On the Motion, Amplification, and Blow-up of Fronts in Burgers-Type Equations with Quadratic and Modular Nonlinearity
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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,** Received May 26, 2020; revised June 8, 2020; accepted June 9, 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
1. INTRODUCTION AND FORMULATION OF THE PROBLEM The Burgers equation (see [1]) and Burgers-type equations have been extensively used for a long time in the modeling of physical systems in gas dynamics, nonlinear acoustics, turbulence, shallow-water waves, and other applications. They are also used in nonlinear partial differential equations describing traveling waves and weak shock waves and in the development of numerical methods for parabolic and hyperbolic equations (see [2–5]). The stability of traveling-wave solutions was analyzed in [6]). Pioneering results concerning the theory of blow-up of solutions can be found in [7] (see also references therein). In this paper, we study solutions with an internal transition layer of the equation
(
)
β 2 u +γu 2 2 ∂ ∂ u u ε 2− = − αf (u, x, t ), ∂t ∂x ∂x x ∈ (0, 1), t ∈ (0, T ], ∂
(1)
where ε is a dimensionless small parameter and α , β, and γ are constants. In the case α = 0 and γ = 0, this is 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 * e-mail: [email protected] ** e-mail: [email protected]
the Burgers equation; in the case α = 0 and β = 0, we have a Burgers-type equation with modular nonlinearity [8, 9]; and in the case α ≠ 0, a Burgers-type equation with amplification (the case of cubic amplification f (u, x, t ) = u3 was considered in [10]). Note that the case of modular nonlinearity is of interest for both applications in mechanics of structurally inhomogeneous multiphase media, composites, and metamaterials [11] and as a simpler model for analytical considerations in transition-layer problems [12]. We consider a formulation similar to the problem for the Burgers equation:
u(0, t, ε) = u (t ), u(1, t, ε) = u (t ), t ∈ [0, T ], u( x, 0, ε) = uinit ( x, ε), x ∈ [0, 1]. 0
1
(2)
The initial function uinit
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