Internal Transition Layer Research Articles

Single transition layer in mass-conserving reaction-diffusion systems with bistable nonlinearity

Abstract Mass-conserving reaction-diffusion systems with bistable nonlinearity are useful models for studying cell polarity formation, which is a key process in cell division and differentiation. We rigorously show the existence and stability of stationary solutions with a single internal transition layer in such reaction-diffusion systems under general assumptions by the singular perturbation theory. Moreover, we present a meaningful model for understanding the existence of an unstable transition layer solution; our numerical simulations show that the unstable solution is a separatrix of the dynamics of the model.

Internal Transition Layer Structure in the Reaction–Diffusion Problem for the Case of a Balanced Reaction with a Weak Discontinuity

Internal Transition Layer Structure in the Reaction–Diffusion Problem for the Case of a Balanced Reaction with a Weak Discontinuity

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.

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

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

Stability of a stationary solution of a system of activator–inhibitor-type equations with a double-scale internal transition layer

Stability of a stationary solution of a system of activator–inhibitor-type equations with a double-scale internal transition layer

ASYMPTOTICS OF THE SOLUTION TO A PIECEWISE-SMOOTH QUASILINEAR SECOND-ORDER DIFFERENTIAL EQUATION

<abstract> We investigate a singularly perturbed boundary value problem for a piecewise-smooth second-order quasilinear differential equation in the case when the discontinuous curve which separates the domain is monotone. Applying the boundary layer function method, the asymptotic expansion of a solution with internal layer appearing in the neighborhoods of some point on the monotone curve and the point itself is constructed. For sufficiently small parameter values, using the matching method, the existence of a smooth solution with an internal transition layer in the neighborhood of a point of the monotone curve is proved. A simple example is given to show the effectiveness of our method. </abstract>

Multidimensional thermal structures in the singularly perturbed stationary models of heat and mass transfer with a nonlinear thermal diffusion coefficient

A new approach to the study of multidimensional singularly perturbed problems of nonlinear heat conduction is proposed, based on the further development and use of asymptotic analysis methods. We study the question of the existence of classical Lyapunov stable stationary solutions with boundary and internal transition layers (stationary thermal structures) of the nonlinear heat transfer equation. We suggest the efficient algorithm for constructing an asymptotic approximation to the localization surface of the transition layer. To justify the constructed formal asymptotics, we use the principle of comparison.We consider the application of the results of asymptotic analysis to solving inverse problem of reconstructing the temperature dependence of the thermal conductivity coefficient from a known position of the internal layer of thermal structure.

On the internal transition layer to some inhomogeneous semilinear problems: Interface location

We are concerned with the location of the internal transition layer for some classes of solutions to an inhomogeneous one-dimensional elliptic problem with Neumann boundary conditions. We generalize and extend some known results with a variational method based on the Young's inequality and the co-area formula. As an application, we establish the location of the internal transition layer to the problem posed on some symmetric Riemannian manifolds.

A multi-layered SiC coating to protect graphite spheres from high temperature oxidation in static air

A multi-layered SiC coating was prepared on graphite spheres to improve its antioxidant performance. The oxidation behaviour of the coating prepared on graphite sphere in static air was studied systematically for the first time. As an antioxidant layer, the dense block layer is effective in protecting the internal transition layer and refined grain layer from oxidation in static air at 1773 K. An amorphous SiO2 layer developing on the surface acts as a protective shell, while some carbon particles developed at the interface of oxide layer and SiC coating. As for the oxidation process of the SiC coated graphite sphere, it is comprised of three stages. Stage I (0−6 h) is characterized by a rapid weight gain, which follows a linear model, and the passive oxidation of SiC occurs in this stage. Stage II (6−50 h) is characterized by a slow weight gain, which follows the parabolic oxidation model. Stage III (50−200 h) is characterized by an invariable weight gain of merely 0.52%, which is accompanied with the interface reaction of SiC and the self-healing of oxide layer.

Profile Analysis of Ferrite–Garnet Films by Glow-Discharge Optical Emission Spectroscopy

The results of experiments on the application of the method of glow discharge optical emission spectroscopy for the profile analysis of vacuum-deposited single- and double-layer films of multicomponent ferrite–garnets of various compositions before and after crystallization annealing have been presented. The presence of internal and external transition layers has been revealed and the modification of their elemental composition during crystallization and at different synthesis sequences of individual components in ferrite–garnet heterostructures have been studied.

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.

Профильный анализ пленок ферритов-гранатов методом оптической эмиссионной спектроскопии тлеющего разряда

Annotation.The paper presents the results of experiments on the application of the method of optical emission spectroscopy of a glow discharge for the profile analysis of vacuum-deposited single and two-layer films of multicomponent garnet ferrites of various compositions before and after crystallization annealing. The presence of internal and external transition layers was revealed, and the modification of their elemental composition during crystallization, as well as the different sequence of synthesis of individual components in ferrite-garnet heterostructures, was investigated.

A note on interface formation in singularly perturbed elliptic problems

In this note, we prove that the equal-area condition is necessary for the formation of internal transition layer by the solutions of a general singularly perturbed elliptic problem. Our result is independent of boundary conditions, it improves previous references in the literature and can be applied to Allen-Cahn, Sine-Gordon and Fisher-KPP equations, for example.

Asymptotic solution of the inverse problem for restoring the modular type source in Burgers’ equation with modular advection

Abstract For a singularly perturbed Burgers’ type equation with modular advection that has a time-periodic solution with an internal transition layer, asymptotic analysis is applied to solve the inverse problem for restoring the function of the source using known information about the observed solution of a direct problem at a given time interval (period).

Existence and stability of periodic contrast structure in reaction-advection-diffusion equation with discontinuous reactive and convective terms

In this project, we study the periodic Dirichlet boundary value problem for a singularly perturbed reaction-advection-diffusion equation on the segment in case of discontinuous reactive and convective terms. Applying the boundary function method, we construct the asymptotic approximation of the periodic solution with internal transition layer located in the vicinity of a curve of discontinuity of the mentioned terms. For the problem here we prove the existence of the periodic solution, estimate the accuracy of the asymptotical approximation and investigate the stability of the periodic solution as solutions of the corresponding initial boundary value problems for the reaction-advection-diffusion equation.

Asymptotic Solution of Coefficient Inverse Problems for Burgers-Type Equations

For a singularly perturbed reaction–diffusion–advection equation, called in applications a Burgers-type equation and having a time-periodic solution with an internal transition layer, asymptotic analysis is used to solve some inverse problems of reconstructing model parameters (determining the linear amplification factor and boundary conditions) from known information about the observed solution of the direct problem in a given time interval (period). Numerical experiments demonstrating the efficiency of the approach proposed are conducted.

Asymptotics of the Solution of a Singularly Perturbed Second-Order Delay Differential Equation

We consider a singularly perturbed boundary value problem for a second-order ordinary differential equation with nonlinear right-hand side containing functions of delayed argument. We prove the existence of a solution with a transition layer that has a more sophisticated structure than the ones studied before and construct a uniform asymptotic approximation to this solution with respect to a small parameter. Vasil’eva’s method is used when constructing the asymptotic approximation, while the existence theorem is proved by combining the matching method and the asymptotic differential inequality method. Conditions for the existence of a solution with monotone internal transition and boundary layers are stated. An example illustrating the class of problems studied here is given.

Existence and Asymptotic Stability of Periodic Two-Dimensional Contrast Structures in the Problem with Weak Linear Advection

We consider the boundary-value singularly perturbed time-periodic problem for the parabolic reaction-advection-diffusion equation in the case of a weak linear advection in a two-dimensional domain. The main result of the present paper is the justification, under certain sufficient assumptions, of the existence of a periodic solution with internal transition layer near some closed curve and the study of the Lyapunov asymptotic stability of such a solution. For this purpose, an asymptotic expansion of the solution is constructed; the justification of the existence of the solution with the constructed asymptotics is carried out by using the method of differential inequalities. The proof of Lyapunov asymptotic stability is based on the application of the so-called method of contraction barriers.

Multidimensional singularly perturbed reaction-diffusion-advection problems with a balanced nonlinearity and their applications in the theory of nonlinear heat conductivity

The stationary Lyapunov-stable solutions with internal transition layers (contrast structures) of multidimensional singularly perturbed reaction-diffusion-advection problems are investigated with the use of the modern methods of asymptotic analysis. On the basis of the modified boundary-function method, the asymptotic approximations of such solutions of an arbitrary order of accuracy in the case of a balanced nonlinearity are obtained. We propose a justified and effective algorithm that allows us to define and describe the localization region of the internal layer of contrast structure. We use this result to describe the thermal structures in the homogeneous nonlinear dissipative media.

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

A two-dimensional reaction–diffusion equation in a medium with discontinuous characteristics is considered; the existence, local uniqueness, and asymptotic stability of its stationary solution, which has a large gradient at the interface, is proved. This paper continues the authors’ works concerning the existence and stability of solutions with internal transition layers of boundary value problems with discontinuous terms to multidimensional problems. The proof of the existence and stability of a solution is based on the method of upper and lower solutions. The methods of analysis proposed in this paper can be generalized to equations of arbitrary dimension of the spatial variables, as well as to more complex problems, e.g., problems for systems of equations. The results of this work can be used to develop numerical algorithms for solving stiff problems with discontinuous coefficients.