C^{1,\alpha}$ regularity of the blow-up curve at non characteristic points for the one dimensional semilinear wave equation

We consider $u(x, t)$ a blow-up solution of $\partial_{tt}u = \partial{xx}u + |u|p−1u$, that blows-up in one space dimension. We consider its blow-up curve $T (x)$ and $R$ the set of non characteristic points. We prove that $T (x)$ is of class $C^{1,\alpha}$ in $R$.