In three spatial dimensions, we study the Cauchy problem for the wave equation
for
. We
exhibit a form of stable Tricomi-type degeneracy formation that has not previously
been studied in more than one spatial dimension. Specifically, using only energy
methods and ODE techniques, we exhibit an open set of data such that
is initially
near
,
while
vanishes in finite time. In fact, generic data, when appropriately rescaled, lead to this
phenomenon. The solution remains regular in the following sense: there is a high-order
-type
energy, featuring degenerate weights only at the top-order, that remains bounded. When
, we show that
any
extension of
to the future of
a point where
must exit the regime of hyperbolicity. Moreover, the Kretschmann scalar of the
Lorentzian metric corresponding to the wave equation blows up at those points.
Thus, our results show that curvature blowup does not always coincide with
singularity formation in the solution variable. Similar phenomena occur when
, where the
vanishing of
corresponds to the failure of strict hyperbolicity, although the equation is hyperbolic at all
values of
.
The data are compactly supported and are allowed to be large or small as measured by
an unweighted Sobolev norm. However, we assume that initially the spatial derivatives of
are nonlinearly
small relative to
,
which allows us to treat the equation as a perturbation of the ODE
. We show that for
appropriate data,
remains quantitatively negative, which simultaneously drives the degeneracy
formation and yields a favorable spacetime integral in the energy estimates that is
crucial for controlling some top-order error terms. Our result complements those of
Alinhac and Lindblad, who showed that if the data are small as measured by a
Sobolev norm with radial weights, then the solution is global.
