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
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.
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