  The strange-metal phase of overdoped cuprate superconductors exhibits a
linear in temperature resistivity in the low temperature, however, the origin
of this remarkable anomaly is still not well understood. Here the linear
temperature dependence of the resistivity in the strange-metal phase of
overdoped cuprate superconductors is investigated. The momentum dependence of
the transport scattering rate arising from the umklapp scattering between
electrons by the exchange of the spin excitations is derived and employed to
calculate the resistivity by making use of the Boltzmann equation. It is shown
that the resistivity is mainly dominated by the antinodal and nodal umklapp
scattering. In particular, a very low temperature $T_{\rm scale}$ scales with
$\Delta^{2}_{p}$, where $\Delta_{p}$ is the minimum umklapp vector at the
antinode. In the low temperature above $T_{\rm scale}$, the resistivity is
linear in temperature with the temperature linear coefficient that decreases
with the increase of doping, however, in the far lower temperature below
$T_{\rm scale}$, the resistivity is instead quadratic in temperature. The
theory also shows that the same spin excitation that acts like a bosonic glue
to hold the electron pairs together also mediates scattering of electrons in
the strange-metal phase responsible for the linear in temperature resistivity
in the low temperature.
