Reducibility of Klein-Gordon equations with maximal order perturbations
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913236350140416 |
|---|---|
| author | Berti, Massimiliano Feola, Roberto Procesi, Michela Terracina, Shulamit |
| author_facet | Berti, Massimiliano Feola, Roberto Procesi, Michela Terracina, Shulamit |
| contents | We prove that all the solutions of a quasi-periodically forced linear Klein-Gordon equation $ψ_{tt}-ψ_{xx}+\mathtt{m}ψ+Q(ωt)ψ=0 $ where $ Q(ωt) := a^{(2)}(ωt, x) \partial_{xx} + a^{(1)}(ωt, x)\partial_x + a^{(0)}(ωt, x) $ is a differential operator of order $ 2 $, parity preserving and reversible, are almost periodic in time and uniformly bounded for all times, provided that the coefficients $ a^{(2) }, a^{(1) }, a^{(0) } $ are small enough and the forcing frequency $ω\in {\mathbb R}^ν$ belongs to a Borel set of asymptotically full measure. This result is obtained by reducing the Klein-Gordon equation to a diagonal constant coefficient system with purely imaginary eigenvalues. The main difficulty is the presence in the perturbation $ Q (ωt) $ of the second order differential operator $ a^{(2)}(ωt, x)\partial_{xx} $. In suitable coordinates the Klein-Gordon equation is the composition of two backward/forward quasi-periodic in time perturbed transport equations with non-constant coefficients, up to lower order pseudo-differential remainders. A key idea is to straighten this first order pseudo-differential operator with bi-characteristics through a novel quantitative Egorov analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11377 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reducibility of Klein-Gordon equations with maximal order perturbations Berti, Massimiliano Feola, Roberto Procesi, Michela Terracina, Shulamit Analysis of PDEs 37K55, 35L05 We prove that all the solutions of a quasi-periodically forced linear Klein-Gordon equation $ψ_{tt}-ψ_{xx}+\mathtt{m}ψ+Q(ωt)ψ=0 $ where $ Q(ωt) := a^{(2)}(ωt, x) \partial_{xx} + a^{(1)}(ωt, x)\partial_x + a^{(0)}(ωt, x) $ is a differential operator of order $ 2 $, parity preserving and reversible, are almost periodic in time and uniformly bounded for all times, provided that the coefficients $ a^{(2) }, a^{(1) }, a^{(0) } $ are small enough and the forcing frequency $ω\in {\mathbb R}^ν$ belongs to a Borel set of asymptotically full measure. This result is obtained by reducing the Klein-Gordon equation to a diagonal constant coefficient system with purely imaginary eigenvalues. The main difficulty is the presence in the perturbation $ Q (ωt) $ of the second order differential operator $ a^{(2)}(ωt, x)\partial_{xx} $. In suitable coordinates the Klein-Gordon equation is the composition of two backward/forward quasi-periodic in time perturbed transport equations with non-constant coefficients, up to lower order pseudo-differential remainders. A key idea is to straighten this first order pseudo-differential operator with bi-characteristics through a novel quantitative Egorov analysis. |
| title | Reducibility of Klein-Gordon equations with maximal order perturbations |
| topic | Analysis of PDEs 37K55, 35L05 |
| url | https://arxiv.org/abs/2402.11377 |