Existence and classification of characteristic points at blow-up for a semilinear wave equation in one space dimension

We consider the semilinear wave equation with power nonlinearity in one space dimension. We first show the existence of a blow-up solution with a characteristic point. Then, we consider an arbitrary blow-up solution $u(x,t)$, the graph $x\mapsto T(x)$ of its blow-up points and $\text{SS}\subset$ the set of all characteristic points, and show that the $\text{SS}$ has an empty interior. Finally, given $x_0\in \text{SS}$, we show that in selfsimilar variables, the solution decomposes into a decoupled sum of (at least two) solitons with alternate signs and that $T(x)$ forms a corner of angle $\frac{\pi }{2}$ at $x_0$.