Oscillations of Stratified Liquid Partially Covered by Crumpling Ice

We study the problem on small motions of ideal stratified fluid with a free surface, partially covered by crumbling ice. By the method of orthogonal projecting the boundary conditions on the moving surface and, with the help of investigation of some auxiliary problems, the original initial-boundary value problem is reduced to the equivalent Cauchy problem for a second order differential equation in a Hilbert space. We find sufficient existence conditions for existence of a strong (with respect to the time variable) solution to the initial-boundary value problem describing evolution of the specified hydrodynamics system.     

Nonstationary nonlinear nonlocal problem of radiative–conductive heat transfer in a system of opaque bodies with properties depending on the radiation frequency

We consider a nonstationary nonlinear initial-boundary value problem governing radiative-conductive heat transfer in a periodic system of grey heat shields. The existence and uniqueness of a weak solution and a regular weak solution are established. Estimates for the solutions in terms of the data of the problem are obtained. Bibliography: 36 titles.     

Solution of Volterra operator-integral equations in the nonregular case by the successive approximation method

The branches of solutions of a nonlinear integral equation of Volterra type in a Banach space are constructed by the successive approximation method. We consider the case in which a solution may have an algebraic branching point. We reduce the equation to a system regular in a neighborhood of the branching point. Continuous and generalized solutions are considered. General existence theorems are used to study an initial-boundary value problem with degeneration in the leading part.     

Existence and Uniqueness in a Theory of Thermoviscoelastic Dielectrics

This work deals with the study of an initial-boundary value problem in linear theory of thermo-viscoelastic dielectrics. A variational formulation of this problem is provided and the existence of its solution is proved by an auxiliary Faedo-Galerkin scheme. Finally, the uniqueness of the solution is obtained as a corollary of the existence proof.     

Asymptotic behavior of the solution of a system of nonlinear integro-differential equations

We study the asymptotic behavior as time tends to infinity of the solution of an initial-boundary value problem for a system of nonlinear integro-differential equations that arises in the mathematical modeling of penetration of electromagnetic field into a medium whose electric conductivity substantially depends on temperature. Both homogeneous and inhomogeneous boundary conditions are considered. The exponential stabilization of the solution is established.     

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.     

On a mixed problem with integral conditions for one-dimensional parabolic equation

We consider an initial-boundary value problem for a one-dimensional parabolic equation with nonlocal boundary conditions. These nonlocal conditions are given in terms of integrals. Based on solution of the Dirichlet problem for the parabolic equation, we constructively establish the well-posedness for the nonlocal problem.     

Optimal boundary force control at one end of a string for a given displacement mode at the other end

We solve the problem of optimal boundary force control at one end of a string for the case of a given displacement mode at the other end. The problem is studied in the sense of a generalized solution of the corresponding mixed initial-boundary value problem in the Sobolev space. We also solve the problem of choosing an optimal boundary control from infinitely many feasible controls. The generalized solution of the mixed initial-boundary value problem is constructed in closed form, and the uniqueness of the solution is proved.     

Immediate regularization of measure-type population densities in a two-dimensional chemotaxis system with signal consumption

This paper deals with the Neumann initial-boundary value problem for a classical chemotaxis system with signal consumption in a disk.In contrast to previous studies which have established a comprehensive theory of global classical solutions for suitably regular nonnegative initial data,the focus in the present work is on the question to which extent initially prescribed singularities can be regularized despite the presence of the nonlinear cross-diffusive interaction.The main result in this paper asserts that at least in the framework of radial solutions immediate regularization occurs under an essentially optimal condition on the initial distribution of the population density.More precisely,it will turn out that for any radially symmetric initial data belonging to the space of regular signed Borel measures for the population density and to L² for the signal density,there exists a classical solution to the Neumann initial-boundary value problem,which is smooth and approaches the given initial data in an appropriate trace sense.     

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.     

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 the blowing up of solutions to quantum hydrodynamic models on bounded domains

The blow-up in finite time for the solutions to the initial-boundary value problem associated to the multi-dimensional quantum hydrodynamic model in a bounded domain is proved. The model consists on conservation of mass equation and a momentum balance equation equivalent to a compressible Euler equations corrected by a dispersion term of the third order in the momentum balance. The proof is based on a priori estimates for the energy functional for a new observable constructed with an auxiliary function, and it is shown that, under suitable boundary conditions and assumptions on the initial data, the solution blows up after a finite time.     

Large-time behavior of the motion of a viscous heat-conducting one-dimensional gas coupled to radiation

We study the large-time behavior of the solution of an initial-boundary value problem for the equations of 1D motions of a compressible viscous heat-conducting gas coupled to radiation through a radiative transfer equation. Assuming suitable hypotheses on the transport coefficients and adapted boundary conditions, we prove that the unique strong solution of this problem converges toward a well-determined equilibrium state at exponential rate.     

Asymptotic expansion of the solution of the first boundary value problem for the Schrödinger system near conical points of the boundary

We study the first initial-boundary value problem for the Schrödinger system in a cylindrical domain. It is assumed that the boundary contains a conical point. We obtain an asymptotic expansion of the solution in a neighborhood of such a point.     

Optimal boundary control by a force at one end of a string with a given force mode at the other end

We consider the problem of boundary control by a force applied to one end of a string in the case of a given force mode at the other end. The problem is studied in the sense of the generalized solution of the corresponding mixed initial-boundary value problem in the Sobolev space. We also solve the problem of choosing an optimal boundary control in the set of all admissible controls. The generalized solution of the mixed initial-boundary value problem is constructed in closed form, and its uniqueness is proved.     

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)     

Inverse problem for the diffusion equation with overdetermination in the form of an external volume potential

An initial-boundary value problem for the diffusion equation with an unknown initial condition is considered. Additional information used for determining the unknown initial condition is an external volume potential whose density is the Laplacian calculated for the solution of the initial-boundary value problem. Uniqueness theorems for the inverse problem are proved in the case when the spatial domain of the initial-boundary value problem is a spherical layer or a parallelepiped.     

Asymptotics of the solution to a singularly perturbed system of parabolic equations in the critical case

A formal asymptotic representation is constructed for the solution of an initial-boundary value problem for a singularly perturbed system of parabolic equations. A specific feature of the problem is that its solution has an internal transition layer.     

Nonlinear dynamics in the initial-boundary value problem on the fluid flow from a ledge for the hydrodynamic approximation to the boltzmann equations

We numerically analyze the laminar-turbulent transition in the problem on the flow of a viscous incompressible fluid from a ledge. To model the fluid flow, we use the Boltzmann integro-differential equations expanded in the Knudsen number. For the fundamental analysis, we use a numerical method of increased accuracy for the integration of the initial-boundary value problem. By analyzing the phase portraits of the behavior of the system, we find that the transition from a stationary solution to an irregular chaotic one takes place in accordance with the Feigenbaum-Sharkovskii-Magnitskii scenario. Moreover, the transition process differs from the results obtained by using the Navier-Stokes equations for solving a similar initial-boundary value problem.