Differential and Integral Equations

Fredholm alternative and solution regularity for time-periodic hyperbolic systems

Irina Kmit and Lutz Recke

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This paper concerns linear first-order hyperbolic systems in one space dimension of the type $$ \partial_tu_j + a_j(x,t)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x,t)u_k = f_j(x,t),\; x \in (0,1),\; j=1,\ldots,n, $$ 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 $\partial_ta_j$), 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.

Article information

Differential Integral Equations, Volume 29, Number 11/12 (2016), 1049-1070.

First available in Project Euclid: 13 October 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B10: Periodic solutions 35L40: First-order hyperbolic systems 47A53: (Semi-) Fredholm operators; index theories [See also 58B15, 58J20]


Kmit, Irina; Recke, Lutz. Fredholm alternative and solution regularity for time-periodic hyperbolic systems. Differential Integral Equations 29 (2016), no. 11/12, 1049--1070. https://projecteuclid.org/euclid.die/1476369329

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