On hyperbolic systems with time-dependent Hölder characteristics

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• 作者：Claudia Garetto ; Michael Ruzhansky
• 关键词：Hyperbolic equations ; Gevrey spaces ; Ultradistributions
• 刊名：Annali di Matematica Pura ed Applicata (1923 -)
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：196
• 期：1
• 页码：155-164
• 全文大小：418KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer Berlin Heidelberg
• ISSN：1618-1891
• 卷排序：196

文摘

In this paper, we study the well-posedness of weakly hyperbolic systems with time-dependent coefficients. We assume that the eigenvalues are low regular, in the sense that they are Hölder with respect to t. In the past, these kinds of systems have been investigated by Yuzawa (J Differ Equ 219(2):363–374, 2005) and Kajitani and Yuzawa (Ann Sc Norm Super Pisa Cl Sci (5) 5(4):465–482, 2006) by employing semigroup techniques (Tanabe–Sobolevski method). Here, under a certain uniform property of the eigenvalues, we improve the Gevrey well-posedness result of Yuzawa (2005) and we obtain well-posedness in spaces of ultradistributions as well. Our main idea is a reduction of the system to block Sylvester form and then the formulation of suitable energy estimates inspired by the treatment of scalar equations in Garetto and Ruzhansky (J Differ Equ 253(5):1317–1340, 2012).