  We consider the global regularity problem for nonlinear wave systems $$ \Box
u = f(u) $$ on Minkowski spacetime ${\bf R}^{1+d}$ with d'Alambertian $\Box :=
-\partial_t^2 + \sum_{i=1}^d \partial_{x_i}^2$, where the field $u \colon {\bf
R}^{1+d} \to {\bf R}^m$ is vector-valued, and the nonlinearity $f \colon {\bf
R}^m \to {\bf R}^m$ is a smooth function with $f(0)=0$ and all derivatives
bounded; the higher-dimensional sine-Gordon equation $\Box u = \sin u$ is a
model example of this class of nonlinear wave system. For dimensions $d \leq
9$, it follows from the work of Heinz, Pecher, Brenner, and von Wahl that one
has smooth solutions to this equation for any smooth choice of initial data.
Perhaps surprisingly, we show that this result is almost sharp, in the sense
that for any $d \geq 11$, there exists an $m$ (in fact we can take $m=2$) and a
nonlinearity $f \colon {\bf R}^m \to {\bf R}^m$ with all derivatives bounded,
for which the above equation admits solutions that blow up in finite time. The
intermediate case $d=10$ remains open.
