Boundary-value problem for a class of nonlinear systems of partial differential equations of higher orders

  • S. S. Kharibegashvili Georgian Technical University, Tbilisi
  • B. G. Midodashvili Ivane Javakhishvili Tbilisi State University
Keywords: 1

Abstract

UDC 517.957

We investigate a boundary problem for a class of nonlinear systems of partial differential equations of higher orders. For this problem, the existence, uniqueness, and absence of solutions is established.

References

S. Kharibegashvili, B. Midodashvili, A boundary value problem for higher-order semilinear partial differential equations, Complex Var. and Elliptic Equat., 64, № 5, 766 – 776 (2019), https://doi.org/10.1080/17476933.2018.1508286 DOI: https://doi.org/10.1080/17476933.2018.1508286

S. Kharibegashvili, B. Midodashvili, On the existence, uniqueness and nonexistence of solutions of one boundary value problem for a semilinear hyperbolic equation, Ukr. Mat. Zh., 71, № 8, 1123 – 1182 (2019). DOI: https://doi.org/10.1007/s11253-019-01713-9

S. Kharibegashvili, B. Midodashvili, Solvability of characteristic boundary-value problems for nonlinear equations with iterated wave operator in the principal part, Electron. J. Different. Equat., № 72, 1 – 12 (2008).

S. Kharibegashvili, B. Midodashvili, On one boundary value problem for a nonlinear equation with iterated wave operator in the principal part, Georgian Math. J., 15, № 3, 541 – 554 (2008). DOI: https://doi.org/10.1515/GMJ.2008.541

S. Kharibegashvili, Boundary value problems for some classes of nonlinear wave equations, Mem. Different. Equat. Math. Phys., 46, 1 – 114 (2009); https://doi.org/10.1186/1687-1847-2013-220. DOI: https://doi.org/10.1186/1687-1847-2013-220

C. Xiangying, Existence and nonexistence of global solutions for nonlinear evolution equation of fourth order, Appl. Math. J. Chinese Univ. Ser. B., 16, № 3, 251 – 258 (2001). DOI: https://doi.org/10.1007/s11766-001-0063-6

C. J. Budd, V. A. Galaktionov, J. F. Williams, Self-similar blow-up in higher-order semilinear parabolic equations,SIAM J. Appl. Math., 64, № 5, 1775 – 1809 (2004), https://doi.org/10.1137/S003613990241552X DOI: https://doi.org/10.1137/S003613990241552X

A. B. Aliev, B. H. Lichaei, Existence and nonexistence of global solutions of the Cauchy problem for higher order semilinear pseudohyperbolic equations, J. Nonlinear Anal.: Theory, Methods & Appl., 72, № 7-8, 3275 – 3288 (2010), https://doi.org/10.1016/j.na.2009.12.006 DOI: https://doi.org/10.1016/j.na.2009.12.006

Y. Z. Wang, Y. X. Wang, Existence and nonexistence of global solutions for a class of nonlinear wave equations of higher order, J. Nonlinear Anal.: Theory, Methods & Appl., 72, № 12, 4500 – 4507 (2010), https://doi.org/10.1016/j.na.2010.02.025 DOI: https://doi.org/10.1016/j.na.2010.02.025

V. A. Galactionov, E. L. Mitidieri, S. I. Pohozhaev, Blow-up for higher-order parabolic, hyperbolic, dispersion and Schrodinger equations, Chapman & Hall/CRC Monogr. and Research Notes Math. (2014). DOI: https://doi.org/10.1201/b17415

T. Ma, J. Gu, L. Li, Asymptotic behaviour of solutions to a class of fourth-order nonlinear evolution equations with dispersive and dissipative terms, J. Inequal. and Appl., 5, Article 318, 1 – 7 (2016), https://doi.org/10.1186/s13660-016-1269-8 DOI: https://doi.org/10.1186/s13660-016-1269-8

G. Lin, Y. Gao, Y. Sun, On local existence and blow-up solutions for nonlinear wave equations of higher-order Kirchhoff type with strong dissipation, Internat. J. Modern Nonlinear Theory and Appl., 6, № 1, 11 – 25 (2017), https://doi.org/10.4236/ijmnta.2017.61002 DOI: https://doi.org/10.4236/ijmnta.2017.61002

O. A. Ladyzhenskaya, Kraevye zadachi matematicheskoj fiziki, Nauka, Moskva (1973).

F. Kufner, S. Fuchik, Nelinejnye differencial'nye uravneniya, Nauka, Moskva (1988).

W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge Univ. Press (2000).

L. Hermander, Linejnye differencial'nye operatory s chastnymi proizvodnymi, Mir, Moskva (1965).

V. A. Trenogin, Funkcional'nyj analiz, Nauka, Moskva (1993).

E. Mitidieri, S. I. Pohozhaev, Apriornye ocenki i otsutstvie reshenij nelinejnyh uravnenij i neravenstv v chastnyh proizvodnyh, Tr. Mat. in-ta RAN, 234 (2001).

G. M. Fihtengol'c, Kurs differencial'nogo i integral'nogo ischisleniya, t. 1, Moskva (1969).

Published
08.07.2022
How to Cite
Kharibegashvili, S. S., and B. G. Midodashvili. “Boundary-Value Problem for a Class of Nonlinear Systems of Partial Differential Equations of Higher Orders”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 6, July 2022, pp. 856 -8, doi:10.37863/umzh.v74i6.6968.
Section
Research articles