Finite time blowup for high dimensional nonlinear wave systems with bounded smooth nonlinearity

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.

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