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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.02715 |
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| _version_ | 1866914303395758080 |
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| author | Kadar, Istvan Li, Warren |
| author_facet | Kadar, Istvan Li, Warren |
| contents | We study the focusing power nonlinear wave equation with any power, in Minkowski space of any spacetime dimension. We present a complete understanding of the local stability and scattering theory (both in high regularity spaces) for solutions exhibiting ODE type blow-up on spacelike hypersurfaces, with the blow-up at each point modelled by the explicit solution $ϕ_{\mathrm{model}} = c_p t^{-α_p}$.
Given a sufficiently regular spacelike hypersurface $Σ_f$, together with auxiliary scattering data $ψ$, we construct the unique corresponding solution to the nonlinear wave equation that (locally) forms an ODE type singularity on $Σ_f$ attaining $ψ$ as scattering data. Conversely, we show that such ODE type singularities are (locally) stable to suitably regular perturbations away from the singularity, and that the blow-up surface and scattering data remain regular, in a continuously dependent manner, following such perturbations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_02715 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Scattering and stability for ODE-type blow-up surfaces for focusing nonlinear wave equations Kadar, Istvan Li, Warren Analysis of PDEs We study the focusing power nonlinear wave equation with any power, in Minkowski space of any spacetime dimension. We present a complete understanding of the local stability and scattering theory (both in high regularity spaces) for solutions exhibiting ODE type blow-up on spacelike hypersurfaces, with the blow-up at each point modelled by the explicit solution $ϕ_{\mathrm{model}} = c_p t^{-α_p}$. Given a sufficiently regular spacelike hypersurface $Σ_f$, together with auxiliary scattering data $ψ$, we construct the unique corresponding solution to the nonlinear wave equation that (locally) forms an ODE type singularity on $Σ_f$ attaining $ψ$ as scattering data. Conversely, we show that such ODE type singularities are (locally) stable to suitably regular perturbations away from the singularity, and that the blow-up surface and scattering data remain regular, in a continuously dependent manner, following such perturbations. |
| title | Scattering and stability for ODE-type blow-up surfaces for focusing nonlinear wave equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.02715 |