  We consider the global regularity problem for defocusing nonlinear wave
systems $$ \Box u = (\nabla_{{\bf R}^m} F)(u) $$ on Minkowski spacetime ${\bf
R}^{1+d}$ with d'Alambertian $\Box := -\partial_t^2 + \sum_{i=1}^d
\partial_{x_i}^2$, the field $u: {\bf R}^{1+d} \to {\bf R}^m$ is vector-valued,
and $F: {\bf R}^m \to {\bf R}$ is a smooth potential which is positive and
homogeneous of order $p+1$ outside of the unit ball, for some $p >1$. This
generalises the scalar defocusing nonlinear wave (NLW) equation, in which $m=1$
and $F(v) = \frac{1}{p+1} |v|^{p+1}$. It is well known that in the energy
sub-critical and energy-critical cases when $d \leq 2$ or $d \geq 3$ and $p
\leq 1+\frac{4}{d-2}$, one has global existence of smooth solutions (for
dimensions $d \leq 7$ at least) from arbitrary smooth initial data $u(0),
\partial_t u(0)$. In this paper we study the supercritical case where $d = 3$
and $p > 5$. We show that in this case, there exists smooth potential $F$ for
some sufficiently large $m$ (in fact we can take $m=40$), positive and
homogeneous of order $p+1$ outside of the unit ball, and a smooth choice of
initial data $u(0), \partial_t u(0)$ 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 $u$, then $u$ itself, and then finally design the potential $F$ in
order to solve the required equation. The Nash embedding theorem is used in the
second step, explaining the need to take $m$ relatively large.
