Stable Self-Similar Blow-Up In Nonlinear Wave Equations With Quadratic Time-Derivative Nonlinearities
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912716079235072 |
|---|---|
| author | Liu, Jie Raees, Faiq |
| author_facet | Liu, Jie Raees, Faiq |
| contents | We study singularity formation in two one-dimensional nonlinear wave models with quadratic time-derivative nonlinearities. The non-null model violates the null condition and typically develops finite-time blow-up; the null-form model is Lorentz-invariant and enjoys small-data global existence, yet still admits blow-up for large data. Building on our earlier work on spatial-derivative nonlinearities, we construct and classify a five-parameter family of generalized self-similar blow-up solutions that captures the observed dynamics. We prove that no smooth exact self-similar profiles exist, while the generalized self-similar solutions -exhibiting logarithmic growth- provide the correct blow-up description inside backward light cones. We further establish asymptotic stability for the relevant branches, including the ODE-type blow-up in both models. These results yield a coherent and unified picture of blow-up mechanisms in time-derivative nonlinear wave equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13504 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stable Self-Similar Blow-Up In Nonlinear Wave Equations With Quadratic Time-Derivative Nonlinearities Liu, Jie Raees, Faiq Analysis of PDEs We study singularity formation in two one-dimensional nonlinear wave models with quadratic time-derivative nonlinearities. The non-null model violates the null condition and typically develops finite-time blow-up; the null-form model is Lorentz-invariant and enjoys small-data global existence, yet still admits blow-up for large data. Building on our earlier work on spatial-derivative nonlinearities, we construct and classify a five-parameter family of generalized self-similar blow-up solutions that captures the observed dynamics. We prove that no smooth exact self-similar profiles exist, while the generalized self-similar solutions -exhibiting logarithmic growth- provide the correct blow-up description inside backward light cones. We further establish asymptotic stability for the relevant branches, including the ODE-type blow-up in both models. These results yield a coherent and unified picture of blow-up mechanisms in time-derivative nonlinear wave equations. |
| title | Stable Self-Similar Blow-Up In Nonlinear Wave Equations With Quadratic Time-Derivative Nonlinearities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.13504 |