In this paper we analyse the well-posedness of the Cauchy problem for a rather general class of hyperbolic systems with space-time dependent coefficients and with multiple characteristics of variable multiplicity. First, we establish a well-posedness result in anisotropic Sobolev spaces for systems with upper triangular principal part under interesting natural conditions on the orders of lower order terms below the diagonal. Namely, the terms below the diagonal at a distance k to it must be of order $$-\,k$$-k. This setting also allows for the Jordan block structure in the system. Second, we give conditions for the Schur type triangularisation of general systems with variable coefficients for reducing them to the form with an upper triangular principal part for which the first result can be applied. We give explicit details for the appearing conditions and constructions for $$2\times 2$$2×2 and $$3\times 3$$3×3 systems, complemented by several examples.

中文翻译：

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具有不可对角化主部分和可变重数的双曲系统，I：适定性

在本文中，我们分析了一类相当一般的具有时空相关系数和变量多重性特征的双曲系统的柯西问题的适定性。首先，我们在有趣的自然条件下，在对角线以下的低阶项阶上，建立了具有上三角主部分的系统的各向异性 Sobolev 空间的适定性结果。也就是说，对角线下方距离 k 的项必须为 $$-\,k$$-k 阶。此设置还允许系统中采用 Jordan 块结构。其次，我们给出了具有可变系数的一般系统的 Schur 型三角化的条件，以将它们简化为可以应用第一个结果的具有上三角主部分的形式。我们给出了 $$2\times 2$$2×2 和 $$3\times 3$$3×3 系统的出现条件和结构的明确细节，并辅以几个例子。