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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2211.16722 |
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| _version_ | 1866918087741145088 |
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| author | Lu, Yu Yin, Huicheng |
| author_facet | Lu, Yu Yin, Huicheng |
| contents | In the previous paper [Ding Bingbing, Lu Yu, Yin Huicheng, 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, Preprint, 2022], for the 3D quasilinear wave equation $-\big(1+(\partial_tϕ)^p\big)\partial_t^2ϕ+Δϕ=0$ with 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 $p\in\mathbb{N}$, $0<\varepsilon_0<1$, under the outgoing constraint condition $(\partial_t+\partial_r)^kϕ(1,x)=O(δ^{2-\varepsilon_0-k\max\{0,1-(1-\varepsilon_0)p\}})$ for $k=1,2$, the authors establish the global existence of smooth large solution $ϕ$ when $p>p_c$ with $p_c=\frac{1}{1-\varepsilon_0}$. In the present paper, under the same outgoing constraint condition, when $1\leq p\leq p_c$, we will show that the smooth solution $ϕ$ blows up and further the outgoing shock is formed in finite time. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_16722 |
| 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. II, shock formation Lu, Yu Yin, Huicheng Analysis of PDEs In the previous paper [Ding Bingbing, Lu Yu, Yin Huicheng, 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, Preprint, 2022], for the 3D quasilinear wave equation $-\big(1+(\partial_tϕ)^p\big)\partial_t^2ϕ+Δϕ=0$ with 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 $p\in\mathbb{N}$, $0<\varepsilon_0<1$, under the outgoing constraint condition $(\partial_t+\partial_r)^kϕ(1,x)=O(δ^{2-\varepsilon_0-k\max\{0,1-(1-\varepsilon_0)p\}})$ for $k=1,2$, the authors establish the global existence of smooth large solution $ϕ$ when $p>p_c$ with $p_c=\frac{1}{1-\varepsilon_0}$. In the present paper, under the same outgoing constraint condition, when $1\leq p\leq p_c$, we will show that the smooth solution $ϕ$ blows up and further the outgoing shock is formed in finite time. |
| 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. II, shock formation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2211.16722 |