phone +7 (3412) 91 60 92
mail imi@udsu.ru
ru-RU Russian
Home » Issues » Archive of Issues » 2022 - 4 issue » pp. 593-614

Archive of Issues

Russia Nizhni Novgorod
Year
2022
Volume
32
Issue
4
Pages
593-614
<<
>>
Section Mathematics
Title On totally global solvability of evolutionary Volterra equation of the second kind
Author(-s) Chernov A.V.ab
Affiliations Nizhni Novgorod State Technical Universitya, Nizhni Novgorod State Universityb
Abstract Let $H$ be a Banach space, $T>0$, $\sigma\in[1;\infty]$ and let $W[0;\tau]$, $\tau\in(0;T)$, be the scale of Banach spaces which is induced by restrictions from a space $W=W[0;T]$; $\mathcal{F}\colon L_\sigma(0,T;H)\to W$ be a Volterra operator (an operator with Volterra property); $f[u] \colon W\to L_\sigma(0,T;H)$ be a controlled Volterra operator depending on a control $u\in U$. We consider the equation as follows $$x=\mathcal{F}\bigl( f[u](x)\bigr),\quad x\in W.$$ For this equation we establish signs of totally (with respect to a set of admissible controls) global solvability subject to global solvability of some functional integral inequality in the space $\mathbb{R}$. In many particular cases the above inequality may be realized as the Cauchy problem associated with an ordinary differential equation. In fact, the analogous result which was obtained by the author formerly is developed, this time under other hypotheses, more convenient for practical usage (although in more particular statement). Separately, we consider the cases of compact embedding of spaces and continuity of the operators $\mathcal{F}$, $f[u]$ (such an approach has not been used by the author formerly), from one hand, and of local integral analogue of the Lipschitz condition with respect to that operators, from another hand. In the second case we prove also the uniqueness of solution. In the first case we use Schauder theorem and in the second case we apply the technique of solution continuation along with the time axis (id est continuation along with a Volterra chain). Finally, as an example, we consider a nonlinear wave equation in the space $\mathbb{R}^n$.
Keywords nonlinear evolutionary Volterra equation in a Banach space, nonlinear wave equation, totally global solvability, uniqueness of solution
UDC 517.957, 517.988, 517.977.56
MSC 47J05, 47J35, 47N10
DOI 10.35634/vm220407
Received 14 September 2022
Language Russian
Citation Chernov A.V. On totally global solvability of evolutionary Volterra equation of the second kind, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 4, pp. 593-614.
References
  1. Chernov A.V. A majorant criterion for the total preservation of global solvability of controlled functional operator equation, Russian Mathematics, 2011, vol. 55, no. 3, pp. 85-95. https://doi.org/10.3103/S1066369X11030108
  2. Chernov A.V. On total preservation of solvability of controlled Hammerstein-type equation with non-isotone and non-majorizable operator, Russian Mathematics, 2017, vol. 61, no. 6, pp. 72-81. https://doi.org/10.3103/S1066369X1706010X
  3. Chernov A.V. Majorant sign of the first order for totally global solvability of a controlled functional operator equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 4, pp. 531-548 (in Russian). https://doi.org/10.20537/vm180407
  4. Chernov A.V. On totally global solvability of controlled second kind operator equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 1, pp. 92-111 (in Russian). https://doi.org/10.35634/vm200107
  5. Chernov A.V. Non-Volterra first-order test for the preservation of solvability of a controlled Hammerstein-type equation, Differential Equations, 2020, vol. 56, no. 2, pp. 264-275. https://doi.org/10.1134/S0012266120020111
  6. Chernov A.V. On totally global solvability of evolutionary equation with monotone nonlinear operator, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 1, pp. 130-149 (in Russian). https://doi.org/10.35634/vm220109
  7. Chernov A.V. Operator equations of the second kind: Theorems on the existence and uniqueness of the solution and on the preservation of solvability, Differential Equations, 2022, vol. 58, no. 5, pp. 649-661. https://doi.org/10.1134/S0012266122050056
  8. Kalantarov V.K., Ladyzhenskaya O.A. The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types, Journal of Soviet Mathematics, 1978, vol. 10, issue 1, pp. 53-70. https://doi.org/10.1007/BF01109723
  9. Sumin V.I. The features of gradient methods for distributed optimal-control problems, USSR Computational Mathematics and Mathematical Physics, 1990, vol. 30, no. 1, pp. 1-15. https://doi.org/10.1016/0041-5553(90)90002-A
  10. Sumin V.I. Funktsional'nye vol'terrovy uravneniya v teorii optimal'nogo upravleniya raspredelennymi sistemami. Chast' 1. Vol'terrovy uravneniya i upravlyaemye nachal'no-kraevye zadachi (Functional Volterra equations in the theory of optimal control of distributed systems. Part I. Volterra equations and controlled initial boundary value problems), Nizhni Novgorod: Nizhni Novgorod State University, 1992.
  11. Korpusov M.O., Sveshnikov A.G. “Destruction” of the solution of a strongly nonlinear equation of pseudoparabolic type with double nonlinearity, Mathematical Notes, 2006, vol. 79, no. 6, pp. 820-840. https://doi.org/10.1007/s11006-006-0093-8
  12. Kobayashi T., Pecher H., Shibata Y. On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity, Mathematische Annalen, 1993, vol. 296, no. 2, pp. 215-234. https://doi.org/10.1007/BF01445103
  13. Lu G. Global existence and blow-up for a class of semilinear parabolic systems: a Cauchy problem, Nonlinear Analysis: Theory, Methods and Applications, 1995, vol. 24, no. 8, pp. 1193-1206. https://doi.org/10.1016/0362-546X(94)00190-S
  14. Cavalcanti M.M., Domingos Cavalcanti V.N., Soriano J.A. On existence and asymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions, Journal of Mathematical Analysis and Applications, 2003, vol. 281, no. 1, pp. 108-124. https://doi.org/10.1016/S0022-247X(02)00558-9
  15. Saito H. Global solvability of the Navier-Stokes equations with a free surface in the maximal regularity $L_p$-$L_q$ class, Journal of Differential Equations, 2018, vol. 264, no. 3, pp. 1475-1520. https://doi.org/10.1016/j.jde.2017.09.045
  16. Seregin G.A., Shilkin T.N. Liouville-type theorems for the Navier-Stokes equations, Russian Mathematical Surveys, 2018, vol. 73, no. 4, pp. 661-724. https://doi.org/10.1070/RM9822
  17. Vulikh B.Z. Kratkii kurs teorii funktsii veshchestvennoi peremennoi (Theory of functions of a real variable), Moscow: Nauka, 1965. https://zbmath.org/?q=an:0142.30203
  18. Kantorovich L.V., Akilov G.P. Functional analysis, Oxford: Pergamon Press, 1982. https://zbmath.org/?q=an:0484.46003
  19. Lions J.-L. Quelques méthodes de résolution des problèmes aux limites non linéaires, Paris: Dunod, 1969. https://zbmath.org/?q=an:0189.40603
  20. Sobolev S.L. Some applications of functional analysis in mathematical physics, Providence, RI: American Mathematical Society, 1991. https://zbmath.org/?q=an:0732.46001
  21. Pavlova M.F., Timerbaev M.R. Prostranstva Soboleva (teoremy vlozheniya) (Sobolev spaces (embedding theorems)), Kazan: Kazan State University, 2010.
  22. Gajewski H., Gröger K., Zacharias K. Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (Nonlinear operator equations and operator differential equations), Berlin: Akademie, 1974. https://zbmath.org/?q=an:0289.47029
Full text
/doc/issues-2022/22-04-07.pdf
<< Previous article
Next article >>