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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.08274 |
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| _version_ | 1866909689853247488 |
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| author | Li, Qianqian Yin, Huicheng |
| author_facet | Li, Qianqian Yin, Huicheng |
| contents | For the $2$-D semilinear wave equation with scale-invariant damping $\square u+\fracμ{t}\partial_tu=|u|^p$, where $t\geq 1$, $μ>0$ and $p>1$, it is conjectured that the global small data weak solution $u$ exists when $p>p_{s}(2+μ) =\frac{μ+3+\sqrt{μ^2+14μ+17}}{2(μ+1)}$ for $0<μ\leq 2$ and $p>p_f(2)=2$ for $μ\geq 2$. In our previous papers, the global small solution $u$ has been obtained for $p>p_{s}(2+μ)$ and $0<μ<2$ but $μ\not=1$. In the present paper, by the vector field method together with the delicate analysis on the Bessel functions, we will show the global existence of small solution $u$ for $p>2$ and $μ>2$. In forthcoming paper, for $μ=1$ and $p>p_{s}(2+μ)=p_{s}(3)=1+\sqrt 2$, the global solution $u$ is also obtained. Therefore, collecting our series of conclusions together with partial results from others, this open question has been solved completely. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_08274 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global small data weak solutions of 2-D semilinear wave equations with scale-invariant damping, III Li, Qianqian Yin, Huicheng Analysis of PDEs For the $2$-D semilinear wave equation with scale-invariant damping $\square u+\fracμ{t}\partial_tu=|u|^p$, where $t\geq 1$, $μ>0$ and $p>1$, it is conjectured that the global small data weak solution $u$ exists when $p>p_{s}(2+μ) =\frac{μ+3+\sqrt{μ^2+14μ+17}}{2(μ+1)}$ for $0<μ\leq 2$ and $p>p_f(2)=2$ for $μ\geq 2$. In our previous papers, the global small solution $u$ has been obtained for $p>p_{s}(2+μ)$ and $0<μ<2$ but $μ\not=1$. In the present paper, by the vector field method together with the delicate analysis on the Bessel functions, we will show the global existence of small solution $u$ for $p>2$ and $μ>2$. In forthcoming paper, for $μ=1$ and $p>p_{s}(2+μ)=p_{s}(3)=1+\sqrt 2$, the global solution $u$ is also obtained. Therefore, collecting our series of conclusions together with partial results from others, this open question has been solved completely. |
| title | Global small data weak solutions of 2-D semilinear wave equations with scale-invariant damping, III |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.08274 |