On the Nonstationary Two-Phase Venttsel Problem in the Transversal Case

We study the solvability of the two-phase Venttsel problem for quasilinear parabolic non-divergence second order equations in the case where the film transversally intersects the boundary of a domain. Bibliography: 15 titles.     

The Dirichlet Problem for Quasilinear Elliptic Equations in Domains with Smooth Closed Edges

The solvability of the Dirichlet problem for quasilinear elliptic second-order equations of nondivergence form are studied in a domain whose boundary contains a conical point or an edge of an arbitrary codimension. Bibliography: 12 titles.     

On the solvability of the boundary-value problem for second-order equations in Hilbert space with an operator coefficient in the boundary condition

We consider the boundary-value problem on a finite interval for a class of second-order operator-differential equations with a linear operator in one of its boundary conditions. We obtain sufficient conditions for the regular solvability of the boundary-value problem under consideration; these conditions are expressed only in terms of its operator coefficients.     

On the unique solvability of a family of two-point boundary-value problems for systems of ordinary differential equations

We consider a family of two-point boundary-value problems for systems of ordinary differential equations with functional parameters. This family is the result of the reduction of a boundary-value problem with nonlocal condition for a system of second-order, quasilinear hyperbolic equations by the introduction of additional functions. Using the parametrization method, we establish necessary and sufficient conditions of the unique solvability of the family of two-point boundary-value problems for a linear system in terms of the initial data. We also prove sufficient conditions of the unique solvability of the problem considered and propose an algorithm for its solution. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 21–39, 2006.     

“Destruction” of the solution of a strongly nonlinear equation of pseudoparabolic type with double nonlinearity

We consider the first initial-boundary value problem for multidimensional strongly nonlinear equations with double nonlinearity of pseudoparabolic type in a bounded domain with sufficiently smooth boundary. We prove the local solvability of this problem in the weak generalized sense. Depending on the nonlinearity and initial conditions under consideration, we prove the solvability of the equation in any finite cylinder (x, t) ∈ Ω × [0, T] or the destruction of the solution in finite time.     

On some classes of coefficient inverse problems for parabolic systems of equations

We examine the question on solvability in the Sobolev spaces of coefficient inverse problems for parabolic systems of equations with the overdetermination conditions on a collection of surfaces. Under certain conditions on the geometry of the domain and the boundary operators, the local solvability of the problem is proven. It is demonstrated that the conditions on the boundary operators are sharp and that, in some cases, the problem is not unconditionally solvable.     

含距离位势的拟线性椭圆方程解的存在性

变分原理证明了一类含距离位势的拟线性椭圆方程齐次Dirichlet边界条件下第一特征值问题的可解性.进一步,利用临界点理论得到了一类含距离位势的非线性椭圆方程非平凡解的存在性.     

On the Solvability of Nonlinear Boundary Value Problems for the System of Differential Equations of Equilibrium of Shallow Anisotropic Timoshenko-Type Shells with Free Edges

Differential Equations - We study the solvability of a nonlinear boundary value problem for a system of five second-order partial differential equations under given boundary conditions. The system...     

On One Nonlocal Problem with Free Boundary

We investigate group-theoretic properties of a nonlocal problem with free boundary for a degenerating quasilinear parabolic equation. We establish conditions for the invariant solvability of this problem, perform its reduction, and obtain an exact self-similar solution.     

Study of the solvability of a boundary value problem for the system of nonlinear differential equations of the theory of shallow shells of the Timoshenko type

We study the solvability of a boundary value problem for a system of nonlinear second-order partial differential equations under given boundary conditions, which describes the equilibrium of elastic shallow shells with hinged edges in the framework of the Timoshenko shear model. The study method implies the reduction of the original system of equations to a single nonlinear differential equation whose solvability is proved with the use of the contraction mapping principle.     

Solvability of a boundary value problem for a stationary model of the magnetic hydrodynamics of a viscous heat-conducting fluid

A boundary value problem is studied for a stationary model of the magnetic hydrodynamics of a viscous heat-conducting fluid under nonhomogeneous boundary conditions on the velocity, electromagnetic field, and temperature. The model consists of the Navier-Stokes equations, the Maxwell equations, the generalized Ohm law, and the convection-diffusion equation for the temperature which are connected nonlinearly with each other. Sufficient conditions on the initial data are established that guarantee the global solvability of the problem under consideration and the local uniqueness of its solution. The properties are studied of the linear operator obtained by linearizing the operator of the original boundary value problem.     

Coerciveness of the discontinuous initial-boundary value problem for parabolic equations

In this paper, the mixed problem for parabolic equations is investigated with the discontinuous coefficient at the highest derivative and with nonstandard boundary conditions. Namely, the boundary conditions contain values of the solution not only on the boundary points, but also on the inner points of the considered domain as well. Moreover, abstract functionals are involved in the boundary conditions. We single out a class of functional spaces in which coercive solvability occurs for the investigated problem.     

Boundary value problem for a second-order elliptic equation in the exterior of an ellipse

We consider a boundary value problem for a second-order linear elliptic differential equation with constant coefficients in a domain that is the exterior of an ellipse. The boundary conditions of the problem contain the values of the function itself and its normal derivative. We give a constructive solution of the problem and find the number of solvability conditions for the inhomogeneous problem as well as the number of linearly independent solutions of the homogeneous problem. We prove the boundary uniqueness theorem for the solutions of this equation.     

On the global continuous solvability of the mixed problem for one-dimensional hyperbolic systems of quasilinear equations

We consider a hyperbolic system of quasilinear equations written in Riemann invariants for the case of one spatial variable. For this system, we obtain sufficient conditions for the global generalized continuous solvability of the mixed problem in the class of functions monotone with respect to x for arbitrary t and with respect to t for x = 0. In contrast to earlier studies, we assume that the boundary conditions may depend not only on time but also on the unknown functions.     

On the “Destruction” of Solutions of Nonlinear Wave Equations of Sobolev Type with Cubic Sources

We consider model three-dimensional wave nonlinear equations of Sobolev type with cubic sources, and foremost, model three-dimensional equations of Benjamin-Bona-Mahony and Rosenau types with model cubic sources. An essentially three-dimensional nonlinear equation of spin waves with cubic source is also studied. For these equations, we investigate the first initial boundary-value problem in a bounded domain with smooth boundary. We prove local solvability in the strong generalized sense and, for an equation of Benjamin-Bona-Mahony type with source, we prove the unique solvability of a “weakened” solution. We obtain sufficient conditions for the “destruction” of the solutions of the problems under consideration. These conditions have the sense of a “large” value of the initial perturbation in the norms of certain Banach spaces. Finally, for an equation of Benjamin-Bona-Mahony type, we prove the “failure” of a “weakened” solution in finite time.     

Solvability conditions of a boundary value problem with operator coefficients and related estimates of the norms of intermediate derivative operators

Sufficient conditions for the proper and unique solvability in the Sobolev space of vector functions of the boundary value problem for a certain class of second-order elliptic operator differential equations on a semiaxis are obtained. The boundary condition at zero involves an abstract linear operator. The solvability conditions are established by using properties of operator coefficients. The norms of intermediate derivative operators, which are closely related to the solvability conditions, are estimated.     

Solvability of Cauchy Problem for a Differential-Algebraic System with Concentrated Delay

We study the Cauchy problem for a linear differential-algebraic system of equations with concentrated delay. Our research continues investigation of solvability of linear Noether boundary value problems for systems of functional-differential equations given in the monographs by A.D. Myshkis, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina, A.M. Samoilenko, and A.A. Boichuk; meanwhile, we use essentially the tool of Moore-Penrose inverse matrices. For a linear differential-algebraic system with concentrated delay, we find sufficient conditions for its solvability and give a construction of generalized Green’s operator for Cauchy’s problem. We also give some examples which illustrate in detail the solvability conditions and the suggested construction.     

Analog of the first Fredholm theorem for higher-order nonlinear differential equations

We study the existence of solutions continuously depending on a parameter for higher-order nonlinear ordinary differential equations with linear boundary conditions. In particular, we prove a theorem of Fredholm type providing tests for the unique solvability of this problem.     

Solvability of Control Problems for Stationary Equations of Magnetohydrodynamics of a Viscous Fluid

We consider the boundary-value problem for the stationary equations of magnetohydrodynamics of a viscous incompressible fluid with nonhomogeneous boundary conditions for the velocity and electromagnetic field. We study global solvability of this problem and establish some sufficient conditions for uniqueness of its solution. We state control problems for the model of magnetohydrodynamics under consideration, study their solvability, give and examine optimality systems for both arbitrary and particular quality functionals.     

On an approach to the study of the solvability of boundary value problems for the system of differential equations of the 3D theory of elasticity

We study the solvability of a boundary value problem for a system of second-order linear partial differential equations. A theorem on the existence of a solution of the problem is proved. The method used in the study is to reduce the original system of equations to a system of 3D singular integral equations, whose solvability can be proved with the use of the notion of symbol of a singular operator.