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Mat. Zametki, 2015, Volume 97, Issue 6, Pages 884–903 (Mi mz10345)  

This article is cited in 8 scientific papers (total in 8 papers)

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

A. A. Panin

M. V. Lomonosov Moscow State University

Abstract: 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.

Keywords: Volterra integral equation, local solvability, noncontinuable solution, solution blow-up.


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English version:
Mathematical Notes, 2015, 97:6, 892–908

Bibliographic databases:

UDC: 517.988.63+517.988.67
Received: 16.07.2013
Revised: 04.02.2014

Citation: A. A. Panin, “On Local Solvability and Blow-Up of Solutions of an Abstract Nonlinear Volterra Integral Equation”, Mat. Zametki, 97:6 (2015), 884–903; Math. Notes, 97:6 (2015), 892–908

Citation in format AMSBIB
\by A.~A.~Panin
\paper On Local Solvability and Blow-Up of Solutions of an Abstract Nonlinear Volterra Integral Equation
\jour Mat. Zametki
\yr 2015
\vol 97
\issue 6
\pages 884--903
\jour Math. Notes
\yr 2015
\vol 97
\issue 6
\pages 892--908

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    This publication is cited in the following articles:
    1. M. O. Korpusov, D. V. Lukyanenko, E. A. Ovsyannikov, A. A. Panin, “Lokalnaya razreshimost i razrushenie resheniya odnogo uravneniya s kvadratichnoi nekoertsitivnoi nelineinostyu”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 10:2 (2017), 107–123  mathnet  crossref  elib
    2. M. O. Korpusov, A. A. Panin, “On the Nonextendable Solution and Blow-Up of the Solution of the One-Dimensional Equation of Ion-Sound Waves in a Plasma”, Math. Notes, 102:3 (2017), 350–360  mathnet  crossref  crossref  mathscinet  isi  elib
    3. A. A. Panin, G. I. Shlyapugin, “Local solvability and solution blow-up of one-dimensional equations of the Yajima–Oikawa–Satsuma type”, Theoret. and Math. Phys., 193:2 (2017), 1561–1573  mathnet  crossref  crossref  adsnasa  isi  elib
    4. M. O. Korpusov, D. V. Lukyanenko, “Instantaneous blow-up versus local solvability for one problem of propagation of nonlinear waves in semiconductors”, J. Math. Anal. Appl., 459:1 (2018), 159–181  crossref  mathscinet  zmath  isi  scopus
    5. M. O. Korpusov, “On an instantaneous blow-up of solutions of evolutionary problems on the half-line”, Izv. Math., 82:5 (2018), 914–930  mathnet  crossref  crossref  adsnasa  isi  elib
    6. M. O. Korpusov, A. V. Ovchinnikov, A. A. Panin, “Instantaneous blow-up versus local solvability of solutions to the Cauchy problem for the equation of a semiconductor in a magnetic field”, Math. Meth. Appl. Sci., 41:17, SI (2018), 8070–8099  crossref  mathscinet  zmath  isi
    7. I. I. Kolotov, A. A. Panin, “O neprodolzhaemykh resheniyakh i razrushenii reshenii psevdoparabolicheskikh uravnenii s koertsitivnoi i znakopostoyannoi nelineinostyami: analiticheskoe i chislennoe issledovanie”, Matem. zametki, 105:5 (2019), 708–723  mathnet  crossref  elib
    8. Korpusov M.O., “Instantaneous Blow-Up of a Weak Solution of a Problem in Plasma Theory on the Half-Line”, Differ. Equ., 55:1 (2019), 60–67  crossref  mathscinet  isi  scopus
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