Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity and a scale-invariant damping
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| Format: | Preprint |
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2026
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| _version_ | 1866915916332138496 |
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| author | Bchatnia, Ahmed Hamouda, Makram Kaabi, Firas Sasaki, Takiko Zaag, Hatem |
| author_facet | Bchatnia, Ahmed Hamouda, Makram Kaabi, Firas Sasaki, Takiko Zaag, Hatem |
| contents | In this article, we investigate the blow-up behavior of solutions to the one-dimensional damped nonlinear wave equation, namely $$ \partial_t^2 u - \partial_x^2 u + \fracμ{1 + t} \partial_t u = |\partial_t u|^p \quad (p > 1). $$ Under the assumption of sufficiently large and smooth initial data, we establish that the blow-up curve is continuously differentiable ($\mathcal{C}^1$). A key step in our analysis involves the characterization of the blow-up profile of the solution. The proof relies on transforming the equation into a first-order system and adapting the techniques of Sasaki in \cite{Sasaki2018,Sasaki2019} which have elegantly extended the method of Caffarelli and Friedman \cite{Caffarelli1986} to nonlinear wave equations with time derivative nonlinearity, but without the scale-invariant term ($μ=0$). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_03848 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity and a scale-invariant damping Bchatnia, Ahmed Hamouda, Makram Kaabi, Firas Sasaki, Takiko Zaag, Hatem Analysis of PDEs 35B40, 35B44, 35L05, 35L51, 35L67, 35L71 In this article, we investigate the blow-up behavior of solutions to the one-dimensional damped nonlinear wave equation, namely $$ \partial_t^2 u - \partial_x^2 u + \fracμ{1 + t} \partial_t u = |\partial_t u|^p \quad (p > 1). $$ Under the assumption of sufficiently large and smooth initial data, we establish that the blow-up curve is continuously differentiable ($\mathcal{C}^1$). A key step in our analysis involves the characterization of the blow-up profile of the solution. The proof relies on transforming the equation into a first-order system and adapting the techniques of Sasaki in \cite{Sasaki2018,Sasaki2019} which have elegantly extended the method of Caffarelli and Friedman \cite{Caffarelli1986} to nonlinear wave equations with time derivative nonlinearity, but without the scale-invariant term ($μ=0$). |
| title | Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity and a scale-invariant damping |
| topic | Analysis of PDEs 35B40, 35B44, 35L05, 35L51, 35L67, 35L71 |
| url | https://arxiv.org/abs/2604.03848 |