Fredholmness and smooth dependence for linear hyperbolic periodic-Dirichlet problems

Please always quote using this URN: urn:nbn:de:0296-matheon-6911
  • This paper concerns $n\times n$ linear one-dimensional hyperbolic systems of the type $$ \om\partial_tu_j + a_j(x)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x)u_k = f_j(x,t),\; j=1,\ldots,n, $$ with periodicity conditions in time and reflection boundary conditions in space. We state sufficient conditions on the data $\om, a_j$ and $b_{jk}$ such that the system has a Fredholm like solvability behavior. Moreover, we state sufficient conditions on the data such that for any right hand side there exists exactly one solution, that the solution survives under small perturbations of the data, and that the corresponding data-to-solution-map is smooth with respect to appropriate function space norms. In particular, those sufficient conditions imply that no small denominator effects occur. Moreover, we show that perturbations of the coefficients $\om$ and $a_j$ lead to essentially different results than perturbations of the coefficients $b_{jk}$, in general.

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Metadaten
Author:Irina Kmit, Lutz Recke
URN:urn:nbn:de:0296-matheon-6911
Referee:Alexander Mielke
Language:English
Date of first Publication:2010/05/17
Institute:Humboldt-Universität zu Berlin
Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)
MSC-Classification:35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Bxx Qualitative properties of solutions / 35B10 Periodic solutions
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Bxx Qualitative properties of solutions / 35B30 Dependence of solutions on initial and boundary data, parameters [See also 37Cxx]
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Lxx Hyperbolic equations and systems [See also 58J45] / 35L40 First-order hyperbolic systems
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Lxx Hyperbolic equations and systems [See also 58J45] / 35L50 Initial-boundary value problems for first-order hyperbolic systems
47-XX OPERATOR THEORY / 47Axx General theory of linear operators / 47A53 (Semi-) Fredholm operators; index theories [See also 58B15, 58J20]
Preprint Number:701