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

2025/10/16 by Oliver Gough, Gough, Oliver · 1 citation
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2510.14815

openalex publication_date 2025/10/16 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

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, (x,t)∈\mathbb R×[0,T). Building on the work of Ghoul, Liu, and Masmoudi \citeghoul2025blow 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 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 \((x₀,T)\). We further prove asymptotic stability of these profiles under small perturbations in the energy topology.

Citations

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Cited by

• Stable Self-Similar Blow-Up In Nonlinear Wave Equations With Quadratic Time-Derivative Nonlinearities

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