Global existence, blowup phenomena, and asymptotic behavior for quasilinear Schrödinger equations
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2018
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| _version_ | 1866908794464763904 |
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| author | An, Xiaowei Song, Xianfa |
| author_facet | An, Xiaowei Song, Xianfa |
| contents | In this paper, we study the Cauchy problem of the quasilinear Schrödinger equation \begin{equation*} \left\{ \begin{array}{lll} iu_t=Δu+2uh'(|u|^2)Δh(|u|^2)+F(|u|^2)u \quad {\rm for} \ x\in \mathbb{R}^N, \ t>0\\ u(x,0)=u_0(x),\quad x\in \mathbb{R}^N. \end{array}\right. \end{equation*} Here $h(s)$ and $F(s)$ are some real-valued functions, with various choices for models from mathematical physics. We examine the interplay between the quasilinear effect of $h$ and nonlinear effect of $F$ for the global existence and blowup phenomena. We provide sufficient conditions on the blowup in finite time and global existence of the solution. In some cases, we can deduce the watershed from these conditions. In the focusing case, we construct the sharp threshold for the blowup in finite time and global existence of the solution and lower bound for blowup rate of the blowup solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1811_05136 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Global existence, blowup phenomena, and asymptotic behavior for quasilinear Schrödinger equations An, Xiaowei Song, Xianfa Analysis of PDEs 35Q55 In this paper, we study the Cauchy problem of the quasilinear Schrödinger equation \begin{equation*} \left\{ \begin{array}{lll} iu_t=Δu+2uh'(|u|^2)Δh(|u|^2)+F(|u|^2)u \quad {\rm for} \ x\in \mathbb{R}^N, \ t>0\\ u(x,0)=u_0(x),\quad x\in \mathbb{R}^N. \end{array}\right. \end{equation*} Here $h(s)$ and $F(s)$ are some real-valued functions, with various choices for models from mathematical physics. We examine the interplay between the quasilinear effect of $h$ and nonlinear effect of $F$ for the global existence and blowup phenomena. We provide sufficient conditions on the blowup in finite time and global existence of the solution. In some cases, we can deduce the watershed from these conditions. In the focusing case, we construct the sharp threshold for the blowup in finite time and global existence of the solution and lower bound for blowup rate of the blowup solution. |
| title | Global existence, blowup phenomena, and asymptotic behavior for quasilinear Schrödinger equations |
| topic | Analysis of PDEs 35Q55 |
| url | https://arxiv.org/abs/1811.05136 |