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.     

Boundary-value problem with mixed conditions for weakly nonlinear hyperbolic equations

In a cylindrical domain, we investigate the unique solvability of a problem with mixed boundary conditions for an inhomogeneous linear hyperbolic equation of higher order with coefficients variable with respect to space coordinates. To estimate from below the small denominators that appear in the construction of a solution of the problem, the metric approach is used. The obtained results are extended to the case where the equation is perturbed by a nonlinear summand.     

ON AN INVERSE INITIAL-BOUNDARY VALUE PROBLEM FOR A ONE-DIMENSIONAL HYPERBOLIC SYSTEM OF DIFFERENTIAL EQUATIONS

1.IntroductionTheproblemstodeterminethecoefficientsindifferentialequationsfromknownfunctionaloftheirsolutionsareoftencalledinverseproblems.Amongtheseproblemsthesimplestisaboutone-dimensionalwaveequations.Thisproblemcanbediscussedinthetimedomainorinthefreq…     

On solvability of non-linear semi-periodic boundary-value problem for system of hyperbolic equations

We study a non-linear semi-periodic boundary-value problem for a system of hyperbolic equations with mixed derivative. At that, the semi-periodic boundary-value problem for a system of hyperbolic equations is reduced to an equivalent problem, consisting of a family of periodic boundary-value problems for ordinary differential equations and functional relation. When solving a family of periodic boundary-value problems of ordinary differential equations we use the method of parameterization. This approach allowed to establish sufficient conditions for the existence of an isolated solution of non-linear semi-periodic boundary-value problem for a system of hyperbolic equations.     

On a system of Klein-Gordon type equations with acoustic boundary conditions

We prove the existence, uniqueness and uniform stabilization of global solutions for a generalized system of Klein-Gordon type equations with acoustic boundary conditions on a portion of the boundary and the Dirichlet boundary condition on the rest.     

On the cauchy problem for a family of quasilinear hyperbolic equations

We prove local well-posedness and give some global existence and blow-up criteria for solutions of a family of quasilinear hyperbolic equations arising in shallow water theory.     

On the Cauchy problem for hyperbolic differential equations with multiple characteristics

A priori estimates leading to existence results for the Cauchy problem are obtained for a class of linear variable coefficient hyperbolic operators with multiple characteristics. These estimates provide an extension of the energy inequalities known for strictly hyperbolic operators. Supported by NSF Grant No. GP-7234.     

On the Cauchy problem for linear hyperbolic functional-differential equations

We study the question of the existence, uniqueness, and continuous dependence on parameters of the Carathéodory solutions to the Cauchy problem for linear partial functional-differential equations of hyperbolic type. A theorem on the Fredholm alternative is also proved. The results obtained are new even in the case of equations without argument deviations, because we do not suppose absolute continuity of the function the Cauchy problem is prescribed on, which is rather usual assumption in the existing literature.     

On a Darboux type multidimensional problem for second-order hyperbolic systems

The correct formulation of a Darboux type multidimensional problem for second-order hyperbolic systems is investigated. The correct formulation of such a problem in the Sobolev space is proved for temporal type surfaces on which the boundary conditions of a Darboux type problem are given.     

Well-posedness of a problem with initial conditions for hyperbolic differential-difference equations with shifts in the time argument

We study a problem with initial conditions on the half-line for a differentialdifference equation of the hyperbolic type with deviations of the time argument. We obtain sufficient conditions for the well-posed solvability of the problem in Sobolev spaces with an exponential weight. In terms of the spectrum of the problem operator, we obtain necessary conditions for the well-posed solvability of the problem, sufficient conditions for the absence of solutions, and sufficient conditions for the nonuniqueness of the solution.     

On the Cauchy problem of degenerate hyperbolic equations

In this paper, we study a class of degenerate hyperbolic equations and prove the existence of smooth solutions for Cauchy problems. The existence result is based on a priori estimates of Sobolev norms of solutions. Such estimates illustrate a loss of derivatives because of the degeneracy.

     

On a first boundary value problem for hyperbolic equations in the plane

In the paper we study a boundary value problem for a hyperbolic equation with two independent variables; this problem is a generalization of the well-known Darboux problem. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 294–306, February, 1999.     

Multipoint problem for hyperbolic equations with variable coefficients

By using the metric approach, we study the problem of classical well-posedness of a problem with multipoint conditions with respect to time in a tube domain for linear hyperbolic equations of order 2n (n ≥ 1) with coefficients depending onx. We prove metric theorems on lower bounds for small denominators appearing in the course of the solution of the problem.     

On a spatial problem of Darboux type for a second-order hyperbolic equation

The theorem of unique solvability of a spatial problem of Darboux type in Sobolev space is proved for a second-order hyperbolic equation.