  The amplitude of oscillations of the freely wobbling kink in the $\phi^4$
theory decays due to the emission of second-harmonic radiation. We study the
compensation of these radiation losses (as well as additional dissipative
losses) by the resonant driving of the kink. We consider both direct and
parametric driving at a range of resonance frequencies. In each case, we derive
the amplitude equations which describe the evolution of the amplitude of the
wobbling and the kink's velocity. These equations predict multistability and
hysteretic transitions in the wobbling amplitude for each driving frequency --
the conclusion verified by numerical simulations of the full partial
differential equation. We show that the strongest parametric resonance occurs
when the driving frequency equals the natural wobbling frequency and not double
that value. For direct driving, the strongest resonance is at half the natural
frequency, but there is also a weaker resonance when the driving frequency
equals the natural wobbling frequency itself. We show that this resonance is
accompanied by translational motion of the kink.
