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

TL;DR

This paper generalizes previous results on nonlinear Jeans-type instabilities, showing that a broad class of quasilinear wave equations exhibit self-increasing blowup solutions and singularities under small perturbations, extending understanding of long-term solution behavior.

Contribution

It extends analysis of nonlinear Jeans-type instabilities to a wider class of quasilinear wave equations with new parameter ranges, demonstrating the existence of self-increasing blowup solutions and singularities.

Findings

Existence of self-increasing blowup solutions for the generalized equations.

Emergence of self-increasing singularities at future null geodesic endpoints.

Stability of these phenomena under small inhomogeneous perturbations.

Abstract

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 $1<\mathsf{a}\leq30$ and $\frac{1}{3}\leq\mathsf{b}\leq\frac{2}{3}$: \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) \Delta \varrho = F(t,\varrho,\partial_{\mu} \varrho) \end{equation*} where $F$ is given by \begin{equation*} F(t,\varrho,\partial_{\mu} \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 $1<\mathsf{a}\leq30$ and $\frac{1}{3}\leq\mathsf{b}\leq\frac{2}{3}$, 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.