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Due to the importance of understanding the formation of the nonlinearstructures in the universe and stellar systems, the nonlinear Jeans instabilitywhich plays the key role in these formation problems is a long-standing openproblem presented by astrophysics more than one hundred years. In this article,we focus on a reduced model of the nonlinear Jeans instability for theexpanding Newtonian universe which is characterized by a class of second ordernonlinear hyperbolic equations  \begin{equation*}  \Box \varrho(x^\mu) +\frac{\mathcal{a} }{t} \partial_{t}\varrho(x^\mu) -  \frac{\mathcal{b}}{t^2} \varrho(x^\mu) (1+ \varrho(x^\mu) )-\frac{\mathcal{c}-\mathcal{k}}{1+\varrho(x^\mu)}(\partial_{t}\varrho(x^\mu))^2= \mathcal{k} F(t).  \end{equation*} We establish a family of nonlinear self-increase blowupsolution (the solution itself becomes infinite, stable ODE-type blowup) to thisequation and give estimates on the growth rate of $\varrho$ which may explainthe much faster growth rate of the nonlinear structures of universe inastrophysical observations than the rate given by the classical Jeansinstability.

Blowups for a class of second order nonlinear hyperbolic equations: A reduced model of nonlinear Jeans instability