Geodesic
A Mixed Problem for a Class of Second-Order Nonlinear Hyperbolic Systems with Dirichlet and Poincar\'e Boundary Conditions
Matematičeskie zametki, Tome 114 (2023) no. 5, pp. 702-720.

Voir la notice de l'article provenant de la source Math-Net.Ru

For a certain class of second-order hyperbolic systems, a mixed problem with Dirichlet and Poincaré boundary conditions is studied. In the linear case, an explicit representation of a soultion of this problem is given and questions related to its uniqueness and existence are studied depending on the character of nonlinearities in the system.
Keywords: semilinear hyperbolic system, mixed problem, a priori estimate
Mots-clés : Laplace invariants. 
@article{MZM_2023_114_5_a4,
     author = {O. M. Dzhokhadze and S. S. Kharibegashvili and N. N. Shavlakadze},
     title = {A {Mixed} {Problem} for a {Class} of {Second-Order} {Nonlinear} {Hyperbolic} {Systems} with {Dirichlet} and {Poincar\'e} {Boundary} {Conditions}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {702--720},
     publisher = {mathdoc},
     volume = {114},
     number = {5},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2023_114_5_a4/}
}
TY  - JOUR
AU  - O. M. Dzhokhadze
AU  - S. S. Kharibegashvili
AU  - N. N. Shavlakadze
TI  - A Mixed Problem for a Class of Second-Order Nonlinear Hyperbolic Systems with Dirichlet and Poincar\'e Boundary Conditions
JO  - Matematičeskie zametki
PY  - 2023
SP  - 702
EP  - 720
VL  - 114
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2023_114_5_a4/
LA  - ru
ID  - MZM_2023_114_5_a4
ER  - 
%0 Journal Article
%A O. M. Dzhokhadze
%A S. S. Kharibegashvili
%A N. N. Shavlakadze
%T A Mixed Problem for a Class of Second-Order Nonlinear Hyperbolic Systems with Dirichlet and Poincar\'e Boundary Conditions
%J Matematičeskie zametki
%D 2023
%P 702-720
%V 114
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2023_114_5_a4/
%G ru
%F MZM_2023_114_5_a4
O. M. Dzhokhadze; S. S. Kharibegashvili; N. N. Shavlakadze. A Mixed Problem for a Class of Second-Order Nonlinear Hyperbolic Systems with Dirichlet and Poincar\'e Boundary Conditions. Matematičeskie zametki, Tome 114 (2023) no. 5, pp. 702-720. http://geodesic.mathdoc.fr/item/MZM_2023_114_5_a4/

[1] R. Kurant, Uravneniya s chastnymi proizvodnymi, Mir, M., 1964 | MR

[2] O. Melnik, “Obschie smeshannye zadachi dlya obschikh dvumernykh giperbolicheskikh sistem”, Differents. uravneniya, 2:7 (1966), 958–966 | MR | Zbl

[3] A. D. Myshkis, A. M. Filimonov, “O globalnoi nepreryvnoi razreshimosti smeshannoi zadachi dlya odnomernykh giperbolicheskikh sistem kvazilineinykh uravnenii”, Differents. uravneniya, 44:3 (2008), 394–407 | DOI | MR

[4] V. A. Terletskii, “Obobschennoe reshenie odnomernykh polulineinykh giperbolicheskikh sistem so smeshannymi usloviyami”, Izv. vuzov. Matem., 2004, no. 12, 75–83 | MR

[5] G. F. D. Duff, “A mixed problem for normal hyperbolic linear partial differential equations of second order”, Canadian J. Math., 9 (1957), 141–160 | DOI | MR

[6] G. Eskin, “Mixed initial-boundary value problems for second order hyperbolic equations”, Comm. Partial Differential Equations, 12:5 (1987), 503–587 | DOI | MR

[7] K. Agre, M. A. Rammaha, “Systems of nonlinear wave equations with damping and source terms”, Differential Integral Equations, 19:11 (2007), 1235–1270 | MR

[8] K. Heinz-Otto, O. E. Ortiz, N. A. Petersson, “Initial-boundary value problems for second order systems of partial differential equations”, ESAIM Math. Model. Numer. Anal., 46:3 (2012), 559–593 | DOI | MR

[9] E. Gursa, Kurs matematicheskogo analiza. T. 3. Ch. 1, M.–L., 1933 | MR

[10] F. Trikomi, Lektsii po uravneniyam s chastnymi proizvodnymi, M., 1957

[11] I. G. Petrovskii, Lektsii po teorii obyknovennykh differentsialnykh uravnenii, Izd-vo MGU, M., 1984 | MR

[12] I. T. Kiguradze, Nachalnye i kraevye zadachi dlya sistem obyknovennykh differentsialnykh uravnenii. I, Metsniereba, Tbilisi, 1997

[13] G. Dekanoidze, S. Kharibegashvili, “On the global solvability of the first Darboux problem for one class of nonlinear second order hyperbolic systems”, Mem. Differ. Equ. Math. Phys., 71 (2017), 51–68 | MR

[14] D. Gilbarg, N. S. Trudinger, Ellipticheskie uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989 | MR

[15] V. A. Trenogin, Funktsionalnyi analiz, Nauka, M., 1993 | MR