Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension
Séminaire Équations aux dérivées partielles (Polytechnique) (2009-2010), Talk no. 11, 10 p.

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 $u\left(x,t\right)$, the graph $x↦T\left(x\right)$ of its blow-up points and $𝒮\subset ℝ$ the set of all characteristic points and show that $𝒮$ is locally finite. Finally, given ${x}_{0}\in 𝒮$, we show that in selfsimilar variables, the solution decomposes into a decoupled sum of (at least two) solitons, with alternate signs and that $T\left(x\right)$ forms a corner of angle $\frac{\pi }{2}$.

@article{SEDP_2009-2010____A11_0,
author = {Merle, Frank and Zaag, Hatem},
title = {Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2009-2010},
note = {talk:11},
language = {en},
url = {http://www.numdam.org/item/SEDP_2009-2010____A11_0}
}

Merle, Frank; Zaag, Hatem. Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension. Séminaire Équations aux dérivées partielles (Polytechnique) (2009-2010), Talk no. 11, 10 p. http://www.numdam.org/item/SEDP_2009-2010____A11_0/

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