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

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. The blowup is driven by a Riccati-type derivative-quadratic semilinear term, and 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 $L^{\infty }$ in an absolute sense, but we assume that their spatial derivatives satisfy a nonlinear smallness condition relative to the size of 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 first-order system whose solutions remain regular, all the way up to the singularity. Using the integrating factor, we construct quasilinear vector fields 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 size assumptions on the initial data and on the structure of the nonlinear 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 nontrivial set of data, and ODE-type blowup for another nontrivial set.