Irina Kmit, Lutz Recke
We study higher regularity and its relation to nonresonant behavior for time-periodic solutions of boundary value problems for one-dimensional linear and nonlinear nonautonomous first-order integro-differential strictly hyperbolic systems. The boundary conditions include integral operators and various types of boundary reflections. We prove that continuous and classical solutions have $C^k$-regularity, provided the coefficients are sufficiently smooth and a suitable number of nonresonance conditions is satisfied. In the linear case, these conditions involve the principal coefficients, the diagonal lower-order coefficients, and the boundary reflection coefficients. In the nonlinear case, they also depend on the nonlinearities and on the solution itself. For nonautonomous hyperbolic systems, higher regularity generally requires additional nonresonance conditions, whose number depends on the desired order of differentiability. These conditions are not only sufficient but, in general, also necessary, revealing a distinctive feature of nonautonomous hyperbolic PDEs. By contrast, in the autonomous case, a single nonresonance condition (if one is needed at all) suffices to obtain arbitrarily high regularity. We also identify a class of nonautonomous hyperbolic problems for which no nonresonance conditions are required. In this case, the higher regularity of solutions is determined solely by the regularity of the data. The main technical tool underlying the proofs is an abstract regularity principle formulated in the setting of vector spaces.