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The existence of stationary solutions of singularly perturbed systems of reaction–diffusion–advection equations is studied in the case of fast and slow reaction–diffusion–advection equations with nonlinearities containing the gradient of the squared sought function (KPZ nonlinearities). The asymptotic method of differential inequalities is used to prove the existence theorems. The boundary layer asymptotics of solutions are constructed in the case of Neumann and Dirichlet boundary conditions. The case of quasimonotone sources and systems without the quasimonotonicity requirement is also considered. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A new method is developed for studying unstable contrast structures (solutions with an internal transition layer), based on the construction of sufficiently accurate unordered upper and lower solutions and the application of a corollary of the Krein–Rutman theorem. Conditions are formulated for the existence of Lyapunov-unstable one-dimensional step-type contrast structures as stationary solutions of singularly perturbed parabolic reaction–diffusion equations with a discontinuous right-hand side. It is shown that the results obtained can be extended to other singularly perturbed one-dimensional reaction–diffusion–advection problems with discontinuous nonlinearities. [ABSTRACT FROM AUTHOR]

In this paper, we study a new class of time-periodic solutions with interior transition layer of reaction-advection-diffusion equations in the case of a fast reaction and a small diffusion. We consider the case of discontinuous sources (i.e., the nonlinearity describing the interaction and reaction) for a certain value of the unknown function that arise in a number of relevant applications. An existence theorem is proved, asymptotic approximations are constructed, and the asymptotic Lyapunov stability of such solutions as solutions of the corresponding initial-boundary-value problems is established. [ABSTRACT FROM AUTHOR]

We study the problem of the existence and asymptotic stability of a stationary solution of an initial boundary value problem for the reaction–diffusion–advection equation assuming that the reaction and advection terms are comparable in size and have a jump along a smooth curve located inside the studied domain. The problem solution has a large gradient in a neighborhood of this curve. We prove theorems on the existence, asymptotic uniqueness, and Lyapunov asymptotic stability for such solutions using the method of upper and lower solutions. To obtain the upper and lower solutions, we use the asymptotic method of differential inequalities that consists in constructing them as modified asymptotic approximations in a small parameter of solutions of these problems. We construct the asymptotic approximation of a solution using a modified Vasil'eva method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We prove the existence and the Lyapunov asymptotic stability of a stationary boundary layer solution of the initial–boundary value problem for a two-dimensional singularly perturbed reaction–diffusion–advection equation. We construct an asymptotic approximation to this solution using the boundary function method. The proofs are based on the applicability of the asymptotic method of differential inequalities. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A singularly perturbed initial boundary value problem is considered for a parabolic equation that is known in application as the reaction-diffusion-advection equation. An asymptotic expansion of solutions with a moving front is constructed. This asymptotics is proved by the method of differential inequalities, which is based on well-known comparison theorems and develops the ideas of formal asymptotics for constructing upper and lower solutions in singularly perturbed problems with internal and boundary layers. [ABSTRACT FROM AUTHOR]

The initial boundary value problem for a nonlinear singularly perturbed integro-parabolic equation is examined. An asymptotic expansion of the solution to this problem containing the temporal, spatial and corner boundary layers is constructed. The existence and local uniqueness of the solution is justified by using the asymptotic method of differential inequalities. [ABSTRACT FROM AUTHOR]

We consider a nonlinear boundary value problem for an equation of reaction-diffusion-advection type whose solutions have internal transition layers (contrast structures). We prove the existence of such solutions, construct their asymptotics, and estimate its accuracy. The analysis is carried out with the use of the asymptotic method of differential inequalities and permits one to prove the Lyapunov stability of the contrast structures. [ABSTRACT FROM AUTHOR]

We consider initial-boundary value problems for a class of singularly perturbed nonlinear integro-differential equations. In applications, they are referred to as nonlocal reactionadvection-diffusion equations, and their solutions have moving interior transition layers (fronts). We construct the asymptotics of such solutions with respect to a small parameter and estimate the accuracy of the asymptotics. To justify the asymptotics, we use the asymptotic differential inequality method. [ABSTRACT FROM AUTHOR]

The article studies the method of differential inequalities for step-like contrast structures in singularly perturbed integro-differential equations in the spatially two-dimensional case. The asymptotic expansions were justified with the use of the asymptotic method of differential inequalities, extended to a new class of problems.