On the critical exponent $p_c$ of the 3D quasilinear wave equation $-\big(1+(\partial_tϕ)^p\big)\partial_t^2ϕ+Δϕ=0$ with short pulse initial data. I, global existence
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866912473197576192 |
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| author | Ding, Bingbing Lu, Yu Yin, Huicheng |
| author_facet | Ding, Bingbing Lu, Yu Yin, Huicheng |
| contents | For the 3D quasilinear wave equation $-\big(1+(\partial_tϕ)^p\big)\partial_t^2ϕ+Δϕ=0$ with the short pulse initial data $(ϕ,\partial_tϕ)(1,x)=\big(δ^{2-\varepsilon_0}ϕ_0(\frac{r-1}δ,ω), δ^{1-\varepsilon_0}ϕ_1(\frac{r-1}δ,ω)\big)$, where $p\in\mathbb N$, $p\geq 2$, $0<\varepsilon_0<1$, $r=|x|$, $ω= \frac{x}{r}\in\mathbb S^2$, and $δ>0$ is sufficiently small, under the outgoing constraint condition $(\partial_t+\partial_r)^kϕ(1,x)=O(δ^{2-\varepsilon_0})$ for $k=1,2$, we will establish the global existence of smooth large data solution $ϕ$ when $p>p_c$ with $p_c=\frac{1}{1-\varepsilon_0}$ being the critical exponent. In the forthcoming paper, when $1\leq p\leq p_c$, we show the formation of the outgoing shock before the time $t=2$ under the suitable assumptions of $(ϕ_0,ϕ_1)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_16153 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the critical exponent $p_c$ of the 3D quasilinear wave equation $-\big(1+(\partial_tϕ)^p\big)\partial_t^2ϕ+Δϕ=0$ with short pulse initial data. I, global existence Ding, Bingbing Lu, Yu Yin, Huicheng Analysis of PDEs For the 3D quasilinear wave equation $-\big(1+(\partial_tϕ)^p\big)\partial_t^2ϕ+Δϕ=0$ with the short pulse initial data $(ϕ,\partial_tϕ)(1,x)=\big(δ^{2-\varepsilon_0}ϕ_0(\frac{r-1}δ,ω), δ^{1-\varepsilon_0}ϕ_1(\frac{r-1}δ,ω)\big)$, where $p\in\mathbb N$, $p\geq 2$, $0<\varepsilon_0<1$, $r=|x|$, $ω= \frac{x}{r}\in\mathbb S^2$, and $δ>0$ is sufficiently small, under the outgoing constraint condition $(\partial_t+\partial_r)^kϕ(1,x)=O(δ^{2-\varepsilon_0})$ for $k=1,2$, we will establish the global existence of smooth large data solution $ϕ$ when $p>p_c$ with $p_c=\frac{1}{1-\varepsilon_0}$ being the critical exponent. In the forthcoming paper, when $1\leq p\leq p_c$, we show the formation of the outgoing shock before the time $t=2$ under the suitable assumptions of $(ϕ_0,ϕ_1)$. |
| title | On the critical exponent $p_c$ of the 3D quasilinear wave equation $-\big(1+(\partial_tϕ)^p\big)\partial_t^2ϕ+Δϕ=0$ with short pulse initial data. I, global existence |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2211.16153 |