  We consider semilinear wave equations with focusing power nonlinearities in
odd space dimensions $d \geq 5$. We prove that for every $p > \frac{d+3}{d-1}$
there exists an open set of radial initial data in $H^{\frac{d+1}{2}} \times
H^{\frac{d-1}{2}}$ such that the corresponding solution exists in a backward
lightcone and approaches the ODE blowup profile. The result covers the entire
range of energy supercritical nonlinearities and extends our previous work for
the three-dimensional radial wave equation to higher space dimensions.
