  Progress on the Intensity Frontier of high energy physics critically depends
on record high intensity charged particles accelerators. Beams in such machines
become operationally limited by coherent beam instabilities, particularly
enhanced in the regime of strong space charge (SC). Usual methods to control
the instabilities, such as octupole magnets, beam feedback dampers and
employment of chromatic effects, become less effective and insufficient. In [1]
it was proposed to employ electron lenses for introduction of sufficient spread
in particle oscillation frequencies needed for beam stabilization and in [2] it
was shown that electron lenses are uniquely effective for Landau damping of
transverse beam instabilities in high energy particle accelerators and their
employment does not compromise incoherent (single particle) stability, dynamic
aperture and the beam lifetime. Here we consider an important issue of
effectiveness of the Landau damping with electron lenses in space-charge
dominated beams and demonstrate that the desired stability can be assured with
proper choice of the electron beam parameters and current distributions.
