We consider the global regularity problem for defocusing nonlinear wave
systems
on Minkowski spacetime
with d’Alembertian
,
where the field
is
vector-valued, and
is a smooth potential which is positive and homogeneous of order
outside of the
unit ball for some
.
This generalises the scalar defocusing nonlinear wave (NLW) equation, in which
and
. It is
well known that in the energy-subcritical and energy-critical cases when
or
and
, one
has global existence of smooth solutions from arbitrary smooth initial data
, at least for dimensions
. We study the
supercritical case where
and
.
We show that in this case, there exists a smooth potential
for some sufficiently
large
(in fact we can
take
), positive and
homogeneous of order
outside of the unit ball, and a smooth choice of initial data
for which
the solution develops a finite-time singularity. In fact the solution is discretely self-similar in
a backwards light cone. The basic strategy is to first select the mass and energy densities
of
, then
itself, and then finally
design the potential
in order to solve the required equation. The Nash embedding
theorem is used in the second step, explaining the need to take
relatively large.
