  LiCuVO$_4$ is a model system of a 1D spin-1/2 chain that enters a planar
spin-spiral ground state below its N\'eel temperature of 2.4 K due to competing
nearest and next nearest neighbor interactions. The spin-spiral state is
multiferroic with an electric polarization along the a axis which has been
proposed to be caused purely by the spin supercurrent mechanism. With external
magnetic fields in c direction T$_N$ can be suppressed down to 0 K at 7.4 T.
Here we report dynamical measurements of the polarization from P(E)-hysteresis
loops, magnetic field dependent pyro-current and non-linear dielectric
spectroscopy as well as thermal expansion and magnetostriction measurements at
very low temperatures. The multiferroic transition is accompanied by strong
anomalies in the thermal expansion and magnetostriction coefficients and we
find slow switching times of electric domain reversal. Both observations
suggest a sizable magnetoelastic coupling in LiCuVO$_4$. By analyzing the
non-linear polarization dynamics we derive domain sizes in the nm range that
are probably caused by Li defects.
