A problem with a nonlocal,with respect to time,condition for multidimensional hyperbolic equations

We study a boundary-value problem for a hyperbolic equation with a nonlocal with respect to time-variable integral condition. We obtain sufficient conditions for unique solvability of the nonlocal problem. The proof is based on reduction of the nonlocal first-type condition to the second-type one. This allows to reduce the nonlocal problem to an operator equation. We show that unique solvability of the operator equation implies the existence of a unique solution to the problem.     

Solvability of a Boundary Value Problem for a Differential Equation of the Boussinesq Type

We study the solvability and the construction of the solution of a boundary value problem with a nonlocal integral boundary condition for a three-dimensional analog of the fourth-order homogeneous Boussinesq type differential equation. Separation of variables is used to derive a criterion for the unique solvability of this nonlocal problem. The problem is also considered in the case of violation of the unique solvability criterion.     

On a nonlinear nonlocal problem of elliptic type

The solvability of a nonlinear nonlocal problem of the elliptic type that is a generalized Bitsadze–Samarskii-type problem is analyzed. Theorems on sufficient solvability conditions are stated. In particular, a nonlocal boundary value problem with p-Laplacian is studied. The results are illustrated by examples considered earlier in the linear theory (for p = 2). The examples show that, in contrast to the linear case under the same “nice” nonlocal boundary conditions, for p > 2, the problem can have one or several solutions, depending on the right-hand side.     

Criteria of unique solvability of nonlocal boundary-value problem for systems of hyperbolic equations with mixed derivatives

We consider nonlocal boundary-value problem for a system of hyperbolic equations with two independent variables. We investigate questions of existence of unique classical solution to problem under consideration. In terms of initial data we propose criteria of unique solvability and suggest algorithms of finding of solutions to nonlocal boundary-value problem. As an application we give conditions of solvability of periodic boundary-value problem for a system of hyperbolic equations.     

Optimal solvability for a nonlocal problem at critical growth

We provide optimal solvability conditions for a nonlocal minimization problem at critical growth involving an external potential function a. Furthermore, we get an existence and uniqueness result for a related nonlocal equation.     

Nonlocal problem with integral conditions for a system of hyperbolic equations in characteristic rectangle

We consider a nonlocal problem with integral conditions for a system of hyperbolic equations in rectangular domain. We investigate the questions of existence of unique classical solution to the problemunder consideration and approaches of its construction. Sufficient conditions of unique solvability to the investigated problem are established in the terms of initial data. The nonlocal problem with integral conditions is reduced to an equivalent problem consisting of the Goursat problem for the system of hyperbolic equations with functional parameters and functional relations. We propose algorithms for finding a solution to the equivalent problem with functional parameters on the characteristics and prove their convergence. We also obtain the conditions of unique solvability to the auxiliary boundary-value problem with an integral condition for the system of ordinary differential equations. As an example, we consider the nonlocal boundary-value problem with integral conditions for a two-dimensional system of hyperbolic equations.     

Nonlocal problem for complete second-order hyperbolic operator-differential equations with variable domains

We prove the well-posed solvability (in the strong sense) of complete second-order hyperbolic operator-differential equations with variable domains of unbounded operator coefficients under nonlocal initial conditions. We are the first to establish the well-posed solvability of the mixed problem for the complete string vibration equation with nonstationary boundary conditions and nonlocal initial conditions.     

Solvability of Nonlocal Elliptic Problems in Dihedral Angles

In this paper, we consider nonlocal elliptic problems in dihedral and plane angles. Such problems arise in the study of nonlocal problems in bounded domains for the case in which the support of nonlocal terms intersects the boundary. We study the Fredholm and unique solvability of this problem in the corresponding weighted spaces. Results are obtained by means of a priori estimates of the solutions and of Green's formula for nonlocal elliptic problems.     

Galerkin method for nonlocal mixed boundary value problem for the Moore‐Gibson‐Thompson equation with integral condition

In this paper, we are going to deal with the nonlocal mixed boundary value problem for the Moore‐Gibson‐Thompson equation. Galerkin method was the main used tool for proving the solvability of the given nonlocal problem.     

On the unique solvability of a nonlocal boundary value problem with data on intersecting lines for systems of hyperbolic equations

We consider a nonlocal boundary value problem for a system of hyperbolic equations with two independent variables with data on intersecting lines one of which is a characteristic. In terms of the data of the nonlocal boundary value problem, we obtain sufficient coefficient conditions for its unique solvability.     

Boundary-value problems for a hyperbolic equation with nonlocal conditions of the I and II kind

In this paper we consider two initial-boundary value problems with nonlocal conditions. The main goal is to propose a method for proving the solvability of nonlocal problems with integral conditions of the first kind. The proposed method is based on the equivalence of a nonlocal problem with an integral condition of the first kind and a nonlocal problem with an integral condition of the second kind in a special form. We prove the unique existence of generalized solutions to both problems.     

A problem with a nonlocal boundary condition on the characteristic for a class of equations of mixed type

We study the boundary-value problem with a nonlocal boundary condition on the characteristic for a class of equations of mixed type. The unique solvability of the problem is proved.     

Solvability of the fractional hyperbolic Keller–Segel system

We study a new nonlocal approach to the mathematical modelling of the chemotaxis problem, which describes the random motion of a certain population due to a substance concentration. Considering the initial–boundary value problem for the fractional hyperbolic Keller–Segel model, we prove the solvability of the problem. The solvability result relies mostly on fractional calculus and kinetic formulation of scalar conservation laws.     

Unique solvability of a non-linear non-local boundary-value problem for systems of non-linear functional differential equations

General conditions for the unique solvability of a non-linear nonlocal boundary-value problem for systems of non-linear functional differential equations are obtained.     

Elliptic Problems with Nonlocal Potential Arising in Models of Nonlinear Optics

Mathematical Notes - The Dirichlet problem in the half-plane for strong elliptic differential-difference equations with nonlocal potentials is considered. The classical solvability of this problem...     

A nonlocal problem for a hyperbolic equation with integral conditions of the 1st kind with time-dependent kernels

We consider a nonlocal problem with integral conditions of the 1st kind. The main goal is to prove the unique solvability of this problem under the assumption that kernels of nonlocal conditions depend both on spatial and time variables. To this end we propose a technique based on the proved equivalence between the nonlocal problem with integral conditions of the 1st kind and a nonlocal problem with integral conditions of the 2nd kind in a special form. We formulate requirements to the initial data guaranteeing the unique existence of a generalized solution to the stated problem.     

On a certain nonlocal problem for a hyperbolic equation

The author proves the existence and uniqueness of a generalized solution of a nonlocal problem with an integral condition for a hyperbolic equation with n spatial variables. This work is a continuation of the studies started in [3–5], where the solvability problem of nonlocal problems with an integral condition was studied for hyperbolic equations on the plane. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 33, Suzdal Conference-2004, Part 1, 2005.     

Nonlocal time-multipoint problem for a class of evolution pseudodifferential equations

We prove the well-posed solvability of a nonlocal time-multipoint problem for evolution equations with pseudodifferential operators with analytic symbols and initial condition in the space of distributions of the type W′.     

An analog of the Tricomi problem with a nonlocal integral conjugate condition

We prove the unique solvability of an analog of the Tricomi problem for an elliptic-hyperbolic equation with a nonlocal integral conjugate condition on the characteristic line.     

Problem with generalized fractional differentiation operators for a hyperbolic equation degenerating in the interior of the domain

We study the unique solvability of a nonlocal problem with generalized fractional differentiation operators in the boundary condition for a hyperbolic equation degenerating in the interior of the domain.