Asymptotics of solutions of spectral problems of hydrodynamics in the neighborhood of angular points

At angular points on the boundary of a domain, we obtain an asymptotic expansion for the eigenfunctions of spectral problems that describe natural oscillations of an ideal liquid that partially fills a cavity in a solid body. We describe cases where the eigenfunctions have singularities at angular points. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 803–811, June, 1998. This work was partially supported by the Ukrainian State Committee on Science and Technology.     

Asymptotics of solutions of discontinuous singularly perturbed boundary-value problems

We construct an asymptotic expansion of a boundary-value problem for a singularly perturbed system of differential equations with the right-hand side discontinuous at certain surface. Odessa University, Odessa. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 6, pp. 861–864, June, 1999.     

Concentration of solutions for some singularly perturbed mixed problems: Asymptotics of minimal energy solutions

In this paper we carry on the study of asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions, started in the first paper [J. Garcia Azorero, A. Malchiodi, L. Montoro, I. Peral, Concentration of solutions for some singularly perturbed mixed problems: Existence results, Arch. Ration. Mech. Anal., in press]. Here we are mainly interested in the analysis of the location and shape of least energy solutions when the singular perturbation parameter tends to zero. We show that in many cases they coincide with the new solutions produced in [J. Garcia Azorero, A. Malchiodi, L. Montoro, I. Peral, Concentration of solutions for some singularly perturbed mixed problems: Existence results, Arch. Ration. Mech. Anal., in press].     

Asymptotics of shell eigenvalue problems

The asymptotic behaviour of the smallest eigenvalue in linear shell problems is studied, as the thickness parameter tends to zero. When pure bending is not inhibited, such a behaviour has been essentially studied by Sanchez-Palencia. When pure bending is inhibited, the situation is more complex and some information can be obtained by using the Real Interpolation Theory. In order to cover the widest range of mid-surface geometry and boundary conditions, an abstract approach has been followed. A result concerning the ratio between the bending and the total elastic energy is also announced. To cite this article: L. Beirão da Veiga, C. Lovadina, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Asymptotics of solutions of boundary value problems in periodically perforated domains with small holes

We consider boundary value problems for operators Δ and Δ² in periodically perforated domains with homogeneous Dirichlet conditions on the boundaries of the holes. The period of perforation and the “size” of the hole with respect to the period of perforation are regarded as two small parameters. We study asymptotic behavior of solutions, eigenvalues, and eigenfunctions for boundary value problems, under various assumptions on the relation between the two parameters. Bibliography: 13 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 17, pp. 153–208, 1994.     

Asymptotics of solutions of a cauchy problem

We consider the Cauchy problem (x)ⁿ=at^r + bx + f(t, x, x), x(0)=0. We prove the existence of continuously differentiable solutions and study their asymptotic behavior for t + 0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 187–193, February, 1991.     

Asymptotics of solutions of matrix integro-differential equations

We obtain an asymptotic expansion of the solution of a system of first-order integro-differential equations with the influence of the roots of the characteristic equation taken into account. A similar expansion is established for a system of Volterra integral equations.     

Asymptotics of solutions of nonlinear parabolic equations

This paper is concerned with the large time behaviour of solutions to the Cauchy problem of the following nonlinear parabolic equations:

     

Asymptotics of solutions of infinite-dimensional homogeneous dynamical systems

In this paper we study the connection between the uniform asymptotic stability and the power-law or exponential asymptotics of the solutions of infinite-dimensional systems (differential equations in Banach spaces, functional differential equations, and completely solvable multidimensional differential equations). Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 115–126, January, 1998.     

Asymptotics of solutions of equations with higher degenerations

We study the asymptotics of homogeneous differential equations with degeneration of the cusp type in the principal symbol. We construct the asymptotic expansion of solutions for the case in which the principal symbol of the operator has simple singularities.     

Asymptotics of solutions of infinite-dimensional homogeneous dynamical systems

In this paper we study the connection between the uniform asymptotic stability and the power-law or exponential asymptotics of the solutions of infinite-dimensional systems (differential equations in Banach spaces, functional differential equations, and completely solvable multidimensional differential equations).     

Asymptotics of solutions of difference equations with delays

We consider a linear scalar difference equation with several variable delays and constant coefficients. The coefficients and maximum admissible values of delays are supposed to be the set of parameters that define a family of equations of the investigated class. We obtain effective necessary and sufficient conditions of the uniform and exponential stability of solutions to all equations of the family, as well as the conditions of the sign-definiteness and monotonicity of stable solutions.     

Asymptotics of solutions to the generalized Ostrovsky equation

We consider the Cauchy problem for the generalized Ostrovsky equation

_utx=u+_(f(u))xx,$u_{tx}=u+{(f(u))}_{xx},$

where f(u)=|u|^ρ−1u$f(u)={|u|}^{\rho -1}u$ if ρ   is not an integer and f(u)=u^ρ$f(u)=u^{\rho }$ if ρ   is an integer. We obtain the L^∞${\mathbf{L}}^{\infty }$ time decay estimates and the large time asymptotics of small solutions under suitable conditions on the initial data and the order of the nonlinearity.