Solvability Theorems Research Articles

Global solvability of initial boundary-value problems for nonlinear analogs of the Boussinesq equation

The solvability of the natural (first, second, and mixed) initial boundary-value problems for nonlinear analogs of the Boussinesq equation is studied. Uniqueness theorems for regular solutions and global solvability theorems are proved.

Solvability of physically and geometrically nonlinear problem of the theory of sandwich plates with transversally-soft core

The paper presents a generalized statement of geometrically and physically nonlinear problem of the equilibrium of sandwich plate with transversally-soft core. Generalized statement is formulated as a problem of finding a saddle point of a functional. We investigate the properties of this functional. These properties allow to prove a theorem of solvability of variational problem under consideration.

Global Existence and Long-Term Behavior of a Nonlinear Schrödinger-Type Equation

We study a global mixed problem for the nonlinear Schrodinger equation with a nonlinear term in which the coefficient is a generalized function. A global solvability theorem for the analyzed problem is proved by using the general solvability theorem from [K. N. Soltanov, Nonlin. Anal.: Theory, Meth., Appl., 72, No. 1 (2010)]. We also investigate the behavior of the solution of the problem under consideration.

Solvability theorems for an inverse nonself-adjoint Sturm-Liouville problem with nonseparated boundary conditions

We prove theorems on the solvability of the inverse Sturm-Liouville problem with nonseparated conditions by two spectra and one eigenvalue and theorems on the unique solvability by two spectra and three eigenvalues. We find exact and approximate solutions of the inverse problems. Related examples and counterexample are given.

V.A. Steklov’s work on equations of mathematical physics and development of his results in this field

This paper is an extended account of the author’s talk at the International Conference “Contemporary Problems of Mathematics, Mechanics, and Mathematical Physics” dedicated to the 150th anniversary of Vladimir Andreevich Steklov. Steklov’s main studies on the solvability of boundary value problems for equations of mathematical physics are briefly described, and the further development of this field of research is surveyed. The main attention is focused on the statements of the Dirichlet problem and the conditions on the domain and given functions under which solvability theorems are valid.

Microlocal Cauchy problem for distribution solutions to systems with regular singularities

For systems of analytic linear differential equation with regular singularities in the sense of Kashiwara–Oshima, the microlocal Cauchy problem for distribution solutions in the framework of algebraic analysis is formulated. Moreover, under a hyperbolicity condition the solvability theorem is obtained.

Simultaneous determination of time-dependent coefficients in the heat equation

In this paper, the determination of time-dependent leading and lower-order thermal coefficients is investigated. We consider the inverse and ill-posed nonlinear problems of simultaneous identification of a couple of these coefficients in the one-dimensional heat equation from Cauchy boundary data. Unique solvability theorems of these inverse problems are supplied and, in one new case where they were not previously provided, are rigorously proved. However, since the problems are still ill-posed the solution needs to be regularized. Therefore, in order to obtain a stable solution, a regularized nonlinear least-squares objective function is minimized in order to retrieve the unknown coefficients. The stability of numerical results is investigated for several test examples with respect to different noise levels and for various regularization parameters. This study will be significant to researchers working on computational and mathematical methods for solving inverse coefficient identification problems with applications in heat transfer and porous media.

The operator approach to Il’yushin’s models of viscoelastic media under the isothermal processes of deformation

Three Il’yushin’s models of viscoelastic media are considered. The theorems of strong unique solvability of the corresponding initial boundary-value problems are proved. All models are uniformly reduced to the Cauchy problems for differential first-order equations in some Hilbert space under some restrictions on physical parameters. Using these equations, we pose the problems of a spectrum of various viscoelastic media.

Petrovskii elliptic systems in the extended Sobolev scale

Petrovskii elliptic systems of linear differential equations given on a closed smooth manifold are investigated in the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to the Hilbert Sobolev scale. Theorems of solvability of the elliptic systems in the extended Sobolev scale are proved. An a priori estimate for solutions is obtained, and their local regularity is studied.

SOME MIXED PROBLEMS FOR SEMILINEAR PARABOLIC TYPE EQUATION

In this paper, some mixed problem with third type boundary value for a semilinear parabolic equation is investigated. Here the solvability theorems for considered problem and the uniqueness theorem for a model case of the problem are showed.

Periodic boundary-value problem for a fourth-order differential equation

In the present paper we obtain sufficient conditions for solvability of a periodic boundary-value problem for a fourth-order ordinary differential equation. The research technique is based on a solvability theorem for a quasi-linear operator equation in the resonance case. We formulate sufficient conditions for existence of periodic solutions in terms of the initial equation. The main result of the paper clarifies the existence theorem established by B. Mehry and D. Shadman in Sci. Iran. 15 (2), 182–185 (2008).

On solvability of the second boundary value problem in a half-space for one elliptic equation

We consider the second boundary value problem for one elliptic equation with a lower term in a half-space. We prove theorems of solvability of the problem in the Sobolev space.

Small motions and eigenoscillations of a “fluid–barotropic gas” hydrosystem

This paper deals the problem of small motions and eigenoscillations of the hydrodynamical system” ideal fluid–barotropic gas” with regard for gravitational and capillary forces. The spectrum structure and the basis property of eigenfunctions are studied. The variational principles for eigenvalues are presented. The theorems of strong solvability of an initial boundary-value problem and the inverse Lagrange’s theorem of stability of the hydrosystem are proved.

Solvability theorem for a model of a unimolecular heterogeneous reaction with adsorbate diffusion

A mathematical model of a unimolecular heterogeneous catalytic reaction is considered in the case where the adsorbate can diffuse along the surface of a catalyst and the desorption of the reaction product from the surface of the adsorbent is instantaneous. The model is described by a coupled parabolic system. The existence and uniqueness of a classical solution are established. Bibliography: 16 titles.

Generalization of B.M. Levitan and M.G. Gasymov’s solvability theorems to the case of indecomposable boundary conditions

Generalization of B.M. Levitan and M.G. Gasymov’s solvability theorems to the case of indecomposable boundary conditions

Some Theorems of Solvability for a Class of g-Comparable Operator Equations

Some Theorems of Solvability for a Class of g-Comparable Operator Equations

Algorithms for a System of General Variational Inequalities in Banach Spaces

The purpose of this paper is using Korpelevich′s extragradient method to study the existence problem of solutions and approximation solvability problem for a class of systems of finite family of general nonlinear variational inequality in Banach spaces, which includes many kinds of variational inequality problems as special cases. Under suitable conditions, some existence theorems and approximation solvability theorems are proved. The results presented in the paper improve and extend some recent results.

A viscosity approach to the Dirichlet problem for complex Monge–Ampère equations

We present a viscosity approach to the Dirichlet problem for the complex Monge–Ampere equation \({\det u_{\bar{j} k} = f (x, u)}\) . Our approach differs from previous viscosity approaches to this equation in several ways: it is based on contact set techniques (the Alexandrov–Bakelman–Pucci estimate), on extensive applications of sup-inf convolutions, and on a relation between real and complex Hessians. More specifically, this paper includes a notion of viscosity solutions; a comparison principle and a solvability theorem; the equivalence between viscosity and pluripotential solutions; an estimate of the modulus of continuity of a solution in terms of that of a given subsolution and of the right-hand side f; and an Alexandrov–Bakelman–Pucci type of L∞-estimate.

Variational inequalities for a system of elliptic–parabolic equations

We create a general framework for mathematical study of variational inequalities for a system of elliptic–parabolic equations. In this paper, we establish a solvability theorem concerning the existence of solutions for the vector-valued elliptic–parabolic variational inequality with time-dependent constraint. Moreover, we give some applications of the system, for example, time-dependent boundary obstacle problem and time-dependent interior obstacle problem.

On a class of nonlocal parabolic equations

We consider a boundary value problem for parabolic equations with nonlocal nonlinearity of such a form that favorably differs from other equations in that it leads to partial differential equations that have important properties of ordinary differential equations. Local solvability and uniqueness theorems are proved, and an analog of the Painleve singular nonfixed points theorem is proved. In this case, there is an alternative—either a solution exists for all t ≥ 0 or it goes to infinity in a finite time t = T (blowup mode). Sufficient conditions for the existence of a blowup mode are given.