Long time solutions of quasilinear Klein-Gordon equations with small weakly decaying initial data
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917210413334528 |
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| author | Hou, Fei Yin, Huicheng |
| author_facet | Hou, Fei Yin, Huicheng |
| contents | It is well known that for the quasilinear Klein-Gordon equation with quadratic nonlinearity and sufficiently decaying small initial data, there exists a global smooth solution if the space dimensions $d\geq2$. When the initial data are of size $\varepsilon>0$ in the Sobolev space, for the semilinear Klein-Gordon equation satisfying the null condition, the authors in the article (J.-M. Delort, Daoyuan Fang, Almost global existence for solutions of semilinear Klein-Gordon equations with small weakly decaying Cauchy data, Comm. Partial Differential Equations 25 (2000), no. 11-12, 2119--2169) prove that the solution exists in time $[0,T_\varepsilon)$ with $T_\varepsilon\ge Ce^{C\varepsilon^{-μ}}$ ($μ=1$ if $d\ge3$, $μ=2/3$ if $d=2$). In the present paper, we will focus on the general quasilinear Klein-Gordon equation without the null condition and further show that the existence time of the solution can be improved to $T_\varepsilon=+\infty$ if $d\geq3$ and $T_\varepsilon\ge e^{C\varepsilon^{-2}}$ if $d=2$. In addition, for $d=2$ and any fixed number $α>0$, if the weighted $L^2$ norm of the initial data with the weight $(1+|x|)^α$ is small, then the solution exists globally and scatters to a free solution. The arguments are based on the introduction of a good unknown, the Strichartz estimate, the weighted $L^2$-norm estimate and the resonance analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_10384 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Long time solutions of quasilinear Klein-Gordon equations with small weakly decaying initial data Hou, Fei Yin, Huicheng Analysis of PDEs It is well known that for the quasilinear Klein-Gordon equation with quadratic nonlinearity and sufficiently decaying small initial data, there exists a global smooth solution if the space dimensions $d\geq2$. When the initial data are of size $\varepsilon>0$ in the Sobolev space, for the semilinear Klein-Gordon equation satisfying the null condition, the authors in the article (J.-M. Delort, Daoyuan Fang, Almost global existence for solutions of semilinear Klein-Gordon equations with small weakly decaying Cauchy data, Comm. Partial Differential Equations 25 (2000), no. 11-12, 2119--2169) prove that the solution exists in time $[0,T_\varepsilon)$ with $T_\varepsilon\ge Ce^{C\varepsilon^{-μ}}$ ($μ=1$ if $d\ge3$, $μ=2/3$ if $d=2$). In the present paper, we will focus on the general quasilinear Klein-Gordon equation without the null condition and further show that the existence time of the solution can be improved to $T_\varepsilon=+\infty$ if $d\geq3$ and $T_\varepsilon\ge e^{C\varepsilon^{-2}}$ if $d=2$. In addition, for $d=2$ and any fixed number $α>0$, if the weighted $L^2$ norm of the initial data with the weight $(1+|x|)^α$ is small, then the solution exists globally and scatters to a free solution. The arguments are based on the introduction of a good unknown, the Strichartz estimate, the weighted $L^2$-norm estimate and the resonance analysis. |
| title | Long time solutions of quasilinear Klein-Gordon equations with small weakly decaying initial data |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2302.10384 |