Existence and classification of characteristic points at blow-up for a semilinear wave equation

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 7→ T (x) of its blow-up points and S ⊂ R the set of all characteristic points and show that S has an empty interior. Finally, given x0 ∈ S, 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 π2 at x0. AMS Classification: 35L05, 35L67