Geodesic
On Local Solvability and Blow-Up of Solutions of an Abstract Nonlinear Volterra Integral Equation
Matematičeskie zametki, Tome 97 (2015) no. 6, pp. 884-903.

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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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A. A. Panin. On Local Solvability and Blow-Up of Solutions of an Abstract Nonlinear Volterra Integral Equation. Matematičeskie zametki, Tome 97 (2015) no. 6, pp. 884-903. http://geodesic.mathdoc.fr/item/MZM_2015_97_6_a6/

[1] E. Mitidieri, S. I. Pokhozhaev, “Apriornye otsenki i otsutstvie reshenii nelineinykh uravnenii i neravenstv v chastnykh proizvodnykh”, Tr. MIAN, 234, Nauka, M., 2001, 3–383 | MR | Zbl

[2] A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov, Rezhimy s obostreniem dlya kvazilineinykh parabolicheskikh uravnenii, Nauka, M., 1987 | MR

[3] H. A. Levine, “Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t=-Au+\mathfrak F(u)$”, Arch. Ration. Mech. Anal., 51 (1973), 371–386 | DOI | MR | Zbl

[4] V. K. Kalantarov, O. A. Ladyzhenskaya, “O vozniknovenii kollapsov dlya kvazilineinykh uravnenii parabolicheskogo i giperbolicheskogo tipov”, Kraevye zadachi matematicheskoi fiziki i smezhnye voprosy teorii funktsii. 10, Zap. nauchn. sem. LOMI, 69, Izd-vo «Nauka», Leningrad. otd., L., 1977, 77–102 | MR | Zbl

[5] V. A. Galaktionov, J. A. Vázqez, “The problem of blow-up in nonlinear parabolic equations”, Discrete Contin. Dyn. Syst., 8:2 (2002), 399–433 | DOI | MR | Zbl

[6] Chi-Cheung Poon, “Blow-up of a degenerate non-linear heat equation”, Taiwanese J. Math., 15:3 (2011), 1201–1225 | MR | Zbl

[7] M. O. Korpusov, “O razrushenii reshenii trekhmernogo uravneniya Rozenau–Byurgersa”, TMF, 170:3 (2012), 342–349 | DOI | Zbl

[8] M. O. Korpusov, “On the blow-up of solutions of the Benjamin–Bona–Mahony–Burgers and Rosenau–Burgers equations”, Nonlinear Anal.: Theory Methods Appl., 75:4 (2012), 1737–1743 | DOI | MR | Zbl

[9] S. Ishida, T. Yokota, “Blow-up in finite or infinite time for quasilinear degenerate Keller–Segel systems of parabolic-parabolic type”, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569–2596 | DOI | MR | Zbl

[10] F. Khartman, Obyknovennye differentsialnye uravneniya, Mir, M., 1970 | MR | Zbl

[11] A. Kartan, Differentsialnoe ischislenie. Differentsialnye formy, Mir, M., 1971 | MR | Zbl

[12] G. Gripenberg, S.-O. Londen, O. Staffans, Volterra Integral and Functional Equations, Encyclopedia Math. Appl., 34, Cambridge Univ. Press, Cambridge, 1990 | MR | Zbl

[13] N. A. Burton, Volterra Integral and Differential Equations, Elsevier Science, Amsterdam, 2005 | MR | Zbl

[14] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge Monogr. Appl. Comput. Math., 15, Cambridge Univ. Press, Cambridge, 2004 | MR | Zbl

[15] W. Mydlarczyk, “A condition for finite blow-up time for a Volterra integral equation”, J. Math. Anal. Appls., 181:1 (1994), 248–253 | DOI | MR | Zbl

[16] W. Mydlarczyk, “The blow-up solutions of integral equations”, Colloq. Math., 79:1 (1999), 147–156 | MR | Zbl

[17] P. J.Bushell, W. Okrasinski, “On the maximal interval of existence for solutions to some non-linear Volterra integral equations with convolutional kernel”, Bull. London Math. Soc., 28:1 (1996), 59–65 | DOI | MR | Zbl

[18] C. A. Roberts, “Characterizing the blow-up solutions for nonlinear Volterra integral equations”, Nonlinear Anal.: Theory Methods Appl., 30:2 (1997), 923–933 | DOI | MR | Zbl

[19] M. R. Arias, R. Benítez, “Properties of solutions for nonlinear Volterra integral equations”, Discrete Contin. Dyn. Syst., 2003, Suppl., 42–47 | MR | Zbl

[20] T. Małolepszy, W. Okrasiński, “Conditions for blow-up of solutions of some nonlinear Volterra integral equations”, J. Comput. Appl. Math., 205:2 (2007), 744–750 | DOI | MR | Zbl

[21] F. Calabrò, G. Capobianco, “Blowing up behavior for a class of nonlinear VIEs connected with parabolic PDEs”, J. Comput. Appl. Math., 228:2 (2009), 580–588 | DOI | MR | Zbl

[22] T. Małolepszy, M. Niedziela, “A note on blow-up solutions to some nonlinear Volterra integral equations”, Appl. Math. Comput., 218:11 (2012), 6401–6406 | DOI | MR | Zbl

[23] D. N. Sidorov, N. A. Sidorov, “Convex majorants method in the theory of nonlinear Volterra equations”, Banach J. Math. Anal., 6:1 (2012), 1–10 | DOI | MR | Zbl

[24] C. A. Roberts, “Analysis of explosion for nonlinear Volterra equations”, J. Comput. Appl. Math., 97:1-2 (1998), 153–166 | DOI | MR | Zbl

[25] C. A. Roberts, “Recent results on blow-up and quenching for nonlinear Volterra equations”, J. Comput. Appl. Math., 205:2 (2007), 736–743 | DOI | MR | Zbl

[26] R. K. Miller, Nonlinear Volterra Integral Equations, W. A. Benjamin, Menlo Park, CA, 1971 | MR | Zbl

[27] Z. Artstein, “Continuous dependence of solutions of Volterra integral equations”, SIAM Jour. Math. Anal., 6 (1975), 446–456 | DOI | MR | Zbl

[28] T. Herdman, “Behavior of maximally defined solutions of a nonlinear Volterra equation”, Proc. Amer. Math. Soc., 67:2 (1977), 297–302 | DOI | MR

[29] A. D. Myshkis, “Obschaya teoriya differentsialnykh uravnenii s zapazdyvayuschim argumentom”, UMN, 4:5 (1949), 99–141 | MR | Zbl

[30] T. L. Herdman, “A note on noncontinuable solution of a delay differential equation”, Differential Equations, Academic Press, New York, 1980, 187–192 | MR | Zbl

[31] L. A. Lyusternik, V. I. Sobolev, Elementy funktsionalnogo analiza, Nauka, M., 1965 | MR | Zbl

[32] V. Komornik, P. Martinez, M. Pierre, J. Vanconsenoble, ““Blow-up” of bounded solutions of differential equations”, Acta Sci. Math. (Szeged), 69:3-4 (2003), 651–657 | MR | Zbl

[33] V. I. Bogachev, O. G. Smolyanov, V. I. Sobolev, Topologicheskie vektornye prostranstva i ikh prilozheniya, RKhD, M.–Izhevsk, 2012