Solution
= Solution
The <Fourier transform> of $u_{xx}$ is $-k^2\widetilde u$: two <integrations by parts> have no boundary terms under the stated decay assumption. Time differentiation commutes with the spatial transform. Therefore
$$
\boxed{\widetilde u_{tt}+2\gamma\widetilde u_t+(\gamma^2+c^2k^2)\widetilde u=0.}
$$
For every fixed spatial <wavenumber> $k$, this is a constant-coefficient <ordinary differential equation> in time.