Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. (English) Zbl 1481.35080
Summary: In this paper, we consider a blow-up solution for the complex-valued semilinear wave equation with power non-linearity in one space dimension. We show that the set of non characteristic points \( I_0 \) is open and that the blow-up curve is of class \( C^{1, \mu_0} \) and the phase \( \theta \) is \( C^{\mu_0} \) on this set. In order to prove this result, we introduce a Liouville Theorem for that equation. Our results hold also for the case of solutions with values in \( \mathbb{R}^m \) with \( m\ge 3 \), with the same proof.
MSC:
| 35B44 | Blow-up in context of PDEs |
| 35B53 | Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35L52 | Initial value problems for second-order hyperbolic systems |
| 35L71 | Second-order semilinear hyperbolic equations |
| 58J45 | Hyperbolic equations on manifolds |
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