Initial-boundary value problem with Dirichlet and Wentzell conditions for a mildly quasilinear biwave equation

V. I. Korzyuk   ; J. V. Rudzko

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 166 (2024) no. 3, pp. 377-394

Résumé

For a nonstrictly hyperbolic mildly quasilinear biwave equation in the first quadrant, an initial-boundary value problem with the Cauchy conditions specified on the spatial half-line and the Dirichlet and Wentzell conditions applied on the time half-line was examined. The solution was constructed in an implicit analytical form as a solution of some integro-differential equations. The solvability of these equations was investigated using the parameter continuation method. For the problem under study, the uniqueness of the solution was proved, and the conditions under which its classical solution exists were established. In the case when the data were not smooth enough, a mild solution was constructed.

Détail

Keywords: method of characteristics, mildly quasilinear biwave equation, nonlinear equation, nonstrictly hyperbolic equation, initial-boundary value problem, matching conditions, classical solution, parameter continuation method, mild solution.

Bibliographie

[1] Korzyuk V., Vinh N.V., Minh N.T., “Classical solution of the Cauchy problem for biwave equation: Application of Fourier transform”, Math. Modell. Anal., 17:5 (2012), 630–641 | DOI

[2] Thomson W.T, Theory of Vibration with Applications, Taylor Francis, London–New York, NY, 2010, 546 pp.

[3] Timoshenko–Ehrenfest beam theory, Wikipedia, the free encyclopedia, https://en.wikipedia.org/wiki/Timoshenko-Ehrenfest_beam_theory

[4] Korzyuk V.I., Konopel'ko O.A., Cheb E.S., “ Boundary-value problems for fourth-order equations of hyperbolic and composite types”, J. Math. Sci., 171:1 (2010), 89–115 | DOI

[5] Korzyuk V.I., Cheb E.S., Thu L.T., “Solution of the first mixed problem for the non-rigorous biwave equation”, Dokl. Nats. Akad. Nauk Belarusi, 55:4 (2011), 5–13 (In Russian)

[6] Korzyuk V.I., Cheb E.S., “Mixed problems for a biwave equation”, Vestn. BGU. Ser. 1, Fiz. Mat. Inf., 2005, no. 1, 63–68 (In Russian)

[7] Korzyuk V.I., Cheb E.S., “Goursat problem for a fourth-order equation with the biwave operator”, Differ. Equations, 45:10 (2009), 1467–1472 | DOI

[8] Fushchych W.I., “Symmetry in problems of mathematical physics”, Algebraic-Theoretical Studies in Mathematical Physics, Inst. Mat. Akad. Nauk USSR, Kyiv, 1981, 6–28 (In Russian)

[9] Fushchych W.I., Roman O.V., Zhdanov R.Z., “Symmetry reduction and exact solutions of nonlinear biwave equations”, Rep. Math. Phys., 37:2 (1996), 267–281 | DOI

[10] Bibilashvili T., Kharibegashvili S., “Darboux type problem for a class of fourth-order nonlinear hyperbolic equations”, Mem. Differ. Equations Math. Phys., 89 (2023), 39–59

[11] Kharibegashvili S., Midodashvili B., “On one boundary value problem for a nonlinear equation with the iterated wave operator in the principal part”, Georgian Math. J., 15:3 (2008), 541–554 | DOI

[12] Kharibegashvili S., “On the solvability of the Cauchy characteristic problem for a nonlinear equation with iterated wave operator in the principal part”, J. Math. Anal. Appl., 338:1 (2008), 71–81 | DOI

[13] Kharibegashvili S., Midodashvili B., “Solvability of characteristic boundary-value problems for nonlinear equations with iterated wave operator in the principal part”, Electron. J. Differ. Equations, 2008:72 (2008), 1–12

[14] Korzyuk V.I., Rudzko J.V., “Classical solution of the first mixed problem for the telegraph equation with a nonlinear potential”, Differ. Equations, 58:2 (2022), 175–186 | DOI

[15] Korzyuk V.I., Rudzko J.V., “Classical solution of the first mixed problem for the telegraph equation with a nonlinear potential in a curvilinear quadrant”, Differ. Equations, 59:8 (2023), 1075–1089 | DOI

[16] Korzyuk V.I., Stolyarchuk I.I., “Classical solution of the first mixed problem for second-order hyperbolic equation in curvilinear half-strip with variable coefficients”, Differ. Equations, 53:1 (2017), 74–85 | DOI

[17] Trenogin V.A., “Global invertibility of nonlinear operators and the method of continuation with respect to a parameter”, Dokl. Math., 54:2 (1996), 730–732

[18] Trenogin V.A., “Invertibility of nonlinear operators and parameter continuation method”, Spectral and Scattering Theory, ed. Ramm A.G., Plenum Press, New York, NY–London, 1998, 189–197 | DOI

[19] Qin Y., Integral and Discrete Inequalities and Their Applications, v. II, Nonlinear inequalities, Birkhäuser, Cham, 2016, xvi+1072 pp. | DOI