The existence and asymptotic stability of periodic solutions with an interior layer of Burgers type equations with modular advection

Нефедов, Н. Н.

doi:10.1051/mmnp/2019009

Cited by 13 publications

(2 citation statements)

References 16 publications

(19 reference statements)

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“…Similar results were obtained in the study of periodic fronts in parabolic periodic boundary value problems, including Burgers-type equations. Various problems of this class are presented in [63][64][65][66][67][68]. In these works, periodic problems, multidimensional in the spatial variable, with an interior transition layer are considered and classes of new problems are singled out in which the multidimensionality leads to new previously unexplored conditions for the existence and stability of a solution with an interior transition layer.…”

Section: Some Relevant Problems With Boundary and Interior Layersmentioning

confidence: 99%

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Development of Methods of Asymptotic Analysis of Transition Layers in Reaction–Diffusion–Advection Equations: Theory and Applications

Нефедов

2021

Comput. Math. and Math. Phys.

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This work presents a review and analysis of modern asymptotic methods for analysis of singularly perturbed problems with interior and boundary layers. The central part of the work is a review of the work of the author and his colleagues and disciples. It highlights boundary and initial-boundary value problems for nonlinear elliptic and parabolic partial differential equations, as well as periodic parabolic problems, which are widely used in applications and are called reaction–diffusion and reaction–diffusion–advection equations. These problems can be interpreted as models in chemical kinetics, synergetics, astrophysics, biology, and other fields. The solutions of these problems often have both narrow boundary regions of rapid change and inner layers of various types (contrasting structures, moving interior layers: fronts), which leads to the need to develop new asymptotic methods in order to study them both formally and rigorously. A general scheme for a rigorous study of contrast structures in singularly perturbed problems for partial differential equations, based on the use of the asymptotic method of differential inequalities, is presented and illustrated on relevant problems. The main achievements of this line of studies of partial differential equations are reflected, and the key directions of its development are indicated.

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Section: Some Relevant Problems With Boundary and Interior Layersmentioning

confidence: 99%

“…). The formal asymptotics of the solution of each of these problems (23) is sought in the form (according to the scheme of Section 2.1; for details, see, e.g., [68]) (24) where and denote the regular and boundary-layer (near ) parts of asymptotics (19).…”

Section: X T K X T X X T X T X T Satisfies the Conditionmentioning

confidence: 99%