Finite-time degeneration of hyperbolicity without blowup for quasilinear wave equations

Abstract

In three spatial dimensions, we study the Cauchy problem for the model wave equation $- \partial_t^2 Ψ+ (1 + Ψ)^P ΔΨ= 0$ for $P \in \lbrace 1,2 \rbrace$. We exhibit a stable form of finite-time Tricomi-type degeneracy formation that has not previously been studied for quasilinear wave equations. Specifically, using only energy methods and ODE-type techniques, we exhibit an open (in an appropriate Sobolev topology) set of data such that $Ψ$ is initially near $0$ while $1 + Ψ$ vanishes in finite time. In fact, generic data profiles, when appropriately rescaled, lead to the vanishing of $1 + Ψ$ in finite time. The solution remains regular up to the degeneracy in the following sense: there is a high-order energy, featuring degenerate weights only at the top order, that remains bounded up to the time of first vanishing. When $P=1$, we show that any $C^1$ extension of $Ψ$ to the future of a point where $1 + Ψ= 0$ must exit the regime of hyperbolicity. Moreover, the Kretschmann scalar (which is a curvature invariant) of the Lorentzian metric corresponding to the wave equation blows up at those points. In particular, our results show that curvature blowup for the metric of a quasilinear wave equation does not always coincide with singularity formation in the solution variable. Similar phenomena occur when $P=2$, but in this case, the vanishing of $1 + Ψ$ corresponds only to a breakdown in the strict hyperbolicity of the equation.