comp-dd\] [**g**]{}(,,E\_[m]{})=0 holds. Then the problem $$\begin{gathered}
-{{\frac{\partial (S_0\phi)}{\partial E}}}+\omega\cdot\nabla_x\phi+CS_0\phi+\Sigma\phi -K_C\phi={\bf f},\ \nonumber\\
{\phi}_{|\Gamma_-}={\bf g},\quad \phi(\cdot,\cdot,E_{\rm m})=0, \label{co3aa-dd}\end{gathered}$$ has a unique solution \[solin\] \_[P\_[C,0]{}]{}(GSI\^)+ L(H\^1(I,T\^2(\_-’)))\_[P\_C]{}(GSI\^),

Substitute $\phi$ in the problem (\[co3aa-dd\]) by $u:=\phi-L{\bf g}$ to obtain $$&
-{{\frac{\partial (S_0u)}{\partial E}}}+ \omega\cdot\nabla_x u+CS_0u+\Sigma u-K_C u,
\nonumber\\
={}&
{\bf f}+{{\frac{\partial (S_0L{\bf g})}{\partial E}}}
-CS_0(L{\bf g})-\Sigma(L{\bf g}) +K_C(L {\bf g})=:\tilde{\bf f}(x,\omega,E),\nonumber$$ where we have used the fact that $\omega\cdot\nabla_x(L{\bf g})=0$ (see Lemma \[trathle1\] with $\Sigma=0$). On the other hand, $$u_{|\Gamma_-}={\phi}_{|\Gamma_-}-(L{\bf g})_{|\Gamma_-}={\bf g}-{\bf g}=0,$$ and, using the compatibility condition (\[comp-dd\]), $$u(\cdot,\cdot,E_{\rm m})=\phi(\cdot,\cdot,E_{\rm m})-{\bf g}(x-t(x,\omega)\omega,\omega,E_{\rm m})=0.$$

We find that the assumptions guarantee that $\tilde {\bf f}\in L^2(G\times S\times I)$ (for more details, see the proof of Corollary \[cdd\] below), and hence by Theorem \[coth2-d\] the problem (for $u$) $$\begin{gathered}
-{{\frac{\partial (S_0u)}{\partial E}}}+ \omega\cdot\nabla_x u+CS_0u+\Sigma u -K_C u=\tilde{\bf f}, \\
u_{|\Gamma_-}=0,\quad u(\cdot,\cdot,E_{\rm m})=0,\end{gathered}$$ has a unique solution $u\in {{{\mathcal{}}}H}_{P_{C,0}}(G\times S\times I^\circ)$. It follows that then $$\phi:=u+L{\bf g}\in {{{\mathcal{}}}H}_{P_{C,0}}(G\times S\times I^\circ) + L\big(H^1(I,T^2(\Gamma_-'))\big),$$ is the wanted unique solution of (\[co3aa-dd\]).

Finally, the last inclusion in is justified by the inclusion (see , , ) $${{{\mathcal{}}}H}_{P_{C,0}}(G\times S\times I^\circ)\subset {{{\mathcal{}}}H}_{P_{C}}(G\times S\times I^\circ),$$ and by the fact that (see the proof of Corollary \[cdd\] below) $$P_C(x,\omega,E,D)L\tilde{{\bf g}}
=
-{{\frac{\p