  We consider a blow-up solution for the semilinear wave equation in $N$
dimensions, with subconformal power nonlinearity. Introducing $\RR_0$ the set
of non-characteristic points with the Lorentz transform of the
space-independent solution as asymptotic profile, we show that $\RR_0$ is open
and that the blow-up surface is of class $C^1$ on $\RR_0$. Then, we show the
stability of $\RR_0$ with respect to initial data.
