Geodesic
Classical solution of the third mixed problem for the telegraph equation with nonlinear potential
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 3, Tome 232 (2024), pp. 37-49.

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For a telegraph equation with a nonlinear potential specified in the first quadrant, we consider a mixed problem with Cauchy conditions on the spatial semi-axis and a condition of the third kind (Robin's condition) on the temporal semi-axis. The solution is constructed by the method of characteristics in an implicit analytical form as a solution of some integral equations. The solvability of these equations and the dependence of their solutions on the initial data are examined. For the problem considered, the uniqueness of the solution is proved and existence conditions for classical solutions are obtained. If the matching conditions are not fulfilled, the problem with matching conditions is constructed, and if the data is not sufficiently smooth, a weak solution is constructed.
Keywords: classical solution, mixed problem, conditions of the third kind, matching conditions, nonlinear wave equation 
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V. I. Korzyuk; J. V. Rudzko. Classical solution of the third mixed problem for the telegraph equation with nonlinear potential. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 3, Tome 232 (2024), pp. 37-49. http://geodesic.mathdoc.fr/item/INTO_2024_232_a3/

[1] Dzhokhadze O. M., “Smeshannaya zadacha s nelineinym granichnym usloviem dlya polulineinogo uravneniya kolebaniya struny”, Differ. uravn., 58:5 (2022), 591–606 | Zbl

[2] Korzyuk V. I., Kozlovskaya I. S., Klassicheskie resheniya zadach dlya giperbolicheskikh uravnenii: kurs lektsii, BGU, Minsk, 2017

[3] Korzyuk V. I., Kozlovskaya I. S. Naumovets S. N., “Klassicheskoe reshenie pervoi smeshannoi zadachi odnomernogo volnovogo uravneniya s usloviyami tipa Koshi”, Izv. NAN Belarusi. Ser. fiz.-mat. nauk., 2015, no. 1, 7–21

[4] Korzyuk V. I., Naumovets S. N., Serikov V. P., “Smeshannaya zadacha dlya odnomernogo volnovogo uravneniya s usloviyami sopryazheniya i proizvodnymi vtorogo poryadka v granichnykh usloviyakh”, Izv. NAN Belarusi. Ser. fiz.-mat. nauk., 56:3 (2020), 287–297 | MR

[5] Korzyuk V. I., Rudko Ya. V., “Klassicheskoe i slaboe reshenie pervoi smeshannoi zadachi dlya telegrafnogo uravneniya s nelineinym potentsialom”, Izv. Irkut. gos. un-ta. Ser. Mat., 43 (2023), 48–63 | MR | Zbl

[6] Korzyuk V. I., Rudko Ya. V., “Klassicheskoe reshenie pervoi smeshannoi zadachi dlya telegrafnogo uravneniya s nelineinym potentsialom”, Differ. uravn., 58:2 (2022), 174–184 | Zbl

[7] Korzyuk V. I., Rudko Ya. V., “Lokalnoe klassicheskoe reshenie zadachi Koshi dlya polulineinogo giperbolicheskogo uravneniya v sluchae dvukh nezavisimykh peremennykh”, Mat. Mezhdunar. nauch. konf. «Ufimskaya osennyaya matematicheskaya shkola». T. 2 (Ufa, 28 sentyabrya"– 1 oktyabrya 2022), RITs BashGU, Ufa, 2022, 48–50

[8] Korzyuk V. I., Stolyarchuk I. I., “Klassicheskoe reshenie pervoi smeshannoi zadachi dlya giperbolicheskogo uravneniya vtorogo poryadka v krivolineinoi polupolose s peremennymi koeffitsientami”, Differ. uravn., 53:1 (2017), 77–88 | DOI | MR | Zbl

[9] Korzyuk V. I., Stolyarchuk I. I., “Klassicheskoe reshenie pervoi smeshannoi zadachi dlya uravneniya tipa Kleina—Gordona—Foka s neodnorodnymi usloviyami soglasovaniya”, Dokl. NAN Belarusi., 63:1 (2019), 7–13 | MR | Zbl

[10] Matematicheskaya entsiklopediya, v. 3, Sovetskaya entsiklopediya, M., 1982

[11] Moiseev E. I., Korzyuk V. I., Kozlovskaya I. S., “Klassicheskoe reshenie zadachi s integralnym usloviem dlya odnomernogo volnovogo uravneniya”, Differ. uravn., 50:10 (2014), 1373–1385 | DOI | MR | Zbl

[12] Prokhorov A. M., Fizicheskaya entsiklopediya. T. 3, M., 1992

[13] Fuschich V. I., “Simmetriya v zadachakh matematicheskoi fiziki”, Teoretiko-algebraicheskie issledovaniya v matematicheskoi fizike, eds. Fuschich V. I., In-t mat. AN USSR, Kiev, 1981, 6–28 | MR

[14] Kharibegashvili S. S., Dzhokhadze O. M., “Zadacha Koshi dlya obobschennogo nelineinogo uravneniya Liuvillya”, Differ. uravn., 47:12, 1741–1753 | MR | Zbl

[15] Evans L. C., Partial Differential Equations, Am. Math. Soc., Providence, 2010 | MR | Zbl

[16] Giai Giang Vo., “A semilinear wave equation with a boundary condition of many-point type: Global existence and stability of weak solutions”, Abstr. Appl. Anal., 2015 (2015), 531872 | MR | Zbl

[17] Korzyuk V. I., Rudzko J. V., On the absence and non-uniqueness of classical solutions of mixed problems for the telegraph equation with a nonlinear potential, arXiv: 2303.17483 [math.AP]

[18] Nesterenko Yu. R., “Mixed problem for the wave equation with Robin boundary conditions”, Dokl. Math., 79:3 (2009), 322–324 | DOI | MR | Zbl

[19] Nikitin A. A., “On the mixed problem for the wave equation with the third and first boundary conditions”, Differ. Equations., 43:12 (2007), 1733–1741 | DOI | MR | Zbl

[20] Le Thi Phuong Ngoc, Le Huu Ky Son, Nguyen Thanh Long, “Existence, blow-up and exponential decay estimates for the nonlinear Kirchhoff–Carrier wave equation in an annular with Robin–Dirichlet conditions”, Kyungpook Math. J., 61 (2021), 859–888 | MR | Zbl

[21] Li T., “Exact boundary controllability for quasilinear wave equations”, J. Comput. Appl. Math., 190 (2006), 127–135 | DOI | MR | Zbl

[22] Nakayama Y., “Liouville field theory: A decade after the revolution”, Int. J. Mod. Phys. A., 19:17–18 (2004), 2771–2930 | DOI | MR | Zbl

[23] Nguyen Huu Nhan, Le Thi Phuong Ngoc, Tran Minh Thuyet, Nguyen Thanh Long, “A Robin–Dirichlet problem for a nonlinear wave equation with the source term containing a nonlinear integral”, Lithuan. Math. J., 57:1 (2017), 80–108 | DOI | MR | Zbl

[24] Van Horssen W. T., Wang Y., Cao G., “On solving wave equations on fixed bounded intervals involving Robin boundary conditions with time-dependent coefficients”, J. Sound Vibr., 424 (2018), 263–271 | DOI

[25] Wu X., Mei L., Liu C., “An analytical expression of solutions to nonlinear wave equations in higher dimensions with Robin boundary conditions”, J. Math. Anal. Appl., 426:2 (2015), 1164–1173 | DOI | MR | Zbl

[26] Xie Y., Tang J., “A unified method for solving sinh-Gordon-type equations”, Nuovo Cim., 121:2 (2006), 1373–1385 | MR