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We consider a version of the Goursat problem for a previously unstudied third-order equation and prove the existence and uniqueness of the solution of this problem. [ABSTRACT FROM AUTHOR]

By reduction to Goursat problems, we obtain various versions of conditions ensuring that the solution of the system in question can be constructed in closed form. [ABSTRACT FROM AUTHOR]

We consider a kind of Goursat problem for a fourth-order equation in a form not studied yet and prove the existence and uniqueness of the solution of this problem. [ABSTRACT FROM AUTHOR]

Six new versions of solvability conditions for the Goursat problem in quadratures are formulated in terms of the coefficients of a linear inhomogeneous hyperbolic equation. [ABSTRACT FROM AUTHOR]

For a generalized fractional telegraph equation, we study an analog of the Goursat problem and construct the Riemann function. [ABSTRACT FROM AUTHOR]

For a hyperbolic equation of a form earlier unstudied, we consider two problems with normal derivatives in the boundary conditions. For each of these problems, we prove the existence of a unique reduction to the Goursat problem. [ABSTRACT FROM AUTHOR]

We present necessary and sufficient conditions for the factorization of equations with dominating partial derivatives and illustrate the possibility of using our results by examples of two forms of equations of the second and third orders. [ABSTRACT FROM AUTHOR]

We prove a theorem on the reduction of the Goursat problem in integral form to an equivalent system of two Fredholm integral equations of the second kind and a uniqueness theorem for its solution. We suggest a method for reducing nonlocal characteristic and mixed problems, including the Samarskii problem for a second-order differential equation, to local problems for partial differential equations of higher order. We present examples illustrating the importance of the condition ensuring the uniqueness of the solution of the problems in question. [ABSTRACT FROM AUTHOR]

We develop a modification of the energy inequality method and use it to prove the well-posedness of the Goursat problem for linear second-order hyperbolic differential equations with operator coefficients whose domains depend on the two-dimensional time. An energy inequality for strong solutions of this Goursat problem is derived with the help of abstract smoothing operators, and we prove that the range of the problem is dense by using properties of a regularizing Cauchy problem whose inverse operator is a family of smoothing operators of a new type. We give an example of a well-posed boundary value problem for a two-dimensional complete second-order hyperbolic partial differential equation with Goursat conditions and with a boundary condition depending on the two-dimensional time. [ABSTRACT FROM AUTHOR]

We consider the nonlinear vector Goursat-Darboux problem of general form with a control on the right-hand side in the differential equation. We consider the case in which its solution necessarily belongs to the class of functions with p-integrable (for some p > 1) mixed and first partial derivatives. In this connection, we use the fact that the first derivatives of such a solution are elements of spaces with a mixed norm; these spaces are much narrower than L. Sufficient conditions for the preservation of the global solvability of the problem under control perturbations are proved. [ABSTRACT FROM AUTHOR]

In a rectangular domain on the plane, we consider the Dirichlet problem for a fourthorder pseudoparabolic equation with double differentiation with respect to each of the variables. To solve the problem, we reduce it to a system of Fredholm equations, whose solvability is established under additional conditions on the coefficients of the equation by the method of a priori estimates. [ABSTRACT FROM AUTHOR]