Varga Kalantarov - Academia.edu (original) (raw)

Papers by Varga Kalantarov

In the article the singular Volterra's integral equation of the second kind is considered, which ... more In the article the singular Volterra's integral equation of the second kind is considered, which because of the «incompressible» of the kernel classical methods of solutions are not applicable. It is shown that the corresponding homogeneous equation at 1   has a continuous spectrum, and the multiplicity of the characteristic numbers grows with increasing.  By the Carleman-Vekua method the equation is reduced to Abelequation. The eigenfunctions of the equation are found in an explicit form.

Mathematical Notes, Oct 1, 2022

Математические заметки, 2022

arXiv (Cornell University), Dec 12, 2019

We study the problem of global exponential stabilization of original Burgers' equations and the B... more We study the problem of global exponential stabilization of original Burgers' equations and the Burgers' equation with nonlocal nonlinearities by controllers depending on finitely many parameters. It is shown that solutions of the controlled equations are steering a concrete solution of the non-controlled system as t → ∞ with an exponential rate.

$Research paper thumbnail of Qualitative properties of solutions for nonlinear Schr\"odinger equations with nonlinear boundary conditions on the half-line$

arXiv (Cornell University), Aug 5, 2015

In this paper, we study the interaction between a nonlinear focusing Robin type boundary source, ... more In this paper, we study the interaction between a nonlinear focusing Robin type boundary source, a nonlinear defocusing interior source, and a weak damping term for nonlinear Schrödinger equations posed on the infinite half line. We construct solutions with negative initial energy satisfying a certain set of conditions which blow-up in finite time in the H 1-sense. We obtain a sufficient condition relating the powers of nonlinearities present in the model which allows construction of blow-up solutions. In addition to the blow-up property, we also discuss the stabilization property and the critical exponent for this model.

Mathematical and Computer Modelling, May 1, 2003

show that, under some conditions, all solutions of the inverse problem for a class of first-order... more show that, under some conditions, all solutions of the inverse problem for a class of first-order and second-order linear differential operator equations in a Hilbert space tend to zero when t-+ co. Applications to inverse problems for the heat equation and damped wave equations are given.

Turkish Journal of Mathematics, Jun 1, 1997

arXiv (Cornell University), Jun 15, 2015

We show that blow up of solutions with arbitrary positive initial energy of the Cauchy problem fo... more We show that blow up of solutions with arbitrary positive initial energy of the Cauchy problem for the abstract wacve eqation of the form P u tt + Au = F (u) (*) in a Hilbert space, where P, A are positive linear operators and F (•) is a continuously differentiable gradient operator can be obtained from the result of H.A. Levine on the growth of solutions of the Cauchy problem for (*). This result is applied to the study of inital boundary value problems for nonlinear Klein-Gordon equations, generalized Boussinesq equations and nonlinear plate equations. A result on blow up of solutions with positive initial energy of the initial boundary value problem for wave equation under nonlinear boundary condition is also obtained.

Communications on Pure and Applied Analysis, 2018

We consider the Cauchy problem for nonlinear abstract wave equations in a Hilbert space. Our main... more We consider the Cauchy problem for nonlinear abstract wave equations in a Hilbert space. Our main goal is to show that this problem has solutions with arbitrary positive initial energy that blow up in a finite time. The main theorem is proved by employing a result on growth of solutions of abstract nonlinear wave equation and the concavity method. A number of examples of nonlinear wave equations are given. A result on blow up of solutions with arbitrary positive initial energy to the initial boundary value problem for the wave equation under nonlinear boundary conditions is also obtained.

Physica D: Nonlinear Phenomena, 2018

The paper is devoted to the study of asymptotic behavior as t → +∞ of solutions of initial bounda... more The paper is devoted to the study of asymptotic behavior as t → +∞ of solutions of initial boundary value problem for structurally damped semi-linear wave equation ∂ 2 t u(x, t) − ∆u(x, t) + γ(−∆) θ ∂ t u(x, t) + f (u) = g(x), θ ∈ (0, 1), x ∈ Ω, t > 0 under homogeneous Dirichlet's boundary condition in a bounded domain Ω ⊂ R 3. We proved that the asymptotic behavior as t → ∞ of solutions of this problem is completely determined by dynamics of the first N Fourier modes, when N is large enough. We also proved that the semigroup generated by this problem when θ ∈ (1 2 , 1) possesses an exponential attractor.

Applicable Analysis, 2014

We show that solutions of the initial boundary value problem for the coupled system of Kuramoto-S... more We show that solutions of the initial boundary value problem for the coupled system of Kuramoto-Sivashinsky and Ginzburg-Landau (KS-GL) equations continuously depend on parameters of the system, and under some restrictions on parameters all solutions of initial boundary value problem for KS-GL equations tend to zero as t → ∞ with an exponential rate.

Uspekhi Matematicheskikh Nauk, 2013

Работа посвящена изучению глобальных аттракторов абстрактных полулинейных параболических уравнени... more Работа посвящена изучению глобальных аттракторов абстрактных полулинейных параболических уравнений и их вложений в конечномерные многообразия. Как известно, достаточным условием существования гладких (как минимум, гладкости C 1) инерциальных многообразий конечной размерности, содержащих глобальный аттрактор, является так называемое условие щели в спектре для соответствующего линейного оператора. Также имеется ряд примеров, показывающих, что если щель в спектре отсутствует, то C 1-гладкого инерциального многообразия может и не быть. С другой стороны, так как аттрактор обычно имеет конечную фрактальную размерность, то, согласно теореме Мане, он проектируется взаимно однозначно и Гёльдер-гомеоморфно в конечномерную плоскость общего положения, если ее размерность достаточно велика. В настоящей работе показано, что при отсутствии щели в спектре существуют аттракторы, которые нельзя вложить ни в какое липшицево или даже лог-липшицево конечномерное многообразие. Таким образом, если щель в спектре отсутствует, то в общем случае нельзя ожидать липшицевости или лог-липшицевости обратной проекции Мане аттрактора. Кроме того, в классе нелинейностей конечной гладкости построены примеры аттракторов с конечной хаусдорфовой и бесконечной фрактальной размерностью. Библиография: 35 названий.

arXiv (Cornell University), Jun 15, 2020

Slightly compressible Brinkman-Forchheimer equations in a bounded 3D domain with Dirichlet bounda... more Slightly compressible Brinkman-Forchheimer equations in a bounded 3D domain with Dirichlet boundary conditions are considered. These equations model fluids motion in porous media. The dissipativity of these equations in higher order energy spaces is obtained and regularity and smoothing properties of the solutions are studied. In addition, the existence of a global and an exponential attractors for these equations in a natural phase space is verified.

Journal of Differential Equations

We study the properties of linear and non-linear determining functionals for dissipative dynamica... more We study the properties of linear and non-linear determining functionals for dissipative dynamical systems generated by PDEs. The main attention is payed to the lower bounds for the number of such functionals. In contradiction to the common paradigm, it is shown that the optimal number of determining functionals (the so-called determining dimension) is strongly related to the proper dimension of the set of equilibria of the considered dynamical system rather than to the dimensions of the global attractors and the complexity of the dynamics on it. In particular, in the generic case where the set of equilibria is finite, the determining dimension equals to one (in complete agreement with the Takens delayed embedding theorem) no matter how complex the underlying dynamics is. The obtained results are illustrated by a number of explicit examples. Contents 16 6. Open problems 21 References 23

Vietnam Journal of Mathematics

The initial boundary value problem for a nonlinear system of equations modeling the chevron patte... more The initial boundary value problem for a nonlinear system of equations modeling the chevron patterns is studied in one and two spatial dimensions. The existence of an exponential attractor and the stabilization of the zero steady state solution through application of a finite-dimensional feedback control is proved in two spatial dimensions. The stabilization of an arbitrary fixed solution is shown in one spatial dimension along with relevant numerical results.

Complex Variables and Elliptic Equations, 2018

Abstract The paper is devoted to the study of asymptotic behavior as of solutions of initial boun... more Abstract The paper is devoted to the study of asymptotic behavior as of solutions of initial boundary value problem for strongly damped nonlinear wave equation and strongly damped Kirchhoff type equation under homogeneous Dirichlet’s boundary conditions. We proved that the asymptotic behavior as of solutions of these problem is completely determined by dynamics of finitely many functionals.

A strongly damped wave equation including the displacement depending nonlinear damping term and n... more A strongly damped wave equation including the displacement depending nonlinear damping term and nonlinear interaction function is considered. The main aim of the note is to show that under the standard dissipativity restrictions on the nonlinearities involved the initial boundary value problem for the considered equation is globally well-posed in the class of sufficiently regular solutions and the semigroup generated by the problem possesses a global attractor in the corresponding phase space. These results are obtained for the nonlinearities of an arbitrary polynomial growth and without the assumption that the considered problem has a global Lyapunov function.

We show that the Helmholtz equation describing the propagation of transverse electric waves in a ... more We show that the Helmholtz equation describing the propagation of transverse electric waves in a Kerr slab with a complex linear permittivity and a complex Kerr coefficient admits blow-up solutions provided that the real part of the Kerr coefficient is negative, i.e., the slab is defocusing. This result applies to homogeneous as well as inhomogeneous Kerr slabs whose linear permittivity and Kerr coefficient are continuous functions of the transverse coordinate. For an inhomogeneous Kerr slab, blow-up solutions exist if the real part of Kerr coefficient is bounded above by a negative number.