Morawetz type estimate for damped wave equation in $\mathbb{R}^n (n\geq 4)$ and its application
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911020869484544 |
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| author | He, Daoyin Lai, Ning-An |
| author_facet | He, Daoyin Lai, Ning-An |
| contents | In this paper we establish a Morawetz type etimate for the linear inhomogeneous wave equation with time-dependent scale invariant damping in $\mathbb{R}^n (n\geq 4)$. The novelty is that we view the differential operator $\Box+\fracμ{t}\partial_t$ as $n+1+μ$ dimensional operator, then a well-matched multiplier is introduced. As an application, a sharp global existence result for the small data Cauchy problem of the semilinear wave equation \[ \partial_t^2u-Δu+\frac{\partial_tu}{t}=|u|^p,~~~t>t_0\geq 0 \] is obtained in $\mathbb{R}^n (n\geq 4)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_05268 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Morawetz type estimate for damped wave equation in $\mathbb{R}^n (n\geq 4)$ and its application He, Daoyin Lai, Ning-An Analysis of PDEs In this paper we establish a Morawetz type etimate for the linear inhomogeneous wave equation with time-dependent scale invariant damping in $\mathbb{R}^n (n\geq 4)$. The novelty is that we view the differential operator $\Box+\fracμ{t}\partial_t$ as $n+1+μ$ dimensional operator, then a well-matched multiplier is introduced. As an application, a sharp global existence result for the small data Cauchy problem of the semilinear wave equation \[ \partial_t^2u-Δu+\frac{\partial_tu}{t}=|u|^p,~~~t>t_0\geq 0 \] is obtained in $\mathbb{R}^n (n\geq 4)$. |
| title | Morawetz type estimate for damped wave equation in $\mathbb{R}^n (n\geq 4)$ and its application |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.05268 |