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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2503.19438 |
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| _version_ | 1866916838014713856 |
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| author | He, Daoyin Li, Qianqian Yin, Huicheng |
| author_facet | He, Daoyin Li, Qianqian Yin, Huicheng |
| contents | For the $2$-D semilinear wave equation with scale-invariant damping $\partial_t^2u-Δu+\fracμ{t}\partial_tu=|u|^p$, where $t\ge 1$ and $p>1$, in the paper [T. Imai, M. Kato, H. Takamura, K. Wakasa, The lifespan of solutions of semilinear wave equations with the scale-invariant damping in two space dimensions, J. Differential Equations 269 (2020), no. 10, 8387-8424], 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 $μ\in (0, 2)$ and $p>p_f(2)=2$ for $μ\geq 2$. In our previous paper, the global small solution $u$ has been obtained for $p_{s}(2+μ)<p<p_{conf}(2,μ)=\frac{μ+5}{μ+1}$ and $μ\in(0,1)\cup(1,2)$. In the present paper, we will show the global existence of small solution $u$ for $p\geq p_{conf}(2,μ)$ and $μ\in(0,1)\cup(1,2)$. In forthcoming papers, we shall show the global existence of small solution $u$ for the remaining cases of $μ>2, p>2$ or $μ=1, p>p_s(μ+2)=1+\sqrt 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19438 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global small data weak solutions of 2-D semilinear wave equations with scale-invariant damping, II He, Daoyin Li, Qianqian Yin, Huicheng Analysis of PDEs For the $2$-D semilinear wave equation with scale-invariant damping $\partial_t^2u-Δu+\fracμ{t}\partial_tu=|u|^p$, where $t\ge 1$ and $p>1$, in the paper [T. Imai, M. Kato, H. Takamura, K. Wakasa, The lifespan of solutions of semilinear wave equations with the scale-invariant damping in two space dimensions, J. Differential Equations 269 (2020), no. 10, 8387-8424], 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 $μ\in (0, 2)$ and $p>p_f(2)=2$ for $μ\geq 2$. In our previous paper, the global small solution $u$ has been obtained for $p_{s}(2+μ)<p<p_{conf}(2,μ)=\frac{μ+5}{μ+1}$ and $μ\in(0,1)\cup(1,2)$. In the present paper, we will show the global existence of small solution $u$ for $p\geq p_{conf}(2,μ)$ and $μ\in(0,1)\cup(1,2)$. In forthcoming papers, we shall show the global existence of small solution $u$ for the remaining cases of $μ>2, p>2$ or $μ=1, p>p_s(μ+2)=1+\sqrt 2$. |
| title | Global small data weak solutions of 2-D semilinear wave equations with scale-invariant damping, II |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.19438 |