Cauchy problem for a degenerate general linear equation of hyperbolic type
Abstract
The paper introduces class <i>R</i><sub>1</sub> of generalized solutions of the Cauchy problem for a general linear equation of hyperbolic type with two lines of degeneracy. The Riemann function <i>V</i>(<i>ξ, η; ξ', η'</i>) is constructed for the considered general equation, and estimates for this Riemann function and its derivatives with respect to <i>ξ</i>, <i>η</i>, are established. The unique solvability of the Cauchy problem for the studied equation is also proved. The existence of a solution for the initial problem is proven via Riemann's method. In the concluding part of the article, examples of given functions satisfying the conditions of the existence and uniqueness theorem for the studied problem are presented. Moreover, the work sets forth the main goals of the research and substantiates the actuality of the problem under consideration.