Classical and mild solution of the first mixed problem for the telegraph equation with a nonlinear potential

Viktor I. Korzyuk ; Jan V. Rudzko

The Bulletin of Irkutsk State University. Series Mathematics, Tome 43 (2023), pp. 48-63

Résumé

We study the first mixed problem for the telegraph equation with a nonlinear potential in the first quadrant. We pose the Cauchy conditions on the lower base of the domain and the Dirichlet condition on the lateral boundary. By the method of characteristics, we obtain an expression for the solution of the problem in an implicit analytical form as a solution of some integral equations. To solve these equations, we use the method of sequential approximations. The existence and uniqueness of the classical solution under specific smoothness and matching conditions for given functions are proved. Under inhomogeneous matching conditions, we consider a problem with conjugation conditions. When the given data is not smooth enough, we construct a mild solution.

Détail

Keywords: nonlinear wave equation, classical solution, mixed problem, matching conditions, generalized solution.

Bibliographie

[1] Bitsadze A. V., Equations of Mathematical Physics, 2nd. ed., Nauka Publ, M., 1982, 336 pp.

[2] Gallagher I., Gérard P., “Profile decomposition for the wave equation outside a convex obstacle”, Journal de Mathé matiques Pures et Appliquées, 80:1 (2001), 1–49 | DOI

[3] Ikehata R., “Two dimensional exterior mixed problem for semilinear damped wave equations”, J. Math. Anal. Appl., 301:2 (2005), 366–377 | DOI

[4] Ikehata R., “Global existence of solutions for semilinear damped wave equation in 2-D exterior domain”, J. Differ. Equations, 200:1 (2004), 53–68 | DOI

[5] Korzyuk V. I., Equations of Mathematical Physics, 2nd ed., Editorial URSS Publ, M., 2021, 410 pp.

[6] Korzyuk V. I., Kozlovskaya I. S., Naumavets S. N., “Classical solution of the first mixed problem of the one-dimensional wave equation with conditions of Cauchy type”, Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series, 2015, no. 1, 7–21

[7] Korzyuk V. I., Naumavets S. N., Serikov V. P., “Mixed problem for a one-dimensional wave equation with conjugation conditions and second-order derivatives in boundary conditions”, Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 56:3 (2020), 287–297 | DOI

[8] Korzyuk V. I., Rudzko J. V., “Classical Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential”, Diff. Equat., 58 (2022), 175–186 | DOI | DOI

[9] Korzyuk V. I., Rudzko J. V., Curvilinear parallelogram identity and mean-value property for a semilinear hyperbolic equation of second-order, 2022, arXiv: 2204.09408 | DOI

[10] 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”, Diff. Equat., 53:1 (2017), 74–85 | DOI | DOI

[11] Korzyuk V. I., Stolyarchuk I. I., “Solving the mixed problem for the Klein–Gordon–Fock type equation with integral conditions in the case of the inhomogeneous matching conditions”, Doklady of the National Academy of Sciences of Belarus, 63:2 (2019), 142–149 | DOI

[12] Lavrenyuk S. P., Panat O. T., “Mixed problem for a nonlinear hyperbolic equation in an unbounded domain”, Reports of the National Academy of Sciences of Ukraine, 2007, no. 1, 12–17

[13] Mitrinović D.S., Pečarić J.E., Fink A. M., Inequalities Involving Functions and Their Integrals and Derivatives, Springer Netherlands, Dordrecht, 1991, 603 pp. | DOI

[14] Polyanin A. D., Zaitsev V. F., Handbook of nonlinear partial differential equations, Chapman Hall/CRC, New York, 2004, 840 pp. | DOI

[15] Staliarchuk I. I., Classical solutions of the problems for Klein–Gordon–Fock equation, Cand. Sci. Phys.-Math. Dis., Grodno, 2020, 124 pp.

[16] M.M. Vainberg, Variational methods for the study of nonlinear operators, Holden-Day series in mathematical physics, Holden-Day Publ, San Francisco, 1964, 323 pp.