Application of nonclassical separation of variables to a nonlocal heat problem

We consider an initial-boundary value problem for the heat equation with a nonlocal two-point boundary condition containing a parameter. By separating the variables in an auxiliary function system, we construct a regular solution. We obtain sufficient conditions for the absolute and uniform convergence of the series in the auxiliary system. We prove conditions close to necessary ones for the existence of a regular solution of the initial-boundary value problem.     

Global C1 Solution to the Initial-boundary Value Problem for Diagonal Hyperbolic Systems with Linearly Degenerate Characteristics

We prove that the Cº boundedness of solution implies the global existence and uniqueness of C¹ solution to the initial-boundary value problem for linearly degenerate quasilinear hyperbolic systems of diagonal form with nonlinear boundary conditions. Thus, if the C¹ solution to the initial-boundary value problem blows up in a finite time, then the solution itself must tend to the infinity at the starting point of singularity.     

Well-posedness of the IBVP for 2-D Euler equations with damping

In this paper we focus on the initial-boundary value problem of the 2-D isentropic Euler equations with damping. We prove the global-in-time existence of classical solution to the initial-boundary value problem for small smooth initial data by the method of local existence of solution combined with a priori energy estimates, where the appropriate boundary condition plays an important role.     

On Solvability of Some Initial-Boundary Problems for Mathematical Models of the Motion of Nonlinearly Viscous and Viscoelastic Fluids

In the paper, the settings of initial-boundary and initial value problems arising in a number of models of movement of nonlinearly viscous or viscoelastic incompressible fluid are considered, and existence theorems for these problems are presented. In particular, the settings of initial-boundary value problems appearing in the regularized model of the movement of viscoelastic fluid with Jeffris constitutive relation are described. The theorems for the existence of weak and strong solutions for these problems in bounded domains are given. The initial value problem for a nonlinearly viscous fluid on the whole space is considered. The estimates on the right-hand side and initial conditions under which there exist local and global solutions of this problem are presented. The modification of Litvinov's model for laminar and turbulent flows with a memory is described. The existence theorem for weak solutions of initial-boundary value problem appearing in this model is given.     

A Neumann problem for the KdV equation with Landau damping on a half-line

We consider the initial-boundary value problem on a half-line for the KdV equation with Landau damping. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.     

Intermediate long-wave equation on a half-line

We consider the initial-boundary value problem for intermediate long-wave equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.     

Ott–Sudan–Ostrovskiy equation on a half-line

We consider the initial-boundary value problem for the Ott-Sudan-Ostrovskiy equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.     

Blow-up and extinction for a sixth-order parabolic equation

In this paper, we consider an initial-boundary value problem for a sixth-order parabolic equation. We use the modified method of potential wells to study the relationship which the equation solutions existence, blow-up and the asymptotic behavior with initial conditions.     

The Love's problem for hyperbolic thermoelasticity

In our paper we investigated the initial-boundary value problem for elastic layer situated on half space of another elastic medium. In this medium the thermomechanical interactions were taken into consideration. The system of equations with initial-boundary conditions describes the phenomenon of wave propagation with finite speed. In our problem there are two surfaces ie. free surface and contact surface between layer and half space. On the free surface are setting boundary conditions for normal and tangent surface force. We consider two types of contact between layer and half-space: rigid contact and slip contact. The initial-boundary value problem was solved by using integral transformations and Cagniard-de Hoope methods. From the solution of this problem follows that in layer and half space exist some kind of thermoelastic waves. We investigated moreover the conditions which should be fullfiled for propagation of Rayleigh and Love's type waves on the contact surface between layers and half space. The results obtained in our investigation were used in technical applications especially engineering design and diagnostics of roads and airfields. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

非线性势力与轴向力联合作用下梁方程的初边值问题

考虑非线性势力的作用下梁方程的初边值问题,利用Galerkin方法证明了非线性项在适当条件下,在不高于6维的空间中该初边值问题整体弱解的存在及唯一性.     

On the existence and uniqueness of a generalized solution of the mixed problem for the wave equation with the second and third boundary conditions

We prove the uniqueness of a generalized solution of an initial-boundary value problem for the wave equation with boundary conditions of the third and second kind. In addition, we find a closed-form expression for the analytic solution of that problem with zero initial data. The result plays an important role in the investigation of the boundary control problem. We show how to use the obtained solution for the investigation of the boundary control problem in the case of subcritical time intervals for which the solution of the boundary control problem, if it exists at all, is unique. We obtain necessary and sufficient conditions for the existence of a unique solution in a class admitting the existence of finite energy.     

Stability and monotonicity of a conservative difference scheme for a multidimensional nonlinear scalar conservation law

We prove the existence, uniqueness, and monotonicity of the solution of an upwind conservative explicit difference scheme approximating an initial-boundary value problem for a many-dimensional nonlinear scalar conservation law with a quadratic nonlinearity under some specific conditions imposed only on the input data of the problem. We show that the resulting solution is not necessarily stable. Under some additional conditions on the input data, which provide the absence of shock waves, we prove the stability of the unique solution of the difference scheme for any finite time.     

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.     

Oscillations of stratified fluids

We study the problem on small motions and normal oscillations of a system of two heavy immiscible stratified fluids partially filling a fixed vessel. The lower fluid is assumed to be viscous, while the upper one is assumed to be ideal. We find sufficient existence conditions for a strong (with respect to the time variable) solution of the initial-boundary value problem describing the evolution of the specified hydraulic system. For the corresponding spectral system, we obtain results about the localization of the spectrum, asymptotic behavior of branches of eigenvalues, and existence of the substantial spectrum of the problem.     

Asymptotic behaviour in a conducting electromagnetic medium

We investigate the initial-boundary value problem for Maxwell'sequations in linear conducting materials together with dissipativeboundary conditions. We show that it is possible to introducethe free energy and derive from it a domain of dependence. Weprove the existence, uniqueness and asymptotic stability ofthe solution when one of Maxwell's equations is considered asa constraint for the electromagnetic fields.     

Boundary value problems to a certain class of non-linear diffusion equations

The first initial-boundary value problem and the free boundary of Stefan's type for a certain class of non-linear diffusion equations which arises in plasma physics are considered. For the first initial-boundary value problem, the global existence of a classical and its large-time behaviour are discussed. The free-boundary problem is studied by the reduction to the first initial-boundary value problem.     

Existence and asymptotic behavior of solutions to a singular parabolic equation

In this paper, we study the initial-boundary value problem for a class of singular parabolic equations. Under some conditions, we obtain the existence and asymptotic behavior of solutions to the problem by parabolic regularization method and the sub-super solutions method. As a byproduct, we prove the existence of solutions to some problems with gradient terms, which blow up on the boundary.     

Mathematical analysis of variable density flows in porous media

We consider a simple model describing the motion of a two-component mixture through a porous medium. We discuss well posedness of the associated initial-boundary value problem, in particular, with respect to the choice of boundary and far-field conditions. The existence of global-in-time solutions is proved in the ideal case when the fluid occupies the whole physical space. Finally, similar results are obtained also for the boundary value problems in the simplified 1-D geometry.     

SEMI-GLOBAL $C^1$ SOLUTION TO THE MIXED INITIAL-BOUNDARY VALUE PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS

By means of an equivalent invariant form of boundary conditions, the authors get the existence and uniqueness of semi-global C^1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems with general nonlinear boundary conditions.     

Initial-boundary value problems for the Vlasov-Poisson equations in a half-space

We consider initial-boundary value problems for the Vlasov-Poisson equations in a half-space that describe evolution of densities for ions and electrons in a rarefied plasma. For sufficiently small initial densities with compact supports and large strength of an external magnetic field, we prove the existence and uniqueness of classical solutions for initial-boundary value problems with different boundary conditions for the electric potential: the Dirichlet conditions, the Neumann conditions, and nonlocal conditions.