Solvability of a nonlocal problem for a hyperbolic equation with degenerate integral conditions

Abstract

In this paper, we consider a nonlocal problem with integral conditions for hyperbolic equation. Close attention focuses on degenerate integral conditions, namely, on the second kind integral conditions which degenerate into the first kind conditions at some points. Such kind of nonlocal conditions inevitably involves some specific difficulties when we try to show solvability of the problem. These difficulties can be overcome by a method suggested in our paper. The essence of this method is the reduction of the problem with degenerate conditions to the problem with dynamical conditions. This technique enables to define effectively a generalized solution to the problem, to obtain a priori estimates and to prove the existence of a unique generalized solution to the problem.

Keywords

• hyperbolic equation
• nonlocal problem
• 1st and 2d kind integral conditions
• degenerate nonlocal conditions
• dynamical boundary conditions
• generalized solution
• sobolev space