Geodesic
On the solvability of the boundary value problem for one class of nonlinear systems of high-order partial differential equations
Sbornik. Mathematics, Tome 215 (2024) no. 6, pp. 841-860 Cet article a éte moissonné depuis la source Math-Net.Ru

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For a class of nonlinear systems of partial differential equations of high order in a cylindrical domain the following boundary value problem is considered: Cauchy-type conditions are prescribed on the upper and lower bases of the cylinder, and a Robin-type condition is prescribed on the lateral part of the boundary. This boundary value problem is reduced in an equivalent way to a nonlinear functional equation in a certain subspace of the Sobolev space. Under certain assumptions about the nonlinear terms an a priori estimate is obtained for the solution of the problem in question and the existence of the solution is established, while in the case when these conditions fail, the absence of a solution is established. The question of uniqueness is also discussed for the solution. Bibliography: 18 titles.
Keywords: high-order nonlinear systems; fixed point principles; existence, uniqueness and nonexistence of a solution. 
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S. S. Kharibegashvili; B. G. Midodashvili. On the solvability of the boundary value problem for one class of nonlinear systems of high-order partial differential equations. Sbornik. Mathematics, Tome 215 (2024) no. 6, pp. 841-860. http://geodesic.mathdoc.fr/item/SM_2024_215_6_a6/

[1] L. Hörmander, The analysis of linear partial differential operators, v. II, Grundlehren Math. Wiss., 257, Differential operators with constant coefficients, Springer-Verlag, Berlin, 1983, ix+391 pp. | MR | Zbl

[2] S. Kharibegashvili and B. Midodashvili, “On the solvability of one boundary value problem for a class of higher-order nonlinear partial differential equations”, Mediterr. J. Math., 18:4 (2021), 131, 18 pp. | DOI | MR | Zbl

[3] S. Kharibegashvili and B. Midodashvili, “The boundary value problem for one class of higher-order semilinear partial differential equations”, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 49:1 (2023), 154–171 | DOI | MR | Zbl

[4] È. Mitidieri and S. I. Pokhozhaev, “A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities”, Proc. Steklov Inst. Math., 234 (2001), 1–362

[5] V. A. Galaktionov, E. L. Mitidieri and S. I. Pohozaev, Blow-up for higher-order parabolic, hyperbolic, dispersion and Schrödinger equations, Monogr. Res. Notes Math., CRC Press, Boca Raton, FL, 2015, xxvi+543 pp. | DOI | MR | Zbl

[6] Guowang Chen, Ruili Song and Shubin Wang, “Local existence and global nonexistence theorems for a damped nonlinear hyperbolic equation”, J. Math. Anal. Appl., 368:1 (2010), 19–31 | DOI | MR | Zbl

[7] Tengyu Ma, Juan Gu and Longsuo Li, “Asymptotic behavior of solutions to a class of fourth-order nonlinear evolution equations with dispersive and dissipative terms”, J. Inequal. Appl., 2016 (2016), 318, 7 pp. | DOI | MR | Zbl

[8] Jiangbo Han, Runzhang Xu and Yanbing Yang, “Asymptotic behavior and finite time blow up for damped fourth order nonlinear evolution equation”, Asymptot. Anal., 122:3–4 (2021), 349–369 | DOI | MR | Zbl

[9] S. Kharibegashvili and B. Midodashvili, “Solvability of characteristic boundary-value problems for nonlinear equations with iterated wave operator in the principal part”, Electron. J. Differential Equations, 2008 (2008), 72, 12 pp. | MR | Zbl

[10] S. Kharibegashvili, “Boundary value problems for some classes of nonlinear wave equations”, Mem. Differential Equations Math. Phys., 46 (2009), 1–114 | MR | Zbl

[11] S. Kharibegashvili and B. Midodashvili, “A boundary value problem for higher-order semilinear partial differential equations”, Complex Var. Elliptic Equ., 64:5 (2019), 766–776 | DOI | MR | Zbl

[12] S. Kharibegashvili and B. Midodashvili, “On the solvability of one boundary value problem for one class of higher-order semilinear hyperbolic systems”, Lith. Math. J., 62:3 (2022), 360–371 | DOI | MR | Zbl

[13] S. S. Kharibegashvili and B. G. Midodashvili, “On the solvability of a special boundary value problem in a cylindrical domain for a class of nonlinear systems of partial differential equations”, Differ. Equ., 58:1 (2022), 81–91 | DOI | MR | Zbl

[14] O. A. Ladyzhenskaya, The boundary value problems of mathematical physics, Appl. Math. Sci., 49, Springer-Verlag, New York, 1985, xxx+322 pp. | DOI | MR | Zbl

[15] S. Fučík and A. Kufner, Nonlinear differential equations, Stud. Appl. Mech., 2, Elsevier Sci. Publ., Amsterdam–New York, 1980, 359 pp. | MR | Zbl

[16] W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge Univ. Press, Cambridge, 2000, xiv+357 pp. | MR | Zbl

[17] V. A. Trenogin, Functional analysis, 2nd ed., Nauka, Moscow, 1993, 440 pp. (Russian) ; French transl. of 1st ed., V. Trénoguine, Analyse fonctionnelle, Traduit Russe Math., Mir, Moscow, 1985, 528 pp. | MR | Zbl | MR

[18] G. M. Fichtehholz, Differential and integral calculus, v. 1, 7th ed., Nauka, Moscow, 1969, 608 pp. (Russian); German transl., Differential- und Integralrechnung, v. I, Hochschulbücher für Math., 61, 12. Aufl., VEB Deutscher Verlag der Wissenschaften, Berlin, 1986, xiv+572 pp. | MR | Zbl