  Propagation of short and ultra-intense laser pulses in a semi-infinite space
of overdense hydrogen plasma is analyzed via fully-relativistic, real geometry
particle-in-cell (PIC) simulations including radiation friction. The
relativistic transparency and hole-boring regimes are found to be sensitive to
the transverse plasma field, backward light reflection, and laser pulse
filamentation. For laser intensities approaching $I\sim10^{24}$ W/cm$^2$ the
direct laser acceleration of protons, along with ion Coulomb explosion, results
in their injection into the acceleration phase of the compressed electron wave
at the front of the laser pulses. The protons are observed to be accelerated up
to 10-20 GeV with densities around a few times the critical density. The effect
strongly depends on initial density and laser intensity disappearing with
initial density increase and intensity decrease.
