We have the equations$\newcommand{\div}{\operatorname{div}}\newcommand{\grad}{\operatorname{grad}}$
$$\rho(x)\frac{\partial^2u}{\partial^2}(t,x)=\div(A(x)\grad u+B(x)\grad \frac{\partial u}{\partial t})(t,x)-\div(\gamma(x) v(t,x))$$
$$\beta(x)\frac{\partial v}{\partial t}(t,x)=\div(C(x)\grad v)(t,x)-\gamma(x).\grad \frac{\partial u}{\partial t}(t,x)$$
$A,B,C$ are are diagonal matrices with strictly positive entries and $\gamma(x)$ is a real vector of length $d$ and $u,v$ are zero at the boundary. This results in an operator
$$A=\begin{pmatrix}0 & 1& 0\\\frac{1}{\rho(x)}\div A(x)\grad &\frac{1}{\rho(x)}\div B(x) \grad&\frac{-1}{\rho(x)}\div\gamma(x)\\0 & \frac{-1}{\beta(x)}\gamma(x)\grad& \frac{1}{\beta(x)}\div C(x)\grad\end{pmatrix}$$
Now $A$ acts on $f=(u_1,u_2,u_3)^T$.We have $\dot{f}=Af$. We define $$(u,v)_H=\int_\Omega u_1 \overline{v_1}+\int_\Omega u_2\rho(x)\overline{v_2}+ \int_\Omega u_3\beta(x) \overline{v_3}$$$$(u,v)_H=\int_\Omega \nabla ~u_1 A(x)\nabla\overline{v_1}+\int_\Omega u_2\rho(x)\overline{v_2}+ \int_\Omega u_3\beta(x) \overline{v_3}$$ $${\|u\|_V}^2=\int_\Omega u_1 \overline{u_1}+\int_\Omega u_2 \overline{u_2}+\int_\Omega u_3 \overline{u_3}$$
What is the associated form $(a,j)$ to the operator $A$?
