Asymptotic Method Of Differential Inequalities Research Articles

Existence and Stability of a Stationary Solution in a Two-Dimensional Reaction-Diffusion System with Slow and Fast Components

In the paper, the existence of a stable stationary solution in a reaction-diffusion system with slow and fast components in a two-dimensional spatial variable case is investigated. The theorem of the existence of a stationary solution with boundary layers in the case of Dirichlet boundary conditions is proven, its asymptotic approximation is constructed, and conditions for Lyapunov asymptotic stability of this solution are obtained. The research is based on the asymptotic method of differential inequalities, applied to a new class of problems. This result is practically important both for various applications described by similar systems and for the application of numerical stationing methods when solving elliptical boundary value problems.

Existence and Stability of Solutions with Internal Transition Layer for the Reaction–Diffusion–Advection Equation with a KPZ-Nonlinearity

We study a boundary value problem for a quasilinear reaction–diffusion–advection ordinary differential equation with a KPZ-nonlinearity containing the squared gradient of the unknown function. The noncritical and critical cases of existence of an internal transition layer are considered. An asymptotic approximation to the solution is constructed, and the asymptotics of the transition layer point is determined. Existence theorems are proved using the asymptotic method of differential inequalities, the Lyapunov asymptotic stability of solutions is proved by the narrowing barrier method, and instability theorems are proved with the use of unordered upper and lower solutions.

Development of Methods of Asymptotic Analysis of Transition Layers in Reaction\u2013Diffusion\u2013Advection Equations: Theory and Applications

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.

Asymptotics of a Steplike Contrast Structure in a Partially Dissipative Stationary System of Equations

A boundary value problem for a system of two ordinary differential equations, one being of the second order, and the other, of the first order, with a small parameter multiplying the derivatives in each equation is considered. The conditions are established under which the problem has a solution involving an internal transition layer near some point within which the solution jumps from a small neighborhood of one root of the corresponding degenerate system to a small neighborhood of its other root. A solution of this type is called a steplike contrast structure (SLCS). An asymptotic approximation of SLCS with respect to the small parameter is constructed and substantiated. It has certain differences from SLCS observed in other singularly perturbed problems. This is concerned primarily with the structure of the solution asymptotics in the transition layer. The constructed asymptotic expansion is substantiated using the asymptotic method of differential inequalities, whose application to the considered problem has a number of qualitative features.

Periodic Boundary Layer Solutions of a Reaction–Diffusion Problem with Singularly Perturbed Boundary Conditions of the Third Kind

We prove the existence and study the stability of time-periodic boundary layer solutions for a two-dimensional reaction–diffusion problem with a small parameter multiplying the parabolic operator for the case of singularly perturbed boundary conditions of the third kind. An asymptotic approximation to such solutions with respect to the small parameter is constructed. Conditions under which these solutions are Lyapunov asymptotically stable, as well as conditions under which such solutions are unstable, are obtained. For the proof, we used results based on applying the asymptotic method of differential inequalities and the Krein–Rutman theorem.

The Existence of a Boundary-Layer Stationary Solution to a Reaction–Diffusion Equation with Singularly Perturbed Neumann Boundary Condition

This paper considers an initial-boundary value problem for a reaction–diffusion equation with a singularly perturbed Neumann boundary condition in a closed, simply connected two-dimensional domain. From a physical point of view, the problem describes processes with an intensive flow through the boundary of a given area. The existence of a stationary solution is proved, its asymptotic is constructed, and the Lyapunov stability conditions for it are established. The asymptotics of the solution are constructed by the classical Vasilieva algorithm using the Lusternik–Vishik method. The existence and stability of the solution are proved using the asymptotic method of differential inequalities.

On a Periodic Inner Layer in the Reaction–Diffusion Problem with a Modular Cubic Source

The article studies a singularly perturbed periodic problem for the parabolic reaction–diffusion equation in the case of a discontinuous source: a nonlinearity describing the reaction (interaction). The case of the existence of an inner transition layer under conditions of an unbalanced and a balanced reaction is considered. An asymptotic approximation is constructed, and the asymptotic Lyapunov stability of periodic solutions in each of the cases is investigated. To prove the existence of a solution and its asymptotic stability, the asymptotic method of differential inequalities is used. The theoretical result is illustrated by an example and numerical calculations.

Singularly Perturbed Stationary Diffusion Model with a Cubic Nonlinearity

We consider a multidimensional singularly perturbed stationary diffusion model with a cubic nonlinearity. For models of this type, a modified asymptotic method of boundary functions, which extends the classical asymptotic analysis methods to the case of multidimensional problems, and the asymptotic method of differential inequalities, which is based on the comparison principle, are used to study the existence of asymptotically Lyapunov stable solutions with internal layers as stationary solutions of the corresponding parabolic problems. Sufficient conditions are established for the existence of such solutions in the form of some conditions on the coefficients of the equation, an asymptotic approximation to the solution of an arbitrary accuracy order with coefficients is constructed in closed form, and the formal constructions are justified. This result can be used for creating efficient numerical algorithms for direct and coefficient inverse problems for stationary equations of the reaction–diffusion–advection type as well as for constructing test examples. Heat and mass transfer problems occurring in chemical industry are pointed out as possible applications of our results.

Asymptotically Stable Stationary Solutions of the Reaction–Diffusion–Advection Equation with Discontinuous Reaction and Advection Terms

We study the Lyapunov asymptotic stability of the stationary solution of the spatially one-dimensional initial–boundary value problem for a nonlinear singularly perturbed differential equation of the reaction–diffusion–advection (RDA) type in the case where the advection and reaction terms undergo a discontinuity of the first kind at some interior point of an interval. Sufficient conditions are derived for the existence of a stable stationary solution with a large gradient near the point of discontinuity. An asymptotic method of differential inequalities is used to prove the existence and stability theorems. The resulting stability conditions can be employed to create mathematical models and develop numerical methods for solving “stiff” problems arising in various applications, for example, when simulating combustion processes and in nonlinear wave theory.

Periodic Solutions with Boundary Layers in the Problem of Modeling the Vertical Transfer of an Anthropogenic Impurity in the Troposphere

The periodic problem that arises in the mathematical modeling of the vertical transfer of an anthropogenic impurity in the lower troposphere is considered for the nonlinear diffusion transfer equation. The model problem in dimensionless variables is classified as a nonlinear singularly perturbed reaction—diffusion—advection problem, which is studied by the methods of asymptotic analysis. Using the method of boundary functions and the asymptotic method of differential inequalities based on the principle of comparison, an asymptotic problem solution of arbitrary-order accuracy is constructed with the further substantiation of constructions and the study of this solution for the Lyapunov asymptotic stability property. The results of this work are illustrated using an example that describes the concentration field of a linear substance sink.

Asymptotic Expansion of the Solution to a Partially Dissipative System of Equations with a Multizone Boundary Layer

An asymptotic expansion with respect to a small parameter is constructed and proved for the solution of the boundary value problem for a singularly perturbed stationary partially dissipative system of equations in the case when one of the equations of the degenerate system has a double root. The multiplicity of this root leads to a multizone boundary layer, so the standard algorithm for constructing an asymptotic expansion of a boundary-layer solution becomes insufficient and requires a substantial modification. The constructed asymptotic expansion is substantiated using the asymptotic method of differential inequalities.

On The Existence and Asymptotic Stability of Periodic Contrast Structures in Quasilinear Reaction-Advection-Diffusion Equations

We consider periodic solutions with internal transition and boundary layers (periodic contrast structures) for a singularly perturbed parabolic equation that is referred to in applications as reaction-advection-diffusion equation. An asymptotic approximation to such solutions is constructed and an existence theorem is proved. An efficient algorithm is developed for constructing an asymptotic approximation to the localization curve of the transition layer. To substantiate the asymptotic thus constructed, we use the asymptotic method of differential inequalities. Moreover, we assert that asymptotic stability of the solution in the sense of Lyapunov occurs.

The Asymptotic Stability of a Stationary Solution with an Internal Transition Layer to a Reaction–Diffusion Problem with a Discontinuous Reactive Term

The problem of the asymptotic stability of a stationary solution with an internal transition layer of a one-dimensional reaction–diffusion equation is considered. What makes this problem peculiar is that it has a discontinuity (of the first kind) of the reactive term (source) at an internal point of the segment on which the problem is stated, making the solutions have large gradients in the narrow transition layer near the interface. The existence, local uniqueness, and asymptotic stability conditions are obtained for the solution with such an internal transition layer. The proof uses the asymptotic method of differential inequalities. The obtained existence and stability conditions of the solution should be taken into account when constructing adequate models that describe phenomena in media with discontinuous characteristics. One can use the results of this work to develop efficient methods for solving differential equations with discontinuous coefficients numerically.

Existence of a solution in the form of a moving front of a reaction-diffusion-advection problem in the case of balanced advection

We consider the initial-boundary value problem for an equation of reaction-diffusion-advection type in the case when the condition of balanced advection is satisfied. We give an algorithm for constructing an asymptotic representation of a solution which has the form of a moving front, obtain the equation of motion for the point of localization of the front, and prove the existence of that solution. The proof uses the asymptotic method of differential inequalities.

Solution of Contrast Structure Type for a Parabolic Reaction–Diffusion Problem in a Medium with Discontinuous Characteristics

We consider a reaction–diffusion-type equation in a two-dimensional domain containing the interface between media with distinct characteristics along which the reactive term has a discontinuity of the first kind. We assume that the interface between the media, as well as the functions describing the reactions, periodically varies in time. We study the existence of a stable periodic solution of a problem with an internal layer. To prove the existence, stability, and local uniqueness of the solution, we use the asymptotic method of differential inequalities, which we generalized to a new class of problems with discontinuous nonlinearities.

On a Singularly Perturbed Problem of the Nonlinear Thermal Conductivity in the Case of Balanced Nonlinearity

On the basis of the modified asymptotic method of boundary functions and the asymptotic method of differential inequalities, the question of the existence of Lyapunov-stable stationary solutions with internal layers of the nonlinear heat equation in the case of nonlinear dependence of the power of thermal sources from temperature is investigated. The main conditions of the existence of such solutions are discussed. We construct an asymptotic approximation of an arbitrary-order accuracy to such solutions and suggest an efficient algorithm for constructing an asymptotic approximation to the localization surface of the transition layer. To justify the constructed formal asymptotics, we use an asymptotic method of differential inequalities. The main complexity is related to the description of the transition surface in whose neighborhood the internal layer is localized. We use a more efficient method for localizing the transition surface, which permits one to develop an approach to a more complicated case of balanced nonlinearity. The results can be used to create a numerical algorithm which uses the asymptotic analyses to construct space-non-uniform meshes while describing internal layer behaviour of the solution. As an illustration, we consider a problem on the plane that allows us to visualize the numerical calculations. Numerical and asymptotic solutions of zero order are compared for different values of the small parameter.

Existence and Stability of the Periodic Solution with an Interior Transitional Layer in the Problem with a Weak Linear Advection

In the paper, we study a singularly perturbed periodic in time problem for the parabolic reaction-advection-diffusion equation with a weak linear advection. The case of the reactive term in the form of a cubic nonlinearity is considered. On the basis of already known results, a more general formulation of the problem is investigated, with weaker sufficient conditions for the existence of a solution with an internal transition layer to be provided than in previous studies. For convenience, the known results are given, which ensure the fulfillment of the existence theorem of the contrast structure. The justification for the existence of a solution with an internal transition layer is based on the use of an asymptotic method of differential inequalities based on the modification of the terms of the constructed asymptotic expansion. Further, sufficient conditions are established to fulfill these requirements, and they have simple and concise formulations in the form of the algebraic equation w ( x 0 ,t ) = 0 and the condition w x ( x 0 ,t ) < 0, which is essentially a condition of simplicity of the root x 0 ( t ) and ensuring the stability of the solution found. The function w is a function of the known functions appearing in the reactive and advective terms of the original problem. The equation w ( x 0 ,t ) = 0 is a problem for finding the zero approximation x 0 ( t ) to determine the localization region of the inner transition layer. In addition, the asymptotic Lyapunov stability of the found periodic solution is investigated, based on the application of the so-called compressible barrier method. The main result of the paper is formulated as a theorem.

Existence and Asymptotic Stability of Periodic Solutions of the Reaction–Diffusion Equations in the Case of a Rapid Reaction

A singularly perturbed periodic in time problem for a parabolic reaction-diffusion equation in a two-dimensional domain is studied. The case of existence of an internal transition layer under the conditions of balanced and unbalanced rapid reaction is considered. An asymptotic expansion of a solution is constructed. To justify the asymptotic expansion thus constructed, the asymptotic method of differential inequalities is used. The Lyapunov asymptotic stability of a periodic solution is investigated.

On one model problem for the reaction–diffusion–advection equation

The asymptotic behavior of the solution with boundary layers in the time-independent mathematical model of reaction–diffusion–advection arising when describing the distribution of greenhouse gases in the surface atmospheric layer is studied. On the basis of the asymptotic method of differential inequalities, the existence of a boundary-layer solution and its asymptotic Lyapunov stability as a steady-state solution of the corresponding parabolic problem is proven. One of the results of this work is the determination of the local domain of the attraction of a boundary-layer solution.

Existence and stability of periodic contrast structures in the reaction–advection–diffusion problem in the case of a balanced nonlinearity

We study a singularly perturbed periodic problem for the parabolic reaction–advection–diffusion equation with small advection. We consider the case in which there exists an internal transition layer under the conditions of balanced nonlinearity. An asymptotic expansion of the solution is constructed. To substantiate this asymptotics, we use the asymptotic method of differential inequalities. The Lyapunov asymptotic stability of the periodic solution is analyzed.