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arXiv · 2510.14815

Stable Type I blow-up for the one-dimensional wave equation with time-derivative nonlinearity

Oliver Gough
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Abstract

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.

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Oliver Gough. 2025-10-16. Stable Type I blow-up for the one-dimensional wave equation with time-derivative nonlinearity. https://arxiv.org/abs/2510.14815

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