The blow-up rate for a log non-scaling invariant semilinear wave equation in the conformal regime
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915957487697920 |
|---|---|
| author | Hamza, Mohamed Ali |
| author_facet | Hamza, Mohamed Ali |
| contents | We consider the blow-up behavior of solutions to the semilinear wave equation $$ \partial_t^2 u - Δu = |u|^{p-1}u \ln^a(u^2+2), \ (x,t)\in \mathbb{R}^n \times [0,T),$$ in the conformal case $ p = p_c = 1 + \frac{4}{n-1}$. Previous results in \cite{HZjmaa2020, HZ2022} show that for $ a \in \mathbb{R} $, solutions in the subconformal regime $ p < p_c $ blow up with a Type~I rate at any non-characteristic point. The objective of this work is to extend this blow-up rate to the conformal regime under the assumption $a<0$. We establish an a priori upper bound for any blow-up solution and construct a Lyapunov functional in similarity variables. The resulting functional exhibits only weak dissipation, which necessitates delicate energy arguments to obtain the sharp blow-up rate in the conformal case. To the best of our knowledge, this provides the first result for the blow-up rate in a critical framework for an evolution problem where the scaling symmetry is broken. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05202 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The blow-up rate for a log non-scaling invariant semilinear wave equation in the conformal regime Hamza, Mohamed Ali Analysis of PDEs 35L05, 35B44, 35L71, 35L67, 35B40 We consider the blow-up behavior of solutions to the semilinear wave equation $$ \partial_t^2 u - Δu = |u|^{p-1}u \ln^a(u^2+2), \ (x,t)\in \mathbb{R}^n \times [0,T),$$ in the conformal case $ p = p_c = 1 + \frac{4}{n-1}$. Previous results in \cite{HZjmaa2020, HZ2022} show that for $ a \in \mathbb{R} $, solutions in the subconformal regime $ p < p_c $ blow up with a Type~I rate at any non-characteristic point. The objective of this work is to extend this blow-up rate to the conformal regime under the assumption $a<0$. We establish an a priori upper bound for any blow-up solution and construct a Lyapunov functional in similarity variables. The resulting functional exhibits only weak dissipation, which necessitates delicate energy arguments to obtain the sharp blow-up rate in the conformal case. To the best of our knowledge, this provides the first result for the blow-up rate in a critical framework for an evolution problem where the scaling symmetry is broken. |
| title | The blow-up rate for a log non-scaling invariant semilinear wave equation in the conformal regime |
| topic | Analysis of PDEs 35L05, 35B44, 35L71, 35L67, 35B40 |
| url | https://arxiv.org/abs/2604.05202 |