Zeitschrift für Analysis und ihre Anwendungen
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Published online: 2002-12-31
An Extended Cauchy-Kovalevskaya Problem and its Solution in Associated SpacesKiyoshi Asano and Wolfgang Tutschke (1) Kyoto University, Japan
(2) Technische Universität Graz, Austria
The classical Cauchy-Kovalevskaya problem with holomorphic intial functions is uniquely solvable provided the right-hand sides of the differential equations are holomorphic in their variables, i.e., they transform holomorphic functions into holomorphic functions. Moreover, the solutions depend holomorphically on the space-like variables. A far-reaching generalization of the Cauchy-Kovalevskaya Theorem is its abstract version which considers an abstract operator equation in a scale of Banach spaces where the behaviour of complex derivatives at the boundary is expressed by a certain mapping property of the operator under consideration in the underlying scale. Another generalization of the Cauchy-Kovalevskaya Theorem replaces the space of holomorphic functions by another so-called associated space which is defined by an elliptic operator. Making use of this second approach, the present short note solves an extended Cauchy-Kovalevskaya problem in which an initial value problem is combined with an implicit equation.
Keywords: Equivalent integro-differential equations, weighted Banach spaces, interior estimates
Asano Kiyoshi, Tutschke Wolfgang: An Extended Cauchy-Kovalevskaya Problem and its Solution in Associated Spaces. Z. Anal. Anwend. 21 (2002), 1055-1060. doi: 10.4171/ZAA/1126