On the Motion, Amplification, and Blow-up of Fronts
in Burgers-Type Equations with Quadratic and Modular Nonlinearity
N. N. Nefedova,* and Academician of the RAS O. V. Rudenkoa,b,c,**
a Faculty of Physics, Lomonosov Moscow State University, Moscow, 119991 Russia
b Prokhorov General Physics Institute, Russian Academy
of Sciences, Moscow, 119991 Russia
c Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, Moscow, 123810 Russia
Correspondence to: * e-mail: nefedov@phys.msu.ru
Correspondence to: ** e-mail: rudenko@acs366.phys.msu.ru
Received 26 May, 2020
Abstract—A singularly perturbed initial-boundary value problem for a parabolic equation, which is called in applications an equation of Burgers type, is considered. Existence conditions are obtained, and an asymptotic approximation of a new class of solutions with a moving front is constructed. The results are applied to problems with quadratic and modular nonlinearity and nonlinear amplification. The influence of nonlinear amplification on the propagation and destruction of fronts is revealed. Estimates for the blow-up localization and blow-up time are obtained.
Keywords: singularly perturbed parabolic problems, equations of Burgers type, reaction–diffusion–advection equations, internal layers, fronts, asymptotic methods, blow-up of solutions
DOI: 10.1134/S1064562420040146