Asymptotic Stability of a Stationary Solution of a Multidimensional Reaction–Diffusion Equation with a Discontinuous Source

Левашова, Н. Т.; Nefedov, N.N.; Orlov, A. S.

doi:10.1134/s0965542519040109

Cited by 22 publications

(5 citation statements)

References 17 publications

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“…In that work, a fairly effective method was proposed for proving the asymptotic stability of periodic solutions, which was then transferred to the analysis of the asymptotic stability of stationary solutions of initial-boundary value problems for reaction-diffusion-type parabolic equations and then generalized to some more complex classes of reaction-diffusionadvection-type quasilinear equations (see [29,45]). In recent years, this approach has been extended to reaction-diffusion-advection problems with discontinuous nonlinearities and sources (see [46] and references therein). An essential development of the method was the results on the asymptotic analysis of boundary and interior layers in various problems for integro-differential equations (see [28], as well as [47] and references therein).…”

Section: Asymptotic Methods Of Differential Inequalitiesmentioning

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: Asymptotic Methods Of Differential Inequalitiesmentioning

confidence: 99%

“…In addition, the local region of attraction of stationary solutions is indicated. These studies are presented in [46,49,50] (see also the references therein).…”

Section: Some Relevant Problems With Boundary and Interior Layersmentioning

confidence: 99%

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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“…Let us introduce the notation U n (x, t, ) = U (∓) n (x, t, ) which describe the sum of the expansion (9)- (11) up to an arbitrary order n and the following definition…”

Section: Q (∓)mentioning

confidence: 99%

“…In [29], Vasilieva proved the existence of a solution with internal transition layer for a system of singularly perturbed equations, which is constructed by the boundary function method and smooth matching condition. The existence and stability of a timeperiodic solution with a internal transition layer for the singularly perturbed twodimensional reaction-diffusion problem in a medium with discontinuous characteristic were shown in [19], and its steady-state equation was studied in [11]. In [9], Levashova, Nefedov and Orlov investigated the boundary problem of a two-dimensional singularly perturbed elliptic equation with discontinuous righthand function, constructed the asymptotic approximation and proved the existence of the solution with internal layer.…”

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confidence: 99%

Solution of contrast structure type for a reaction-diffusion equation with discontinuous reactive term

Wu¹,

Ni²

2021

DCDS-S

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In this paper, we consider the Dirichlet boundary value problem for a singularly perturbed reaction-diffusion equation with discontinuous reactive term. We use the asymptotic analysis to construct the formal asymptotic approximation of the solution with internal and boundary layers. The internal layer is located in the vicinity of a curve of the discontinuous reactive term. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic solution and estimate the accuracy of its asymptotic approximation.

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“…The existence theorem for moving front solutions of FitzHugh-Nagumo-type systems of equations was proved in [48]. The existence conditions for stable solutions with a large gradient at the boundaries of media discontinuities were formulated in [49,50].…”

Section: Theoretical Background Of a Spatio-temporal Model Of Shanghamentioning

confidence: 99%

A Spatio-Temporal Autowave Model of Shanghai Territory Development

et al. 2019

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A spatio-temporal model of megacity development that treats the megacity as an active medium is presented. From our point of view, it is advisable to consider the process of urban ecosystem development from the standpoint of the theory of autowave self-organization in active media. According to this concept, the urban ecosystem is considered as interacting with each other’s natural and anthropogenic subsystems with significant heterogeneity of areas affected by human intervention and urban geobiocoenoses. The model is based on the general principles of active medium dynamics; therefore, it is universal for any object to be considered an active medium. The only difference when using the model to predict the development of urban ecosystems in countries with different socio-economic and political prerequisites is the variety of parameters included in the model, i.e., the activation parameter, the autowave process inhibitors, and the characteristic scales of the activator and inhibitor. The model was tested on the example of Moscow expansion in the period of 1952–1968 and showed good agreement with the map data. By means of the model, a prediction of Shanghai and surrounding territory development until 2030 was made.

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Asymptotic Stability of a Stationary Solution of a Multidimensional Reaction–Diffusion Equation with a Discontinuous Source

Cited by 22 publications

References 17 publications

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

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

Solution of contrast structure type for a reaction-diffusion equation with discontinuous reactive term

A Spatio-Temporal Autowave Model of Shanghai Territory Development

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