I have a wave equation in PDE form, defined as
$$ \frac{\partial P}{\partial t} + div(u) = 0, \\ \frac{\partial u_i}{\partial t} + \nabla_i P - \mu \Delta u_i = 0. $$ Here $(P, u_i)$ are the acoustic pressure and velocity perturbation fields. I also supplement all the necessary boundary conditions.
There is another (yet equivalent) form of the equation above: $$ \frac{\partial^2 u_i}{\partial t^2} - \Delta u_i - \mu \frac{\partial}{\partial t} \Delta u_i = 0. $$ This is a Helmholtz equation with a dissipative term, proportional to $\mu$.
As we can expect, the spectrum of these equations are complex, with negative real part representing the decay rate (since $\mu > 0$). And therefore the operator $R u_i = \left( \frac{\partial^2 }{\partial t^2} - \Delta - \mu \frac{\partial}{\partial t} \Delta \right) u_i$ is non-normal.
Since the equation is linear, let's perform a Fourier transform. We get:
$$ - \omega^2 u_i - \Delta u_i - i \omega \mu\Delta u_i = 0. $$
Now, let's define the following inner product:
$$ \left\langle a, b \right\rangle = \int_\Omega a b \ \mathrm{d}x, $$ which involves no complex conjugation. Under this inner product, the adjoint operator $$ R^\dagger \lambda_i = \omega^2 \lambda_i - \Delta \lambda_i - i \omega \mu\Delta \lambda_i = 0 $$ is clearly the same as the direct one.
My questions is:
- Am I correct that the operator $R$ is self-adjoint under this inner product?
- Is it normal then?
- What problems can I expect in my further analysis if I consider this inner product?