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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2507.06897 |
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| _version_ | 1866911047514849280 |
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| author | Ning-An, Lai Cui, Ren Wei, Xu |
| author_facet | Ning-An, Lai Cui, Ren Wei, Xu |
| contents | In this paper, we study the long-time existence result for small data solutions of quasilinear wave equations exterior to star-shaped regions in two space dimensions. The key novelty is that we establish a Morawetz type energy estimate for the perturbed inhomogeneous wave equation in the exterior domain, which yields $t^{-\frac12}$ decay inside the cone. In addition, two new weighted $L^2$ product estimates are established to produce $t^{-\frac12}$ decay close to the cone. We then show that the existence lifespan $T_\e$ for the quasilinear wave equations with general quadratic nonlinearity satisfies \begin{equation*} \varepsilon^2T_{\varepsilon}\ln^3T_{\varepsilon}=A, \end{equation*} for some fixed positive constant $A$, which is almost sharp (with some logarithmic loss) comparing to the known result of the corresponding Cauchy problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06897 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Long-Time Existence of Quasilinear Wave Equations Exterior to Star-shaped Obstacle in $2\mathbf{D}$ Ning-An, Lai Cui, Ren Wei, Xu Analysis of PDEs In this paper, we study the long-time existence result for small data solutions of quasilinear wave equations exterior to star-shaped regions in two space dimensions. The key novelty is that we establish a Morawetz type energy estimate for the perturbed inhomogeneous wave equation in the exterior domain, which yields $t^{-\frac12}$ decay inside the cone. In addition, two new weighted $L^2$ product estimates are established to produce $t^{-\frac12}$ decay close to the cone. We then show that the existence lifespan $T_\e$ for the quasilinear wave equations with general quadratic nonlinearity satisfies \begin{equation*} \varepsilon^2T_{\varepsilon}\ln^3T_{\varepsilon}=A, \end{equation*} for some fixed positive constant $A$, which is almost sharp (with some logarithmic loss) comparing to the known result of the corresponding Cauchy problem. |
| title | Long-Time Existence of Quasilinear Wave Equations Exterior to Star-shaped Obstacle in $2\mathbf{D}$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.06897 |