  This expository article is intended to give an overview about recently
achieved results on asymptotic properties of solutions to the Cauchy problem
$u_{tt}-\Delta u+b(t)u_t =0,\qquad u(0,\cdot)=u_1,\quad
\mathrm{D}_tu(0,\cdot)=u_2$ for a wave equation with time-dependent dissipation
term. The results are based on structural properties of the Fourier multipliers
representing its solution. The article explains the general philosophy behind
the approach.
