=0$. It can be solved explicitly, using the method of characteristics, and we obtain $$\begin{aligned}
\label{010304}
&W^{\rm un}(t,y,k)
=  
e^{-2{\gamma}_0R(k)t}\left\{\vphantom{\int_0^1}W_0\left(y-\bar{{\omega}}'(k)t,k\right)
  1_{[0,\bar{{\omega}}'(k)t]^c}(y)
  +p_+(k)W_0\left(y-\bar{{\omega}}'(k)t,k\right)1_{[0,\bar{{\omega}}'(k)t]}(y)
\right. \nonumber
\\
