Fredholm Solvability of Dissipative Hyperbolic Problems with Periodic Boundary Conditions

Details Fredholm Solvability of Dissipative Hyperbolic Problems with Periodic Boundary Conditions Fredholm Solvability of Dissipative Hyperbolic Problems with Periodic Boundary Conditions

2011-08-14

arxiv.org/abs/1108.2882v1

Mathematics

Analysis of PDEs

24 pages

Scientific paper

This paper concerns a linear first-order hyperbolic system in one space dimension of the type $$ \partial_tu_j + a_j(x,t)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x,t)u_k = f_j(x,t),\; 1\le j\le n, $$ with periodicity conditions in time and reflection boundary conditions in space. We state conditions on the coefficients $a_j$ and $b_{jk}$ and the reflection coefficients such that the system is Fredholm solvable in the space of continuous functions. In particular, these conditions exclude the small denominators effect. Our results cover non-strictly hyperbolic systems, but they are new even in the case of strict hyperbolicity.

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