Stable ODE-type blowup for some quasilinear wave equations with derivative-quadratic nonlinearities

TL;DR

This paper demonstrates a constructive method to prove stable ODE-type blowup in certain quasilinear wave equations, revealing solutions that blow up more severely than shocks and behave like ODE solutions near singularities.

Contribution

The authors develop a novel approach using quasilinear integrating factors and energy estimates to establish stable ODE-type blowup in quasilinear wave equations with derivative-quadratic nonlinearities.

Findings

Existence of solutions with ODE-type blowup behavior.

Construction of a quasilinear integrating factor for reformulating the wave equation.

Identification of conditions leading to shock versus ODE-type blowup.

Abstract

We prove a constructive stable ODE-type blowup result for open sets of solutions to a family of quasilinear wave equations in three spatial dimensions featuring a Riccati-type derivative-quadratic semilinear term. The singularity is more severe than a shock in that the solution itself blows up like the log of the distance to the blowup-time. We assume that the quasilinear terms satisfy certain structural assumptions, which in particular ensure that the "elliptic part" of the wave operator vanishes precisely at the singular points. The initial data are compactly supported and can be small or large in $L^{\infty}$, but the spatial derivatives must initially satisfy a nonlinear smallness condition compared to the time derivative. The first main idea of the proof is to construct a quasilinear integrating factor, which allows us to reformulate the wave equation as a system whose solutions remain regular, all the way up to the singularity. This is equivalent to constructing quasilinear vectorfields adapted to the nonlinear flow. The second main idea is to exploit some crucial monotonic terms in various estimates, especially the energy estimates, that feature the integrating factor. The availability of the monotonicity is tied to our assumptions on the data and on the structure of the quasilinear terms. The third main idea is to propagate the relative smallness of the spatial derivatives all the way up to the singularity so that the solution behaves, in many ways, like an ODE solution. As a corollary of our main results, we show that there are quasilinear wave equations that exhibit two distinct kinds of blowup: the formation of shocks for one non-trivial set of data, and ODE-type blowup for another non-trivial set.