On the Solvability of the Multidimensional Version of the First Darboux Problem for a Model Second-Order Degenerating Hyperbolic Equation

A multidimensional version of the first Darboux problem is considered for a model second-order degenerating hyperbolic equation. Using the technique of functional spaces with a negative norm, the correct formulation of this problem in the Sobolev weighted space is proved.     

First Darboux problem for nonlinear hyperbolic equations of second order

We study the first Darboux problem for hyperbolic equations of second order with power nonlinearity. We consider the question of the existence and nonexistence of global solutions to this problem depending on the sign of the parameter before the nonlinear term and the degree of its nonlinearity. We also discuss the question of local solvability of the problem.     

The Cauchy-Goursat problem for wave equations with nonlinear dissipative term

The Cauchy-Goursat problem for wave equations with nonlinear dissipative term is studied. The existence, uniqueness, and blow-up of global solutions of this problem are considered. The local solvability of this problem is also discussed.     

On the solvability of one boundary value problem for one class of higher-order semilinear hyperbolic systems

Lithuanian Mathematical Journal - In this paper, we consider the boundary value problem for one class of higher-order semilinear hyperbolic systems. We prove the theorems on the existence,...     

On the solvability of a Darboux type non-characteristic spatial problem for the wave equation

The question of the correct formulation of a Darboux type non-characteristic spatial problem for the wave equation is investigated. The correct solvability of the problem is proved in the Sobolev space for surfaces of the temporal type on which Darboux type boundary conditions are given.     

On the Solvability of a Mixed Problem for an One-dimensional Semilinear Wave Equation with a Nonlinear Boundary Condition

In this paper, for an one-dimensional semilinear wave equation we study a mixed problem with a nonlinear boundary condition. The questions of uniqueness and existence of global and blow-up solutions of this problem are investigated, depending on the nonlinearity nature appearing both in the equation and in the boundary condition.     

Investigation of hyperbolic systems with order degeneration arising in I. Vekua's hierarchical models

We study the well-posedness of an initial-boundary value problem corresponding to the zeroth approximation of I. Vekua's hierarchical models for elastic cusped prismatic shells. The mathematical model is described by a two-dimensional order-degenerating hyperbolic system. We formulate the problem in the weak setting and prove the uniqueness and existence theorems. We show that the sequence of corresponding explicit Galerkin approximations converges to the exact solution in an appropriate weighted Lebesgue space.     

On the solvability of a boundary value problem for nonlinear wave equations in angular domains

For a one-dimensional wave equation with a weak nonlinearity, we study the Darboux boundary value problem in angular domains, for which we analyze the existence and uniqueness of a global solution and the existence of local solutions as well as the absence of global solutions.     

On the Existence,Uniqueness, and Nonexistence of Solutions of One Boundary-Value Problem for a Semilinear Hyperbolic Equation

Ukrainian Mathematical Journal - We consider a boundary-value problem for a semilinear hyperbolic equation with iterated multidimensional wave operator in the principal part. The theorems on...