Nonlocal Initial-Boundary Value Problems for a Degenerate Hyperbolic Equation

The paper considers initial-boundary value problems for the degenerate hyperbolic equation \(y^m u_{xx}-u_{yy}-b^2 y^m u=0\) in the rectangular domain \(\Omega =\{(x,y): 0<x_1_ _="" where="" _m="">0\) , \(b\ge 0\) , \(T>0\) are given real numbers. We study problems with classical initial conditions \(u(x,0)=\tau (x)\) , \(u_y(x,0)=\nu (x)\) , \(0\le x \le 1\) , and nonlocal boundary conditions \(u_x(0,y)=\alpha u_x(1,y)\) , \(u(0,y)=u(1,y)\) , or \(u_x(0,y)=u_x(1,y)\) , \(u(0,y)= \beta u(1,y)\) with \(0\le y\le T\) . Using the method of spectral analysis, we prove uniqueness and existence theorems for solutions of these problems.