omega,E',E) d\omega' dE'
\geq c, \label{csda4aa}\end{aligned}$$ hold for a.e. $(x,\omega,E)\in G\times S\times I$.

Note that if $\sigma_{kj}$ were (cf. Remark \[cosdare1\]) of the form $\sigma_{kj}(x,\omega',\omega,E',E)=\tilde\sigma_{kj}(x,\omega',\omega,E)\delta(E-E')$ then $$\int_{S\times I}\sigma_{kj,C}(x,\omega',\omega,E',E) d\omega' dE'=
\int_S\tilde\sigma_{kj}(x,\omega',\omega,E) d\omega',$$ for any $C$, and hence the conditions (\[csda3aa\]), (\[csda4aa\]) would be independent of $C$. All the considerations below, after obvious adaptations, are valid for this simplified case.

At first, we apply the variational formulations to deduce existence of solutions. Recall that the inner product in $L^2(G\times S\times I)^3$ is given by $${\left\langle}\phi,v{\right\rangle}_{L^2(G\times S\times I)^3}
=\sum_{j=1}^3 {\left\langle}\phi_j,v_j{\right\rangle}_{L^2(G\times S\times I)},$$ and analogously in other products of inner product spaces. Integrating by parts and applying the Green’s formula (\[green\]) we find (similarly as in section \[esols\]) that the bilinear form ${\bf B}(\cdot,\cdot):C^1(\ol G\times S\times I)^3\times C^1(\ol G\times S\times I)^3\to{\mathbb{R}}$ and the linear form ${\bf F}:
C^1(\ol G\times S\times I)^3 \to{\mathbb{R}}$ corresponding to the problem (\[cosyst1\])- (\[cosyst4\]) are \[cosyst5\] [**B**]{}(,v)=& \_[j=2,3]{}\_j,S\_j[E]{}\_[L\^2(GSI)]{} -,\_x v\_[L\^2(GSI)\^3]{}\
& +\_[j=2,3]{} C\_j,S\_jv\_j\_[L\^2(GSI)]{} +,(\^\*-K\_C\^\*) v\_[L\^2(GSI)\^3]{}\
& +\_+(), \_+(v) \_[T\^2(\_+)\^3]{} +\_[j=2,3]{} \_j(,,0),S\_j(,0) v\_j(,,0)\_[L\^2(GS)]{} , and \[cosyst6\] [**F**]{}(v) = [**f**]{},v\_[L\^2(GSI)\^3]{} +[**g**]{},\_-(v)\_[T\^2(\_-)\^3]{}.

For $j=2,3$, let $$q_j:={1\over 2}{\mathop{\mathrm{ess\hspace{1mm}sup}}}_{(x,E)\in G\times I}{{\frac{\partial S_j}{\partial E}}}(x,E).$$ Moreover, define $$\begin{aligned}
\label{cosyst6a}
&C_j:={{\max\{q_j,0\}}\over\kappa_j},\quad j=2,3, \nonumber \\
&C:=\max\{C_1,C_2\}. \end{aligned}$$

The appropriate Hilbert spaces are defined as &[H]{}:=HH\_1H