Sergo Kharibegashvili

FINITE DIFFERENCE SOLUTION OF A NONLINEAR KLEIN-GORDON EQUATION WITH AN EXTERNAL SOURCE

In this paper, we consider the Darboux problem for a (1+1)-dimensional cubic nonlinear Klein-Gordon equation with an external source. Stable finite difference scheme is constructed on a four-point stencil, which does not require additional iterations for passing from one level to another. It is proved, that the finite difference scheme converges with the rate O(h 2 ), when the exact solution belongs to the Sobolev space W 2 2 .

Second Darboux problem for the wave equation with a power-law nonlinearity

For the one-dimensional wave equation with a power-law nonlinearity, we consider the second Darboux problem and study the existence and uniqueness of a global solution, the existence of local solutions, and the absence of global solutions.

On the existence and absence of global solutions of the first Darboux problem for nonlinear wave equations

For the one-dimensional wave equation with a power-law nonlinearity, we consider the first Darboux problem, for which we study issues related to the existence and absence of local and global solutions.

Cauchy problem for a generalized nonlinear liouville equation

We consider the Cauchy problem for a generalized Liouville equation. We study the existence, uniqueness, and absence of a global solution of this problem. We also discuss the local solvability of the problem.

On the Cauchy and Cauchy–Darboux problems for semilinear wave equations

Abstract The Cauchy and Cauchy–Darboux problems for semilinear wave equations in the class of continuous functions are investigated. The questions of existence, uniqueness and nonexistence of global solutions of the problems are considered. The local solvability of the problems is also discussed.

On the Global and Local Solution of the Multidimensional Darboux Problem for Some Nonlinear Wave Equations

Abstract We consider a multidimensional analogue of the Darboux problem for wave equations with power nonlinearity. Depending on the spatial dimension of an equation, a power nonlinearity exponent and the sign in front of a nonlinear term, it is proved that the Darboux problem is globally solvable in some cases, but has no global solution in other cases though the local solvability of this problem remains in force.

On solvability of a periodic problem for a nonlinear telegraph equation

The time-periodic problem is studied for a nonlinear telegraph equation with the Dirichlet–Poincaré boundary conditions. The questions are considered of existence and smoothness of solutions to this problem.

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.

Time-periodic problem for a weakly nonlinear telegraph equation with directional derivative in the boundary condition

We study a time-periodic problem for the wave equation with a power-law nonlinearity and with a directional derivative in the boundary condition. We study the existence, uniqueness, and absence of solutions of the problem.

The Cauchy-Darboux problem for the one-dimensional wave equation with power nonlinearity

The questions are studied of existence and uniqueness of a global solution to the Cauchy-Darboux problem for the one-dimensional wave equation with power nonlinearity. Under consideration are the existence of local solutions and the absence of global solutions.