Stabilizing Energy-Critical Wave Equation to a Finite Dimensional Attractor via Nonlinear Damping
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918187863375872 |
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| author | Lasiecka, Irena Narciso, Vando |
| author_facet | Lasiecka, Irena Narciso, Vando |
| contents | The wave equation with energy critical sources and nonlinear damping defined on a 3D bounded domain is considered. It is shown that the resulting dynamical system admits a global attractor. Under the additional assumption of strong monotonicity of the damping at the origin, it is shown that the originally unstable quintic wave is uniformly stabilised to a finite dimensional and smooth set. Moreover, the existence of exponential attractor is established. In order to handle \enquote{energy criticality} of both sources and damping, the methods used depend on enhanced dissipation \cite{Bociu-lasiecka-jde}, energy {\it identity} for weak solutions \cite{Koch-lasiecka}, an adaptation of Ball's method \cite{ball}, and the theory of quasi-stable systems \cite{chueshov-white}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18042 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stabilizing Energy-Critical Wave Equation to a Finite Dimensional Attractor via Nonlinear Damping Lasiecka, Irena Narciso, Vando Dynamical Systems 35B33, 35B40, 35B41, 35L05, 35L70 The wave equation with energy critical sources and nonlinear damping defined on a 3D bounded domain is considered. It is shown that the resulting dynamical system admits a global attractor. Under the additional assumption of strong monotonicity of the damping at the origin, it is shown that the originally unstable quintic wave is uniformly stabilised to a finite dimensional and smooth set. Moreover, the existence of exponential attractor is established. In order to handle \enquote{energy criticality} of both sources and damping, the methods used depend on enhanced dissipation \cite{Bociu-lasiecka-jde}, energy {\it identity} for weak solutions \cite{Koch-lasiecka}, an adaptation of Ball's method \cite{ball}, and the theory of quasi-stable systems \cite{chueshov-white}. |
| title | Stabilizing Energy-Critical Wave Equation to a Finite Dimensional Attractor via Nonlinear Damping |
| topic | Dynamical Systems 35B33, 35B40, 35B41, 35L05, 35L70 |
| url | https://arxiv.org/abs/2510.18042 |