Geodesic
On the asymptotics of the solution to a singularly perturbed hyperbolic system of equations with several spatial variables in the critical case
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 3, pp. 450-462 
T. V. Pavlyuk; A. V. Nesterov. On the asymptotics of the solution to a singularly perturbed hyperbolic system of equations with several spatial variables in the critical case. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 3, pp. 450-462. http://geodesic.mathdoc.fr/item/ZVMMF_2014_54_3_a8/
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     title = {On the asymptotics of the solution to a singularly perturbed hyperbolic system of equations with several spatial variables in the critical case},
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Voir la notice de l'article provenant de la source Math-Net.Ru

A complete asymptotic expansion of the solution to an initial value problem for a singularly perturbed hyperbolic system of equations in several spatial variables is constructed and justified. A specific feature of the problem is that its solution has a spike zone in a neighborhood of which the asymptotics is described by a parabolic equation.

[1] Nesterov A. V., Shuliko O. V., “Asimptotika resheniya singulyarno vozmuschennoi sistemy differentsialnykh uravnenii pervogo poryadka v chastnykh proizvodnykh s maloi nelineinostyu v kriticheskom sluchae”, Zh. vychisl. matem. i matem. fiz., 47:3 (2007), 438–444 | MR | Zbl

[2] Nesterov A. V., Shuliko O. V., “Ob asimptotike resheniya singulyarno vozmuschennoi sistemy parabolicheskikh uravnenii v kriticheskom sluchae”, Zh. vychisl. matem. i matem. fiz., 50:2 (2010), 268–275 | MR | Zbl

[3] Nesterov A. V., “On the asymptotics of the solution to a singularly perturbed system of first-order partial differential equations with small nonlinearity in the critical case”, Zh. vychisl. matem. i matem. fiz., 52:7 (2012), 1267–1276 | Zbl

[4] Vasileva A. B., Butuzov V. F., Singulyarno vozmuschennye uravneniya v kriticheskikh sluchayakh, Izd-vo MGU, M., 1978 | MR

[5] Kalyakin L. A., “Dlinnovolnovye asimptotiki. Integriruemye uravneniya kak asimptoticheskii predel nelineinykh sistem”, Uspekhi matem. nauk, 44:1(265) (1989), 5–27 | MR

[6] Babich V. M., Buldyrev V. S., Asimptoticheskie metody v zadachakh difraktsii korotkikh voln, Nauka, M., 1972 | MR

[7] Talanov V. I., “O samofokusirovke volnovykh puchkov v nelineinykh sredakh”, Pisma v zhurn. eksperim. i teor. fiz., 1965, no. 2, 218–222

[8] Kulikovskii A. G., Sveshnikova E. I., Chugainova A. P., Matematicheskie metody izucheniya razryvnykh reshenii nelineinykh giperbolicheskikh sistem uravnenii, Lektsionnye kursy NOTs, 16, MIAN, M., 2010 | DOI

[9] Ryskin N. M., Trubetskov D. I., Nelineinye volny, Ucheb. posobie, Fizmatlit, M., 2000

[10] Rozhdestvenskii B. L., Yanenko N. N., Sistemy kvazilineinykh uravnenii, Nauka, M., 1978 | MR

[11] Ilin A. M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989 | MR

[12] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967 | MR