Asymptotics of solutions of partial differential equations with higher degenerations and the Laplace equation on a manifold with a cuspidal singularity

We study the asymptotics of solutions of partial differential equations with higher degenerations. Such equations arise, for example, when studying solutions of elliptic equations on manifolds with cuspidal singular points. We construct the asymptotics of a solution of the Laplace equation defined on a manifold with a cuspidal singularity of order k.     

Exact Asymptotics of the Density of the Transition Probability for Discontinuous Markov Processes

In this paper we consider some Kolmogorov–Feller equations with a small parameter h. We present a method for constructing the exact (exponential) asymptotics of the fundamental solution of these equations for finite time intervals uniformly with respect to h. This means that we construct an asymptotics of the density of the transition probability for discontinuous Markov processes. We justify the asymptotic solutions constructed. We also present an algorithm for constructing all terms of the asymptotics of the logarithmic limit (logarithmic asymptotics) of the fundamental solution as t → +0 uniformly with respect to h. We write formulas of the asymptotics of the logarithmic limit for some special cases as t → +0. The method presented in this paper also allows us to construct exact asymptotics of solutions of initial–boundary value problems that are of probability meaning.     

Computation of the asymptotics of solutions for equations with polynomial degeneration of the coefficients

We study linear differential equations with holomorphic coefficients. We establish the reducibility of such equations to equations with degeneration in the principal symbol. For the case of cuspidal degeneration, we show that the solutions of such equations are resurgent whenever so are their right-hand sides. We also refine earlier-obtained asymptotics of solutions for some equations of this type.     

On the asymptotics of some systems of difference equations

We prove an inclusion theorem regarding a system of difference equations and apply it in getting the asymptotics of some solutions of some concrete difference equations.     

Asymptotic analysis for the eikonal equation with the dynamical boundary conditions

We study the dynamical boundary value problem for Hamilton‐Jacobi equations of the eikonal type with a small parameter. We establish two results concerning the asymptotic behavior of solutions of the Hamilton‐Jacobi equations: one concerns with the convergence of solutions as the parameter goes to zero and the other with the large‐time asymptotics of solutions of the limit equation.     

Asymptotics of Eigenvalues of Differential Operator With Alternating Weight Function

We study a differential operator of the sixth order with an alternating weight function. The potential of the operator has a first-order discontinuity at some point of the segment, where the operator is being considered. The boundary conditions are separated. We study the asymptotics of solutions to the corresponding differential equations and the asymptotics of eigenvalues of the considered differential operator.     

On behavior at infinity of the solutions to equations with dominating mixed derivative

We prove a theorem on the polynomial asymptotics at infinity for the solutions to differential equations with dominating mixed derivative with constant coefficients.     

Nonresonance case for differential equations with degeneration

We study what form the asymptotics of solutions of degenerate elliptic equations have in the nonresonance case under the condition that the asymptotics of the right-hand side have the form corresponding to the class of asymptotics of the nonresonance case, that is, the case in which the singularities of the right-hand side do not coincide with the zeros of the left-hand side. We show that, in the resonance case, the problem is no longer closed in this class.     

Repeated quantization method and its applications to the construction of asymptotics of solutions of equations with degeneration

We study the asymptotics of solutions of homogeneous nth-order differential equations with a cusp degeneration for the case in which the principal symbol has multiple roots. We describe a new method for constructing the asymptotics, which we call the repeated quantization method. Examples of application of the method are given.     

On the asymptotics of some difference equations

We prove two inclusion theorems regarding a difference equation and apply them in getting the asymptotics of positive solutions of some concrete difference equations.     

Dispersive estimates for 1D discrete Schrödinger and Klein–Gordon equations

We derive the long-time asymptotics for solutions of the discrete 1D Schrödinger and Klein–Gordon equations.     

Construction of Asymptotics of Solutions of Differential Equations with Cusp-Type Degeneration in the Coefficients in the Case of Multiple Roots of the Highest-Order Symbol

The asymptotics of linear differential equations with cusp-type degeneration are studied. The problem of constructing asymptotics at infinity for equations with holomorphic coefficients can be reduced to that problem. The main result is the construction of asymptotics of solutions of such equations in the case of multiple roots of the highest-order symbol under certain additional conditions on the lower-order symbol of the differential operator.     

On dispersive properties of discrete 2D Schrödinger and Klein-Gordon equations

We derive the long-time asymptotics for solutions of the discrete 2D Schrödinger and Klein-Gordon equations.     

Asymptotic analysis of linear differential equations with a large parameter: the resonance case

We construct completely justified asymptotics for solutions to some classes of linear ordinary differential equations with slow and fast coefficients in presence of resonance conditions.     

Mellin and Green symbols for boundary value problems on manifolds with edges

We introduce the algebra of smoothing Mellin and Green symbols in a pseudodifferential calculus for manifolds with edges. In addition, we define scales of weighted Sobolev spaces with asymptotics based on the Mellin transform and analyze the mapping properties of the operators on these spaces. This will allow us to obtain complete information on the regularity and asymptotics of solutions to elliptic equations on these spaces.     

Asymptotics of solutions of inhomogeneous equations with higher-order degeneration

We study inhomogeneous differential equations with higher-order degeneration in the coefficients in the resonance-free case and construct the asymptotics of their solutions.     

Boundary-contact problems for domains with edge singularities

We study boundary-contact problems for elliptic equations (and systems) with interfaces that have edge singularities. Such problems represent continuous operators between weighted edge spaces and subspaces with asymptotics. Ellipticity is formulated in terms of a principal symbolic hierarchy, containing interior, transmission, and edge symbols. We construct parametrices, show regularity with asymptotics of solutions in weighted edge spaces and illustrate the results by boundary-contact problems for the Laplacian with jumping coefficients.     

Problem of Conjugation of Solutions of the Lame Wave Equation in Domains with Piecewise-Smooth Boundaries

We study the problem of conjugation of solutions of the Lame wave equation in domains containing singular lines (sets of angular points) and conic points. We show that solutions of the Lame wave equation have power-type singularities near nonsmoothnesses of boundary surfaces and determine their asymptotics. Taking these asymptotics into account and using the introduced simple-layer, double-layer, and volume elastic retarded potentials, we reduce the problem to a system of functional equations and formulate conditions for its solvability.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 32–46, January, 2005.     

A simple global model for the general circulation of the atmosphere

In this article we present a new method of Rossby asymptotics for the equations of the atmosphere similar to the geostrophic asymptotics. We depart from the classical geostrophics (see J. G. Charney [5] and our previous article [29]) by considering an asymptotics valid for the whole atmosphere, not only in midlatitude regions, and by taking into account the spherical form of the earth. We obtain in this way a very simple global circulation model of the atmosphere for which the equations of motion for wind and temperature are linear evolution equations similar to the linear Stokes equations. Furthermore, the solutions are independent of longitude, and winds travel exactly to the east or to the west. In this mathematically oriented article, we do not discuss the physical significance of the model that we derive except for observing that this picture coincides in general terms with the classically averaged data obtained by experimental measurements. We also note that different global geostrophic asymptotics (called planetary geostrophic asymptotics) are considered elsewhere in the literature, for example, in J. Pedlosky [33]. In a less mathematically rigorous way, H. Lamb [21] observed that the only globally valid geostrophic flow that maintains a slow time scale is zonally symmetric (see the comments in N. Phillips [35] and an explicit derivation in H. Jeffreys [19]). © 1997 John Wiley & Sons, Inc.     

On the solution of boundary-value problems of Kolmogorov-Petrovskii-Piskunov type

We describe a number of properties of solutions of boundary-value problems for nonlinear ordinary differential equations of the same type as those studied by Kolmogorov, Petrovskii, and Piskunov in their well-known paper on waves described by the parabolic equation. We construct and justify the asymptotics of such solutions for large values of the modulus of the independent variable.