Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.01670 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929657185566720 |
|---|---|
| author | He, Daoyin Sun, Yaqing Zhang, Kangqun |
| author_facet | He, Daoyin Sun, Yaqing Zhang, Kangqun |
| contents | In this paper we prove a sharp global existence result for semilinear wave equations with time-dependent scale-invariant damping terms if the initial data is small. More specifically, we consider Cauchy problem of $\partial_t^2u-Δu+\fracμ{t}\partial_tu=|u|^p$, where $n\ge 3$, $t\ge 1$ and $μ\in(0,1)\cup(1,2)$. For critical exponent $p_{crit}(n,μ)$ which is the positive root of $(n+μ-1)p^2-(n+μ+1)p-2=0$ and conformal exponent $p_{conf}(n,μ)=\frac{n+μ+3}{n+μ-1}$, we establish global existence for $n\geq3$ and $p_{crit}(n,μ)<p\leq p_{conf}(n,μ)$. The proof is based on changing the wave equation into the semilinear generalized Tricomi equation $\partial_t^2u-t^mΔu=t^{α(m)}|u|^p$, where $m=m(μ)>0$ and $α(m)\in\Bbb R$ are two suitable constants, then we investigate more general semilinear Tricomi equation $\partial_t^2v-t^mΔv=t^α|v|^p$ and establish related weighted Strichartz estimates. Returning to the original wave equation, the corresponding global existence results on the small data solution $u$ can be obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01670 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global existence for small amplitude semilinear wave equations with time-dependent scale-invariant damping He, Daoyin Sun, Yaqing Zhang, Kangqun Analysis of PDEs In this paper we prove a sharp global existence result for semilinear wave equations with time-dependent scale-invariant damping terms if the initial data is small. More specifically, we consider Cauchy problem of $\partial_t^2u-Δu+\fracμ{t}\partial_tu=|u|^p$, where $n\ge 3$, $t\ge 1$ and $μ\in(0,1)\cup(1,2)$. For critical exponent $p_{crit}(n,μ)$ which is the positive root of $(n+μ-1)p^2-(n+μ+1)p-2=0$ and conformal exponent $p_{conf}(n,μ)=\frac{n+μ+3}{n+μ-1}$, we establish global existence for $n\geq3$ and $p_{crit}(n,μ)<p\leq p_{conf}(n,μ)$. The proof is based on changing the wave equation into the semilinear generalized Tricomi equation $\partial_t^2u-t^mΔu=t^{α(m)}|u|^p$, where $m=m(μ)>0$ and $α(m)\in\Bbb R$ are two suitable constants, then we investigate more general semilinear Tricomi equation $\partial_t^2v-t^mΔv=t^α|v|^p$ and establish related weighted Strichartz estimates. Returning to the original wave equation, the corresponding global existence results on the small data solution $u$ can be obtained. |
| title | Global existence for small amplitude semilinear wave equations with time-dependent scale-invariant damping |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.01670 |