发表机构
Institute of Mathematics, Humboldt University of Berlin; Institute for Applied Problems of Mechanics and Mathematics, Ukrainian National Academy of Sciences（柏林洪堡大学数学研究所; 乌克兰国家科学院应用力学与数学问题研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对一维非自治双曲系统边值问题的时间周期解，探讨其更高正则性与非共振条件的关系，揭示非自治双曲PDE的共振特性，确定无需非共振条件的问题类并给出抽象正则性原理作为证明工具。
AI 中文摘要
我们研究一维线性和非线性非自治一阶积分微分严格双曲系统的边值问题的时间周期解的更高正则性及其与非共振行为的关系。边界条件包含积分算子和各类边界反射。我们证明，只要系数足够光滑且满足适当数量的非共振条件，连续解和经典解就具有$C^k$正则性。在线性情形下，这些条件涉及主系数、对角低阶系数和边界反射系数；在非线性情形下，它们还取决于非线性项和解本身。对于非自治双曲系统，更高正则性通常需要额外的非共振条件，其数量取决于所需的可微阶数。这些条件不仅是充分的，而且一般来说也是必要的，这揭示了非自治双曲偏微分方程的一个显著特征。相比之下，在自治情形下，仅需一个非共振条件（若确实需要的话）即可获得任意高的正则性。我们还确定了一类不需要非共振条件的非自治双曲问题，在此情形下，解的更高正则性仅由数据的正则性决定。证明的主要技术工具是在向量空间框架下提出的抽象正则性原理。
英文摘要
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.
Comments36 pages
