A note on blow-up solutions to some nonlinear Volterra integral equations

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 nth kind (with a linear leading part) are special cases of the theorems proved in this paper.Obviously, this space is complete. We assume that the mapping A(t, u): R + × B → B has the following properties A1 and A2. Property A1. The mapping A(t, u) is continuous in the sense of the metric (2.2). Property A2. There is a function μ(t, s): RWe immediately note that property A1 implies the following property A3.Indeed, it follows from A1 that the numerical function A(t, θ) is continuous for all t ≥ 0. Further, properties A2 and A3 imply the following property.Property A4. There is a function λ(t, s): R 2 + → R + , bounded on each rectangle [0; T ] × [0; S], T, S > 0, such that ∀ t ≥ 0 ∀ u ∈ B A(t, u) ≤ λ(t, u ).

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О Локальной Разрешимости И Разрушении Решения Абстрактного Нелинейного Интегрального Уравнения Вольтерра

Panin¹

2015

Matematicheskie Zametki

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Доказана теорема о непродолжаемом решении для абстрактных интегральных уравнений Вольтерра с операторнозначными ядрами (непрерывным и полярным). Показано, что в случае отсутствия глобальной разрешимости $C$-норма решения не ограниченна, но, вообще говоря, не стремится к бесконечности. Построен пример уравнения Вольтерра, непродолжаемое решение которого является неограниченным, но не бесконечно большим. Показано, что теоремы о непродолжаемом решении задачи Коши для абстрактных уравнений первого и $n$-го (с линейной главной частью) порядков являются частными случаями доказанных в работе теорем. Библиография: 33 названия.

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A note on blow-up solutions to some nonlinear Volterra integral equations

Cited by 3 publications

References 7 publications

Analysis of blow-up behavior of solutions to CVIEs

Analysis of blow-up behavior of solutions to CVIEs

On local solvability and blow-up of solutions of an abstract nonlinear Volterra integral equation

О Локальной Разрешимости И Разрушении Решения Абстрактного Нелинейного Интегрального Уравнения Вольтерра

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