On the analytic continuation of the solution of the Cauchy problem for certain differential operators of principal part with polynomial coefficients
The author studies the domain of holomorphy of the solutions of the Cauchy problem for linear partial differential operators in Kowalewskian form. Namely, the question is whether the local holomorphy given by the Cauchy-Kowalewsky theorem can be extended to a given complex domain, cf. \textit{Y. Hamada}, \textit{J. Leray} and \textit{A. Takeuchi} [J. Math Pures Appl., IX. Sér. 64, 257--319 (1985; Zbl 0581.32017)]. Attention is here mainly fixed on the interesting two-dimensional example \[ (D_t^m-at^px^qD_x^m)u= P_{m-1}u+v, \] \[ D^k_tu(0,x)= w_k(x), \quad 0\leq k\leq m-1, \quad (t,x)\in \mathbb{C}^2, \] where \(a\in\mathbb{C}\) and \(P_{m-1}\) is a lower order operator. Assuming that the data \(v\), \(w_k\) and the coefficients of \(P_{m-1}\) are holomorphic in \(\mathbb{C}^2\), the author gives a precise study of the domain of holomorhy of the corresponding solution \(u(t,x)\).
scientific article; zbMATH DE number 2091871