  Nobel Prize laureate P.J.E. Peebles [24] has emphasized the importance and
difficulties of studying the large scale clustering of matter in cosmology.
Nonlinear gravitational instability plays a central role in understanding the
clustering of matter and the formation of nonlinear structures in the universe
and stellar systems. However, there is no rigorous result on the nonlinear
analysis of this instability except for some particular exact solutions without
pressure, and numerical and phenomenological approaches. Both Rendall [26] and
Mukhanov [21] have highlighted the challenge posed by nonlinear gravitational
instability with effective pressure. This has been a longstanding open problem
in astrophysics for over a century since the occurrence of linearized Jeans
instabilities in Newtonian universes in 1902. This article contributes to a
fully nonlinear analysis of the gravitational instability for the Euler-Possion
system which models expanding Newtonian universes with inhomogeneous density
and pressure. The exponential or finite-time increasing blowups of the density
contrast $\varrho$ can be determined, which may account for the considerably
faster growth rate of nonlinear structures observed in astrophysics than that
suggested by the classical Jeans instability. We believe this is the first
rigorous result for the nonlinear Jeans instability with effective pressure and
the method is concise and robust.
