The critical power of short pulse initial data on the global existence or blowup of smooth solutions to 3-D semilinear Klein-Gordon equations
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| Format: | Preprint |
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2025
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| _version_ | 1866917997283639296 |
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| author | Shen, Jindou Yin, Huicheng |
| author_facet | Shen, Jindou Yin, Huicheng |
| contents | It is well-known that there are global small data smooth solutions for the 3-D semilinear Klein-Gordon equations $\square u + u = F(u,{\partial u})$ with cubic nonlinearities. However, for the short pulse initial data $(u, \partial_tu)(0, x)=({δ^{ν+1}}{u_0}({\frac{x}δ}),{δ^ν}{u_1}({\frac{x}δ}))$ with $ν\in\Bbb R$ and $(u_0, u_1)\in C_0^{\infty}(\Bbb R)$, which are a class of large initial data, we establish that when $ν\le -\frac{1}{2}$, the solution $u$ can blow up in finite time for some suitable choices of $(u_0, u_1)$ and cubic nonlinearity $F(u,{\partial u})$; when $ν>-\frac{1}{2}$, the smooth solution $u$ exists globally. Therefore, $ν=-\frac{1}{2}$ is just the critical power corresponding to the global existence or blowup of smooth
short pulse solutions for the cubic semilinear Klein-Gordon equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_17169 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The critical power of short pulse initial data on the global existence or blowup of smooth solutions to 3-D semilinear Klein-Gordon equations Shen, Jindou Yin, Huicheng Analysis of PDEs It is well-known that there are global small data smooth solutions for the 3-D semilinear Klein-Gordon equations $\square u + u = F(u,{\partial u})$ with cubic nonlinearities. However, for the short pulse initial data $(u, \partial_tu)(0, x)=({δ^{ν+1}}{u_0}({\frac{x}δ}),{δ^ν}{u_1}({\frac{x}δ}))$ with $ν\in\Bbb R$ and $(u_0, u_1)\in C_0^{\infty}(\Bbb R)$, which are a class of large initial data, we establish that when $ν\le -\frac{1}{2}$, the solution $u$ can blow up in finite time for some suitable choices of $(u_0, u_1)$ and cubic nonlinearity $F(u,{\partial u})$; when $ν>-\frac{1}{2}$, the smooth solution $u$ exists globally. Therefore, $ν=-\frac{1}{2}$ is just the critical power corresponding to the global existence or blowup of smooth short pulse solutions for the cubic semilinear Klein-Gordon equations. |
| title | The critical power of short pulse initial data on the global existence or blowup of smooth solutions to 3-D semilinear Klein-Gordon equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.17169 |