The blow-up boundary for nonlinear wave equations

Consider the Cauchy problem for a nonlinear wave equation $\square u = F(u)$ in $N$ space dimensions, $N \leqslant 3$, with $F$ superlinear and nonnegative. It is well known that, in general, the solution blows up in finite time. In this paper it is shown, under some assumptions on the Cauchy data, that the blow-up set is a space-like surface $t = \phi (x)$ with $\phi (x)$ continuously differentiable.