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Izv. RAN. Ser. Mat., 2017, Volume 81, Issue 6, Pages 232–242 (Mi izv8471)  

This article is cited in 1 scientific paper (total in 1 paper)

Gradient blow-up in generalized Burgers and Boussinesq equations

E. V. Yushkov, M. O. Korpusov

Lomonosov Moscow State University, Faculty of Physics

Abstract: We study the influence of gradient non-linearity on the global solubility of initial-boundary value problems for a generalized Burgers equation and an improved Boussinesq equation which are used for describing one-dimensional wave processes in dissipative and dispersive media. For a large class of initial data, we obtain sufficient conditions for global insolubility and a bound for blow-up times. Using the Boussinesq equation as an example, we suggest a modification of the method of non-linear capacity which is convenient from a practical point of view and enables us to estimate the blow-up rate. We use the method of contraction mappings to study the possibility of instantaneous blow-up and short-time existence of solutions.

Keywords: gradient non-linearity, Burgers equation and generalized Boussinesq equations, blow-up phenomena, method of non-linear capacity.

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-12460
This paper was written with the support of the Russian Foundation for Basic Research (grant no. 13-01-12460).


DOI: https://doi.org/10.4213/im8471

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English version:
Izvestiya: Mathematics, 2017, 81:6, 1286–1296

Bibliographic databases:

UDC: 517.957
MSC: 35B44, 35A01, 35G31
Received: 12.11.2015

Citation: E. V. Yushkov, M. O. Korpusov, “Gradient blow-up in generalized Burgers and Boussinesq equations”, Izv. RAN. Ser. Mat., 81:6 (2017), 232–242; Izv. Math., 81:6 (2017), 1286–1296

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. I. I. Kolotov, A. A. Panin, “O neprodolzhaemykh resheniyakh i razrushenii reshenii psevdoparabolicheskikh uravnenii s koertsitivnoi i znakopostoyannoi nelineinostyami: analiticheskoe i chislennoe issledovanie”, Matem. zametki, 105:5 (2019), 708–723  mathnet  crossref  elib
  • Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya Izvestiya: Mathematics
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