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

Understanding the formation of nonlinear structures in the universe and stellar systems is crucial. The nonlinear Jeans instability plays a key role in these formation processes. It has been a long-standing open problem in astrophysics for more than a century. In this article, we focus on a reduced model of the nonlinear Jeans instability in an expanding Newtonian universe, which is described by a class of second-order nonlinear 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-increasing blowup solutions (where the solution itself becomes infinite in a stable ODE-type blowup) for this equation. Furthermore, we provide estimates on the growth rate of $\varrho$, which may help explain why the nonlinear structures in the universe grow much faster in astrophysical observations than predicted by the classical Jeans instability.