The unique solvability of a certain nonlocal nonlinear problem with a spatial operator strongly monotone with respect to the gradient

We consider a nonlinear degenerate parabolic equation whose spatial operator depends on a nonlocal characteristic of the solution. We prove the uniqueness of the solution in the class of vector-valued functions that take on values in Sobolev spaces.     

On the solvability of nonlocal nonstationary problems with double degeneration

We study the solvability in Sobolev spaces of the first boundary value problem for a nonlinear evolution equation degenerating both on the solution and on the solution gradient. We consider the case in which the spatial operator can depend on a nonlocal characteristic of a solution, for example, on an integral characteristic. The theorem is proved with the use of the time discretization method. To study the solvability of the spatial problems arising in the course of the proof, we use the Galerkin method.     

The nonstationary venttsel' problem with quadratic growth with respect to the gradient

A priori estimates are established for solutions to initial/boundary-value problems for quasilinear parabolic equations of nondivergence type with the Venttsel' boundary condition. These estimates are used in proving the existence theorems in Sobolev spaces. Bibliography: 7 titles. Translated fromProblemy Matematicheskogo Analiza. No. 15. 1995, pp. 33–46.     

Uniform convergence of the difference scheme for one nonstationary nonlocal boundary-value problem

Translated from Aktual'nye Voprosy Prikladnoi Matematiki, pp. 61–68, 1989.     

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.     

On the solvability of a nonlinear pseudoparabolic problem

This paper is concerned with the study of an initial boundary value problem for a nonlinear second order pseudoparabolic equation arising in the unidirectional flow of a thermodynamic compatible third grade fluid. We establish some a priori bounds for the solution and prove its existence.     

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.     

On the solvability of one multidimensional version of the first Darboux problem for some nonlinear wave equations

In the present paper, for wave equations with power nonlinearity we investigate the problem of the existence or nonexistence of global solutions of a multidimensional version of the first Darboux problem in the conic domain.     

General conditions for the unique solvability of initial-value problem for nonlinear functional differential equations

We establish general conditions for the unique solvability of the Cauchy problem for systems of nonlinear functional differential equations. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 167–172, February, 2008.     

Limiting smoothness of the solution of a nonstationary problem with one or two obstacles

The regularity of the solution of a nonstationary problem with an obstable for various forms of parabolic operators has been thoroughly investigated. Under the condition of sufficient smoothness of the data of the problem, one proves that the solutionW _q ^2,1 (Q) belongs to the Sobolev space In the present paper one establishes that the limiting possible smoothness of the solution of a nonstationary problem with one or two obstacles is the boundedness of the second derivatives of the solution with respect to the spatial variables and of the first derivatives with respect to t. One assumes that the operator is linear and the functions defining the obstacles have the minimal possible smoothness.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 149–157, 1984.     

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.     

On solvability of a space-periodic problem for the nonstationary VT-equation of transonic gas dynamics

We consider the problem of existence and uniqueness of a solution to the nonstationary VT-equation of transonic gas dynamics which is periodic in an appropriate space variable and study the question of smoothness of this solution in time.     

On the solvability of the first nonlocal boundary value problem for an elliptic equation

We consider problems of the existence, uniqueness, and sign-definiteness of the classical solutions of the problem

$(Lu)(x) = f(x)(x \in D),u(x) - \beta (x)u(\sigma x) = \psi (x)(x \in S),$

where L is a linear second-order operator elliptic in the closure of a domain D ? R ⁿ and σ is a single-valued continuous mapping of S ≡ ?D into \(\bar D\).

We show that, under natural assumptions on the smoothness of β, σ, and the coefficients of L, this problem is Fredholm provided that either σ has no attractors on S or σ generates an attractor Θ on S and the spectral radius of the operator A defined on η(x) ∈ C(Θ) by the formula (Aη)(x) = |β(x)|η(σx) is less than unity.We obtain semieffective (in terms of a test function) conditions for the unique solvability of the problem.     

On one proof of the classical solvability of the Hele-Shaw problem with free boundary

We consider the Stefan problem for a parabolic equation with a small parameter as the coefficient of the derivative with respect to time. We justify the limit transition as the small parameter tends to zero, which enables us to prove the classical solvability of the Hele-Shaw problem with free boundary in the small with respect to time. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50; No. 11, pp. 1452–1462, November, 1998.     

一类非线性快慢系统非局部问题的摄动解

主要讨论了一类非线性快慢系统非局部问题的摄动解,在适当的条件下,根据不同边界层利用伸长变量和幂级数展开理论,构造了问题的形式渐近解,并利用微分不等式理论在整个区间上证明了形式渐近解的一致有效性,把奇摄动问题的摄动解推广到快慢系统非局部问题的摄动解.     

On a nonlocal problem for nonlinear pseudoparabolic equations

For a nonlinear pseudoparabolic equation with one space dimension we consider its initial boundary value problem on an interval. The boundary condition on the left end is of Dirichlet type, the right end condition is replaced by a nonlocal one. Because it is given by an integral, the function involved could exhibit singularities, which distinguishes this nonlocal condition from its Dirichlet counterpart. Based on an elliptic estimate and an iteration method we established the well-posedness of solutions in a weighted Sobolev space.