A new critical exponent for semi-linear damped wave equations with the initial data from Sobolev spaces of negative order
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914082131542016 |
|---|---|
| author | Van Duong, Dinh Dao, Tuan Anh |
| author_facet | Van Duong, Dinh Dao, Tuan Anh |
| contents | In this paper, we would like to study the critical exponent for semi-linear damped wave equations with power nonlinearity and the initial data belonging to Sobolev spaces of negative order $\dot{H}_m^{-γ}$. Precisely, we obtain a new critical exponent $$p_{\rm c}(m,γ,n): = 1 + \frac{2m}{n+mγ} $$ for $m \in (1, 2], \,γ\in \big[0, \frac{n(m-1)}{m}\big)$ by proving the global (in time) existence of small data Sobolev solutions when $p \geq p_{\rm c}(m,γ,n)$ and the blow-up result for weak solutions in finite time even for small data if $1 < p < p_{\rm c}(m,γ,n)$. In addition, a novelty of this paper is that the critical value $p= p_{\rm c}(m,γ,n)$ belongs to the global existence range. Furthermore, we are going to provide lifespan estimates for solutions when a blow-up phenomenon occurs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07802 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new critical exponent for semi-linear damped wave equations with the initial data from Sobolev spaces of negative order Van Duong, Dinh Dao, Tuan Anh Analysis of PDEs 35A01, 35B33, 35B44, 35L15, 35L71 In this paper, we would like to study the critical exponent for semi-linear damped wave equations with power nonlinearity and the initial data belonging to Sobolev spaces of negative order $\dot{H}_m^{-γ}$. Precisely, we obtain a new critical exponent $$p_{\rm c}(m,γ,n): = 1 + \frac{2m}{n+mγ} $$ for $m \in (1, 2], \,γ\in \big[0, \frac{n(m-1)}{m}\big)$ by proving the global (in time) existence of small data Sobolev solutions when $p \geq p_{\rm c}(m,γ,n)$ and the blow-up result for weak solutions in finite time even for small data if $1 < p < p_{\rm c}(m,γ,n)$. In addition, a novelty of this paper is that the critical value $p= p_{\rm c}(m,γ,n)$ belongs to the global existence range. Furthermore, we are going to provide lifespan estimates for solutions when a blow-up phenomenon occurs. |
| title | A new critical exponent for semi-linear damped wave equations with the initial data from Sobolev spaces of negative order |
| topic | Analysis of PDEs 35A01, 35B33, 35B44, 35L15, 35L71 |
| url | https://arxiv.org/abs/2508.07802 |