  We study electron acceleration within a sub-critical plasma channel
irradiated by an ultra-intense laser pulse ($a_0>100$ or
$I>10^{22}~\mathrm{W/cm^2}$). In this regime, radiation reaction significantly
alters the electron dynamics. This has an effect not only on the maximum
attainable electron energy but also on the phase-matching process between
betatron motion and electron oscillations in the laser field. Our study
encompasses analytical description, test-particle calculations and
2-dimensional particle-in-cell simulations. We show single-stage electron
acceleration to multi-GeV energies within a 0.5 mm-long channel and provide
guidelines how to obtain energies beyond 10 GeV using optimal initial
configurations. We present the required conditions in a form of explicit
analytical scaling laws that can be applied to plan the future electron
acceleration experiments.
