  We present a preliminary study of a new phenomena associated with the
Euler-Poisson equations -- the so called critical threshold phenomena, where
the answer to questions of global smoothness vs. finite time breakdown depends
on whether the initial configuration crosses an intrinsic, ${O}(1)$ critical
threshold.
  We investigate a class of Euler-Poisson equations, ranging from
one-dimensional problem with or without various forcing mechanisms to
multi-dimensional isotropic models with geometrical symmetry. These models are
shown to admit a critical threshold which is reminiscent of the conditional
breakdown of waves on the beach; only waves above certain initial critical
threshold experience finite-time breakdown, but otherwise they propagate
smoothly. At the same time,the asymptotic long time behavior of the solutions
remains the same, independent of crossing these initial thresholds.
  A case in point is the simple one-dimensional problem where the unforced
inviscid Burgers' solution always forms a shock discontinuity except for the
non-generic case of increasing initial profile, $u_0' \geq 0$. In contrast, we
show that the corresponding one dimensional Euler-Poisson equation with zero
background has global smooth solutions as long as its initial $(\rho_0,u_0)$-
configuration satisfies $u_0'\geq -\sqrt{2k\rho_0}$, allowing a finite,
critical negative velocity gradient. As is typical for such nonlinear
convection problems one is led to a Ricatti equation which is balanced here by
a forcing acting as a 'nonlinear resonance', and which in turn is responsible
for this critical threshold phenomena.
