From Polynomial Stability to Periodic Well-posedness in Partially Dissipative Systems
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917490377883648 |
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| author | Galdi, Giovanni P. Muha, Boris Webster, Justin T. |
| author_facet | Galdi, Giovanni P. Muha, Boris Webster, Justin T. |
| contents | The study of resonances (and well-posedness) for complex systems under time-periodic loading is of broad interest in application. The work of Galdi et al.~(2014) connects asymptotic stability of solutions to an unforced Cauchy problem to solvability of the time-periodic forced problem. Uniform stability of the solution semigroup gives periodic well-posedness for all forces in the natural mild forcing class, whereas strong stability yields only existence of a dense set of forcings for which resonance can be excluded. We address an intermediate regime for polynomial (also: rational or semiuniform) stability. Working with a Fourier decomposition in Hilbert space, we demonstrate that polynomial stability of the semigroup yields an explicit characterization of the dense forcing set on which periodic well-posedness holds. More precisely, resolvent bounds translate directly into certain losses of time derivatives on the forcing required to ensure well-posedness. Our result is motivated by partially dissipative models -- including the famous heat-wave interaction problem idealizing fluid-structure interactions, as well as some thermoelastic, viscoelastic, and weakly damped hyperbolic systems -- for which polynomial decay is the natural regime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12892 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | From Polynomial Stability to Periodic Well-posedness in Partially Dissipative Systems Galdi, Giovanni P. Muha, Boris Webster, Justin T. Analysis of PDEs Dynamical Systems The study of resonances (and well-posedness) for complex systems under time-periodic loading is of broad interest in application. The work of Galdi et al.~(2014) connects asymptotic stability of solutions to an unforced Cauchy problem to solvability of the time-periodic forced problem. Uniform stability of the solution semigroup gives periodic well-posedness for all forces in the natural mild forcing class, whereas strong stability yields only existence of a dense set of forcings for which resonance can be excluded. We address an intermediate regime for polynomial (also: rational or semiuniform) stability. Working with a Fourier decomposition in Hilbert space, we demonstrate that polynomial stability of the semigroup yields an explicit characterization of the dense forcing set on which periodic well-posedness holds. More precisely, resolvent bounds translate directly into certain losses of time derivatives on the forcing required to ensure well-posedness. Our result is motivated by partially dissipative models -- including the famous heat-wave interaction problem idealizing fluid-structure interactions, as well as some thermoelastic, viscoelastic, and weakly damped hyperbolic systems -- for which polynomial decay is the natural regime. |
| title | From Polynomial Stability to Periodic Well-posedness in Partially Dissipative Systems |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2605.12892 |