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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.14815 |
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| _version_ | 1866908678580338688 |
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| author | Gough, Oliver |
| author_facet | Gough, Oliver |
| contents | We study finite-time blow-up for the one-dimensional nonlinear wave equation with a quadratic time-derivative nonlinearity, \[ u_{tt}-u_{xx}=(u_t)^2,\qquad (x,t)\in\mathbb R\times[0,T). \] Building on the work of Ghoul, Liu, and Masmoudi \cite{ghoul2025blow} on the spatial-derivative analogue, we establish the non-existence of smooth, exact self-similar blow-up profiles. Instead we construct an explicit family of \emph{generalised self-similar} solutions, bifurcating from the ODE blow-up, that are smooth within the past light cone and exhibit type-I blow-up at a prescribed point \((x_0,T)\). We further prove asymptotic stability of these profiles under small perturbations in the energy topology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_14815 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stable Type I blow-up for the one-dimensional wave equation with time-derivative nonlinearity Gough, Oliver Analysis of PDEs We study finite-time blow-up for the one-dimensional nonlinear wave equation with a quadratic time-derivative nonlinearity, \[ u_{tt}-u_{xx}=(u_t)^2,\qquad (x,t)\in\mathbb R\times[0,T). \] Building on the work of Ghoul, Liu, and Masmoudi \cite{ghoul2025blow} on the spatial-derivative analogue, we establish the non-existence of smooth, exact self-similar blow-up profiles. Instead we construct an explicit family of \emph{generalised self-similar} solutions, bifurcating from the ODE blow-up, that are smooth within the past light cone and exhibit type-I blow-up at a prescribed point \((x_0,T)\). We further prove asymptotic stability of these profiles under small perturbations in the energy topology. |
| title | Stable Type I blow-up for the one-dimensional wave equation with time-derivative nonlinearity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.14815 |