  Nonlinear gravitational instability is a crucial way to comprehend the
clustering of matter and the formation of nonlinear structures in both the
Universe and stellar systems. However, with the exception of a few exact
particular solutions for pressureless matter, there are only some
approximations and numerical and phenomenological approaches to study the
nonlinear gravitational instability instead of mathematically rigorous
analysis. We construct a family of particular solutions of the Euler-Poisson
system that exhibits the nonlinear gravitational instability of matter with
inhomogeneous pressure and entropy (i.e., the cold center and hot rim) in the
expanding Newtonian universe. Despite the density perturbations being
homogeneous, the pressure is not, resulting in significant nonlinear effects.
By making use of our prior work on nonlinear analysis of a class of
differential equations \cite{Liu2022b}, we estimate that the growth rate of the
density contrast is approximately $\sim \exp(t^{\frac{2}{3}})$, much faster
than the growth rate anticipated by classical linear Jeans instability ($\sim
t^{\frac{2}{3}}$). Our main motivation for constructing this family of
solutions is to provide a family of reference solutions for conducting a fully
nonlinear analysis of inhomogeneous perturbations of density contrast. We will
present the general results in a mathematical article \cite{Liu2023b}
separately. Additionally, we emphasize that our model does not feature any
shell-crossing singularities before mass accretion singularities since we are
specifically interested in analyzing the mathematical mechanics of a pure mass
accretion model, which poses limitations on the applicability of our model for
understanding the realistic nonlinear structure formation.
