This article is cited in 2 papers

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.

UDC: 517.957

MSC: 35B44, 35A01, 35G31

Received: 12.11.2015

DOI: 10.4213/im8471

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

Bibliographic databases:

• 
• 
• 
• 
•