  We demonstrate that the injection of the ballistic electrons into the
two-dimensional electron plasma in lateral n$^+$-i-n-n$^+$ graphene
field-effect transistors (G-FET) might lead to a substantial Coulomb drag of
the quasi-equilibrium electrons due the violation of the Galilean and Lorentz
invariance in the systems with a linear electron dispersion. This effect can
result in the S-shaped current-voltage characteristics (IVs). The resulting
negative differential conductivity enables the hysteresis effects and current
filamentation that can be used for the implementation of voltage switching
devices. Due to a strong nonlinearity of the IVs, the G-FETs can be used for an
effective frequency multiplication and detection of terahertz radiation.
