On blow-up for the supercritical defocusing nonlinear wave equation

In this paper, we consider the defocusing nonlinear wave equation $-{\mathrm{\partial }}_t^{2}u+\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$ in $\mathbb{R}\times {\mathbb{R}}^d$. Building on our companion work ({\it \small Self-similar imploding solutions of the relativistic Euler equations}), we prove that for $d=4,p\ge 29$ and $d\ge 5,p\ge 17$, there exists a smooth complex-valued solution that blows up in finite time.