Ein abstraktes nichtlineares Cauchy-Kowalewskaja-Theorem mit singulären Koeffizienten. II. (An abstract nonlinear Cauchy-Kowalewskaja theorem with singular coefficients. II) (Q1121433)

[For part I see ibid. 6, 35-41 (1987; Zbl 0643.35009).] In the last years many authors used abstract Cauchy-Kovalevsky theorems as an essential tool for dealing with differential equations. By the aid of such theorems one can prove existence and uniqueness results for time- dependent problems of different mathematical nature. In the paper by \textit{T. Nishida} [J. Differ. Geom. 12, 629-633 (1977; Zbl 0368.35007)] the abstract nonlinear Cauchy problem \(dw/dt=L(t,w),\) \(w(0)=0\), is studied where the operator L acts continuously in a scale of Banach spaces \(\{B_ s,\| \cdot \|_ s\}_{0<s<1}\) with a singularity behaviour of first order (that means \(\| L\|_{s\to s'}\leq C/(s-s')\) for all \(0<s'<s<1).\) In the present paper under similar assumptions new qualitative and quantitative properties of abstract Cauchy-Kovalevsky problems with singular coefficients are presented. Applying the method of Banach space scales the existence and uniqueness of the solution for the Cauchy problem \[ t^ 1(dw/dt)+cw=t^ 1 L_ 1(t,w)+L_ 2(t,w),\quad w(T)=w_ T,\quad T>0, \] in scales can be shown. In a subscale there exists the continuous solution up to singularity \(t=0\).