The emergence of nonlinear Jeans-type instabilities for quasilinear wave equations. II: Generalizations

This work extends the previous work by the first author [arXiv:2409.02516] and [Math. Ann. 393 (2025), 317-363], analyzing the long-term behavior of solutions to a broader class of quasilinear wave equations with parameter $1<\mathsf{a}\le 30$ and $\frac{1}{3}\le \mathsf{b}\le \frac{2}{3}$: \begin{equation*} \partial^2_t \varrho- \biggl( \frac{ $\mathsf{m}$^2 (\partial_{t}\varrho )^2}{(1+\varrho )^2} + 4($\mathsf{k}$-$\mathsf{m}$^2)(1+\varrho )\biggr) Δ\varrho = F(t,\varrho,\partial_μ \varrho) \end{equation*} where $F$ is given by \begin{equation*} F(t,\varrho,\partial_μ \varrho):= $\mathsf{b}$ \varrho (1+ \varrho ) -($\mathsf{a}$-1) \partial_{t}\varrho + \frac{4}{3} \frac{(\partial_{t}\varrho )^2}{1+\varrho } + \biggl($\mathsf{m}$^2 \frac{ (\partial_{t}\varrho )^2}{(1+\varrho )^2} + 4($\mathsf{k}$-$\mathsf{m}$^2) (1+\varrho ) \biggr) q^i \partial_{i}\varrho - $\mathtt{K}$^{ij} \partial_{i}\varrho\partial_{j}\varrho . \end{equation*} The results demonstrate that for this extensive family of quasilinear wave equations satisfying $1<\mathsf{a}\le 30$ and $\frac{1}{3}\le \mathsf{b}\le \frac{2}{3}$, self-increasing blowup solutions also exist, and self-increasing singularities emerge at certain future endpoints of null geodesics provided the inhomogeneous perturbations of data are sufficiently small.