Solution of a modified Cauchy problem by the Riemann method for a certain spatial analog of the Euler-Darboux equation with a negative parameter (Q5930626)

The equation \[ u_{xyz}- (\gamma/(z+ y+ x)) u_{xy}= 0,\qquad 0<\gamma<1,\tag{1} \] in the domain \(G\), bounded by the planes \(z= x- y\), \(x= h\), \(y= 0\), and \(z=0\) is considered. A solution to equation (1) with the conditions \[ u(x,y,z)|_{z= x-y-0}= \tau(x, y),\qquad y\leq x\leq h,\quad 0\leq y\leq h, \] \[ u_y|_{z= x-y-0}= \omega(x, y),\quad (x- y- z)^{-\gamma} u_{xy}|_{z= x-y-0}= \nu(x,y),\;y< x< h,\;0< y< h, \] is given explicitly. In the proof, the Riemann function of equation (1) is used. Since the coefficient of equation (1) and the Riemann function become infinite on the plane \(z= x-y\) the pyramid \(G\) is approximated by pyramids \(H_\varepsilon\), \(\varepsilon> 0\).