Blow-up of the one-dimensional wave equation with quadratic spatial derivative nonlinearity
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2025
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| author | Ghoul, Tej-eddine Liu, Jie Masmoudi, Nader |
| author_facet | Ghoul, Tej-eddine Liu, Jie Masmoudi, Nader |
| contents | We investigate the blow-up dynamics of smooth solutions to the one-dimensional wave equation with a quadratic spatial derivative nonlinearity, motivated by its applications in Effective Field Theory (EFT) in cosmology. Despite its relevance, explicit blow-up solutions for this equation have not been documented in the literature. In this work, we establish the non-existence of smooth, exact self-similar blow-up solutions and construct a five-parameter family of generalized self-similar solutions exhibiting logarithmic growth. Moreover, we prove the asymptotic stability of these blow-up solutions.
Our proof tackles several significant challenges, including the non-self-adjoint nature of the linearized operator, the presence of unstable eigenvalues, and, most notably, the treatment of non-compact perturbations. By substantially advancing Donninger's spectral-theoretic framework, we develop a robust methodology that effectively handles non-compact perturbations. Key innovations include the incorporation of the Lorentz transformation in self-similar variables, an adaptation of the functional framework in [Merle-Raphael-Rodnianski-Szeftel, Invent.Math., 2022], and a novel resolvent estimate. This approach is general and robust, allowing for straightforward extensions to higher dimensions and applications to a wide range of nonlinear equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07887 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Blow-up of the one-dimensional wave equation with quadratic spatial derivative nonlinearity Ghoul, Tej-eddine Liu, Jie Masmoudi, Nader Analysis of PDEs We investigate the blow-up dynamics of smooth solutions to the one-dimensional wave equation with a quadratic spatial derivative nonlinearity, motivated by its applications in Effective Field Theory (EFT) in cosmology. Despite its relevance, explicit blow-up solutions for this equation have not been documented in the literature. In this work, we establish the non-existence of smooth, exact self-similar blow-up solutions and construct a five-parameter family of generalized self-similar solutions exhibiting logarithmic growth. Moreover, we prove the asymptotic stability of these blow-up solutions. Our proof tackles several significant challenges, including the non-self-adjoint nature of the linearized operator, the presence of unstable eigenvalues, and, most notably, the treatment of non-compact perturbations. By substantially advancing Donninger's spectral-theoretic framework, we develop a robust methodology that effectively handles non-compact perturbations. Key innovations include the incorporation of the Lorentz transformation in self-similar variables, an adaptation of the functional framework in [Merle-Raphael-Rodnianski-Szeftel, Invent.Math., 2022], and a novel resolvent estimate. This approach is general and robust, allowing for straightforward extensions to higher dimensions and applications to a wide range of nonlinear equations. |
| title | Blow-up of the one-dimensional wave equation with quadratic spatial derivative nonlinearity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.07887 |