Time-Periodic Second-Order Hyperbolic Equations: Fredholmness, Regularity, and Smooth Dependence
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arXiv
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| Format: | Preprint |
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2014
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| _version_ | 1866917134448197632 |
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| author | Kmit, Irina Recke, Lutz |
| author_facet | Kmit, Irina Recke, Lutz |
| contents | The paper concerns the general linear one-dimensional second-order hyperbolic equation $$ \partial^2_tu - a^2(x,t)\partial^2_xu + a_1(x,t)\partial_tu + a_2(x,t)\partial_xu + a_3(x,t)u=f(x,t), \quad x\in(0,1) $$ with periodic conditions in time and Robin boundary conditions in space. Under a non-resonance condition (formulated in terms of the coefficients $a$, $a_1$, and $a_2$) ruling out the small divisors effect, we prove the Fredholm alternative. Moreover, we show that the solutions have higher regularity if the data have higher regularity and if additional non-resonance conditions are fulfilled. Finally, we state a result about smooth dependence on the data, where perturbations of the coefficient $a$ lead to the known loss of smoothness while perturbations of the coefficients $a_1$, $a_2$, and $a_3$ do not. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1411_5556 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Time-Periodic Second-Order Hyperbolic Equations: Fredholmness, Regularity, and Smooth Dependence Kmit, Irina Recke, Lutz Analysis of PDEs The paper concerns the general linear one-dimensional second-order hyperbolic equation $$ \partial^2_tu - a^2(x,t)\partial^2_xu + a_1(x,t)\partial_tu + a_2(x,t)\partial_xu + a_3(x,t)u=f(x,t), \quad x\in(0,1) $$ with periodic conditions in time and Robin boundary conditions in space. Under a non-resonance condition (formulated in terms of the coefficients $a$, $a_1$, and $a_2$) ruling out the small divisors effect, we prove the Fredholm alternative. Moreover, we show that the solutions have higher regularity if the data have higher regularity and if additional non-resonance conditions are fulfilled. Finally, we state a result about smooth dependence on the data, where perturbations of the coefficient $a$ lead to the known loss of smoothness while perturbations of the coefficients $a_1$, $a_2$, and $a_3$ do not. |
| title | Time-Periodic Second-Order Hyperbolic Equations: Fredholmness, Regularity, and Smooth Dependence |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1411.5556 |