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The method of Frobenius to Fuchsian partial differential equations. (English) Zbl 0993.35004

Let \(t\in \mathbb C\), \(x=(x_1,\ldots, x_n) \in \mathbb C^n\). The following Fuchsian differential operator is considered: \[ \begin{gathered} P=t^m D_t ^m+ P_1(t,x,D_x) t^{m-1}D_t ^{m-1}+ \ldots + P_m(t,x,D_x), \\ P_j(t,x,D_x)=\sum_{\alpha\leq j} a_{j,\alpha}(t,x) D_x ^{\alpha}, \quad (1\leq j \leq m), \\ {\text{ord}}_{D_x}P_j(0,x,D_x) \leq 0, \quad (1\leq j \leq m), \end{gathered} \] where \(m\) is a positive integer, and \(D_t=\partial/\partial t\), \(D_{x_j}=\partial/\partial x_j\). The coefficients \(a_{j,\alpha}\) are supposed to be holomorphic in a neighborhood of \((t,x)=(0,0)\). Let \(a_j(x)=P_j(0,x,D_x)\). The polynomial \[ C(x,\lambda)= (\lambda)_m +a_1(x) (\lambda)_{m-1}+ \ldots +a_m (x) \] is called a indicial polynomial, and its roots are called the characteristic exponents of \(P\) in \(x\). H. Tahara investigated the solutions of such equations under the assumption that the characterictic exponents \(\lambda_k(0)\), \(k=1,\ldots, m\), do not differ by an integer [J. Math. Soc. Japan 36, 449-473 (1984; Zbl 0545.35015)]. The aim of the paper under review is to construct the solutions of the Fuchsian equation without any assumption on the characteristic exponents, and using the same Frobenius method originally used for ordinary differential equations with regular singular points and than used by H. Tahara.
The structure of the paper: in section 2 the review and modification of the Frobenius method are given, in section 3 the statement of a main theorem is presented, sections 4-8 contain its proof. In conclusion some variants of the result are discussed. For example, its Nagumo-type version, that is, the Fuchsian operators with not only holomorphic coefficients but also with \(C^{\infty}\) coefficients. Besides, the well-posedness of the characteristic Cauchy problem is considered.

MSC:

35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35C20 Asymptotic expansions of solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs

Citations:

Zbl 0545.35015
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