  Kink oscillations of a coronal loop are observed and studied in detail
because they provide a unique probe into the structure of coronal loops through
MHD seismology and a potential test of coronal heating through the phase-mixing
of Alfv\'en waves. In particular, recent observations show that standing
oscillations of loops often involve also higher harmonics, beside the
fundamental mode. The damping of these kink oscillations is explained by mode
coupling with Alfv\'en waves. We investigate the consequences for wave-based
coronal heating of higher harmonics and what coronal heating observational
signatures we may use to infer the presence of higher harmonic kink
oscillations. We perform a set of non-ideal MHD simulations where the damping
of the kink oscillation of a flux tube via mode coupling is modelled. Our MHD
simulation parameters are based on the seismological inversion of an
observation for which the first three harmonics are detected. We study the
phase-mixing of Alfv\'en waves that leads to the deposition of heat in the
system, and we apply the seismological inversion techniques to the MHD
simulation output. We find that the heating due to phase-mixing of the Alfv\'en
waves triggered by the damping of the kink oscillation is relatively small,
however we can illustrate i) how the heating location drifts due to the
subsequent damping of lower order harmonics. We also address the role of the
higher order harmonics and the width of the boundary shell in the energy
deposition. We conclude that the coronal heating due to phase-mixing seems not
to provide enough energy to maintain the thermal structure of the solar corona
even when multi-harmonics oscillations are included, and these oscillations
play an inhibiting role in the development of smaller scale structures.
