In my master course of operator theory of infinite dimensional linear systems, I am studying the equation $\mathcal{M}x=0$, where $$ \mathcal{M}:=\left( \begin{array}{rcccc} \frac{d}{dt}& 0 & 0 & \frac{1}{c_1} & \frac{-1}{c_1} \\ 0 & \partial_t & \partial_x & 0 & 0 \\ 0 & c_2\partial_x & \partial_t+\gamma \partial_{xx}^2 & 0 & 0 \\ cD & (\cdot)|_{b^+} \cdot \beta c \, - (\cdot)|_{a^-} \cdot \alpha c & \alpha c_3\partial_x (\cdot)|_{a^-} - \beta c_3\partial_x (\cdot)|_{b^+} & \frac{d}{dt}+\lambda D & -\lambda D \\ -cD & (\cdot)|_{b^+}\cdot \alpha c - (\cdot)|_{a^-} \cdot \beta c & \beta c_3\partial_x (\cdot)|_{a^-} - \alpha c_3\partial_x (\cdot)|_{b^+} & -\lambda D & \frac{d}{dt} +\lambda D \end{array} \right) , $$ $x=(f_1(t),f_2(x,t),f_3(x,t),f_4(t),f_5(t))^T$, and $c$, $c_1,\,c_2,\,c_3,\,D,\, a,\,b,\alpha,\beta,\lambda,\gamma $ are positive constants and $b>a$. I am trying to compute its adjoint operator, but I cannot.
I know the procedure, but here like appear evaluate terms as for instance $(\cdot)|_{a^-}$, I do not know how the procedure follows in this kind of cases.