On the existence and uniqueness of a generalized solution of the Protter problem for \((3+1)\) -D Keldysh-type equations

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• 作者：Nedyu Popivanov ; Tsvetan Hristov ; Aleksey Nikolov…
• 关键词：weakly hyperbolic equations ; boundary value problems ; generalized solutions ; uniqueness ; behavior of solution
• 刊名：Boundary Value Problems
• 出版年：2017
• 出版时间：December 2017
• 年：2017
• 卷：2017
• 期：1
• 全文大小：1744KB
• 刊物主题：Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations; Analysis; Approximations and Expansions; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1687-2770
• 卷排序：2017

文摘

A \((3+1)\)-dimensional boundary value problem for equations of Keldysh type (the second kind) is studied. Such problems for equations of Tricomi type (the first kind) or for the wave equation were formulated by M.H. Protter (1954) as multidimensional analogues of Darboux or Cauchy-Goursat plane problems. Now, it is well known that Protter problems are not correctly set, and they have singular generalized solutions, even for smooth right-hand sides. In this paper an analogue of the Protter problem for equations of Keldysh type is given. An appropriate generalized solution with possible singularity is defined. Results for uniqueness and existence of such a generalized solution are obtained. Some a priori estimates are stated.