A Class of Degenerate Hyperbolic Equations with Neumann Boundary Conditions and Its Application to Observability
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911550928846848 |
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| author | Yang, Dong-Hui Zhong, Jie |
| author_facet | Yang, Dong-Hui Zhong, Jie |
| contents | We establish a mixed observability inequality for a class of degenerate hyperbolic equations on the cylindrical domain $Ω= \mathbb{T} \times (0,1)$ with mixed Neumann Dirichlet boundary conditions. The degeneracy acts only in the radial variable, whereas the periodic angular variable allows propagation with a strong tangential component, making a direct top boundary observation delicate. For $α\in [1,2)$, we prove that the solution can be controlled by a boundary observation on the top boundary together with an interior observation on a narrow strip. The proof combines a weighted functional framework, improved regularity, a cutoff decomposition in the angular variable, a multiplier argument for the localized component, and an energy estimate for the remainder. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_27525 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Class of Degenerate Hyperbolic Equations with Neumann Boundary Conditions and Its Application to Observability Yang, Dong-Hui Zhong, Jie Analysis of PDEs Optimization and Control 35L80, 93B07, 35L05, 35L20 We establish a mixed observability inequality for a class of degenerate hyperbolic equations on the cylindrical domain $Ω= \mathbb{T} \times (0,1)$ with mixed Neumann Dirichlet boundary conditions. The degeneracy acts only in the radial variable, whereas the periodic angular variable allows propagation with a strong tangential component, making a direct top boundary observation delicate. For $α\in [1,2)$, we prove that the solution can be controlled by a boundary observation on the top boundary together with an interior observation on a narrow strip. The proof combines a weighted functional framework, improved regularity, a cutoff decomposition in the angular variable, a multiplier argument for the localized component, and an energy estimate for the remainder. |
| title | A Class of Degenerate Hyperbolic Equations with Neumann Boundary Conditions and Its Application to Observability |
| topic | Analysis of PDEs Optimization and Control 35L80, 93B07, 35L05, 35L20 |
| url | https://arxiv.org/abs/2603.27525 |