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.
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Kiyoshi Asano, Wolfgang Tutschke, An Extended Cauchy-Kovalevskaya Problem and its Solution in Associated Spaces. Z. Anal. Anwend. 21 (2002), no. 4, pp. 1055–1060DOI 10.4171/ZAA/1126