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

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

2008-11-25

arxiv.org/abs/0811.4068v2

Mathematics

Analysis of PDEs

60 pages

Scientific paper

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 $\SS\subset $ the set of all characteristic points, and show that the $\SS$ has an empty interior. Finally, given $x_0\in \SS$, 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 $x_0$.

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