  We calculate the quasiparticle scattering rate in the superconducting state
of the overdoped cuprates, in the context of the Eliashberg formalism for a
Fermi liquid with strong van Hove singularities close to the chemical
potential. For a $d_{x^2-y^2}$ superconducting gap, we demonstrate analytically
that the scattering rate is linear in the maximum of temperature or energy, but
with different intercepts and momentum dependence. We discuss our results in
view of angle-resolved photoemission experiments. We also discuss the case of a
s-wave gap.
