Abstract
The boundary value problem for a class of higher-order nonlinear partial differential equations is considered. The theorems on existence, uniqueness and nonexistence of solutions of this problem are proved.
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Aliev, A.B., Lichaei, B.H.: 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)
Evans, L.C.: Partial differential equations. In: Graduate Student of Mathematics, vol 19. American Mathematical Society, Providence, RI (1998)
Galactionov, V.A., Mitidieri, E.L., Pohozaev, S.I: Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrodinger Equations. Series: Chapman & Hall/CRC Monographs and Research Notes in Mathematics (2014)
Hörmander, L.: The Analysis of Linear Partial Differential Operators II. Differential Operators with Constant Coefficients. Springer, Berlin (1983)
Kharibegashvili, S.: Boundary value problems for some classes of nonlinear wave equations. Mem. Differ. Equ. Math. Phys. 46, 1–114 (2009)
Kharibegashvili, S.: The boundary value problem for one class of semilinear partial differential equations. In: International Workshop QUALITDE—2017, December 24–26, Tbilisi, Georgia, pp. 81–82 (2017)
Kharibegashvili, S.: The solvability of the boundary value problem for one class of higher-order nonlinear partial differential equations. In: International Workshop QUALITDE—2019: December 7–9, Tbilisi, Georgia, pp. 96–98 (2019)
Kharibegashvili, S., Midodashvili, B.: Solvability of characteristic boundary-value problems for nonlinear equations with iterated wave operator in the principal part. Electron. J. Differ. Equ. 2008(72), 1–12 (2008)
Kharibegashvili, S., Midodashvili, B.: On one boundary value problem for a nonlinear equation with iterated wave operator in the principal part. Ga. Math. J. 15(3), 541–554 (2008)
Kiguradze, T.: Raja Ben-Rabha: on strong well-posedness of initial-boundary value problems for higher order nonlinear hyperbolic equations with two independent variables. Ga. Math. J. 24(3), 409–428 (2017)
Kufner, A., Fučik, S.: Nonlinear Differential Equations. Elsevier, Amsterdam (1980)
Ladyzhenskaya, O.A.: The Boundary Value Problems of Mathematical Physics. Springer, New York (1985)
Lin, G., Gao, Y., Sun, Y.: On local existence and blow-up solutions for nonlinear wave equations of higher-order Kirchhoff type with strong dissipation. IJMNTA 6(1), 11–25 (2017)
Ma, T., Gu, J., Li, L.: Asymptotic behaviour of solutions to a class of fourth-order nonlinear evolution equations with dispersive and dissipative terms. J. Inequal. Appl. 318(1), 1–7 (2016)
Mitidieri, E., Pohozhaev, S.I.: A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities (Russian). Tr. Mat. Inst. Steklova 234, 1–384 (2001) [English transl.: Proc. Steklov Inst. Math. 2001, No. 3 (234), pp. 1–362]
Trenogin, V.A.: Functional Analysis (Russian), 2nd edn. Nauka, Moscow (1993)
Vulikh, B.Z.: Concise Course of the Theory of Functions of a Real Variable (Russian). Nauka, Moscow (1973)
Wang, Y.Z., Wang, Y.X.: 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)
Xiangying, C.: Existence and nonexistence of global solutions for nonlinear evolution equation of fourth order. Appl. Math. J. Chin. Univ. Ser. B 16(3), 251–258 (2001)
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Kharibegashvili, S., Midodashvili, B. On the Solvability of One Boundary Value Problem for a Class of Higher-Order Nonlinear Partial Differential Equations. Mediterr. J. Math. 18, 131 (2021). https://doi.org/10.1007/s00009-021-01752-2
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DOI: https://doi.org/10.1007/s00009-021-01752-2
Keywords
- Nonlinear higher-order equations
- Schaefer’s fixed point theorem
- existence
- uniqueness and nonexistence of solutions