Periodic Solutions to Dissipative Hyperbolic Systems. I: Fredholm Solvability of Linear Problems

Abstract

This paper concerns linear first-order hyperbolic systems in one space dimension of the type $${\partial }_t{u}_j+a_j(x,t){\partial }_x{u}_j+\sum _{k=1}^n{b}_{jk}(x,t)u_k=f_j(x,t),x\in (0,1),j=1,\dots ,n,$$ with periodicity conditions in time and reflection boundary conditions in space. We state a non-resonance condition (depending on the coefficients $a_j$ and $b_{jj}$ and the boundary reflection coefficients), which implies Fredholm solvability of the problem in the space of continuous functions. Further, we state one more non-resonance condition (depending also on ${\partial }_t{a}_j$), which implies $C^1$-solution regularity. Moreover, we give examples showing that both non-resonance conditions cannot be dropped, in general. Those conditions are robust under small perturbations of the problem data. Our results work for many non-strictly hyperbolic systems, but they are new even in the case of strict hyperbolicity.

20 pages, added Remark 1.4

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