On Local Solvability and Blow-Up of Solutions of an Abstract Nonlinear Volterra Integral Equation

A theorem on noncontinuable solutions is proved for abstract Volterra integral equations with operator-valued kernels (continuous and polar). It is shown that if there is no global solvability, then the $C$-norm of the solution is unbounded but does not tend to infinity in general. An example of Volterra equations whose noncontinuable solutions are unbounded but not infinitely large is constructed. It is shown that the theorems on noncontinuable solutions of the Cauchy problem for abstract equations of the first and $n$th kind (with a linear leading part) are special cases of the theorems proved in this paper.