  The aim of this paper is to understand the influence of a dissipative term
which is small in the sense that it is asymptotically below scaling on the
asymptotic properties of solutions.
  A diagonalization procedure is applied in order to construct the main terms
of the solution representation and this information is used to prove
$L^p$--$L^q$ decay estimates. Furthermore, the question of sharpness is
considered and formulated in terms of a modified scattering theory.
