  This work revisits the electrostatic instability for blazar-induced pair
beams propagating through IGM with the methods of linear analysis and PIC
simulations. We study the impact of the realistic distribution function of
pairs resulting from interaction of high-energy gamma-rays with the
extragalactic background light. We present analytical and numerical
calculations of the linear growth rate of the instability for arbitrary
orientation of wave vectors. Our results explicitly demonstrate that the finite
angular spread of the beam dramatically affects the growth rate of the waves,
leading to fastest growth for wave vectors quasi-parallel to the beam direction
and a growth rate at oblique directions that is only by a factor of 2-4 smaller
compared to the maximum. To study the non-linear beam relaxation, we performed
PIC simulations that take into account a realistic wide-energy distribution of
beam particles. The parameters of the simulated beam-plasma system provide an
adequate physical picture that can be extrapolated to realistic blazar-induced
pairs. In our simulations the beam looses only 1\% percent of its energy, and
we analytically estimate that the beam would lose its total energy over about
$100$ simulation times. Analytical scaling is then used to extrapolate to the
parameters of realistic blazar-induced pair beams. We find that they can
dissipate their energy slightly faster by the electrostatic instability than
through inverse-Compton scattering. The uncertainties arising from, e.g.,
details of the primary gamma-ray spectrum are too large to make firm statements
for individual blazars, and an analysis based on their specific properties is
required.
