Boundary value problem with integral conditions and conjunction conditions on the non-characteristic plane (Q2768775)

The author obtains on the set \(G=G_1\cup G_2\), NEWLINE\[NEWLINE\begin{aligned} G_1&=\left\{(x,y,z): 0<x<z<a,\;0<y<b\right\},\\ G_2&=\left\{(x,y,z): 0<z<x<a, 0<y<b\right\}\end{aligned}NEWLINE\]NEWLINE continuous on \(G_1\) and \(G_2\) unique solution of the equation NEWLINE\[NEWLINEu_{xyz}+a_1u_{xy}+a_2u_{xz}+a_3u_{yz}+b_1u_{x}+b_2u_{y}+b_3u_{z}+cu=0,NEWLINE\]NEWLINE that satisfies conditions \(\int_{x}^{a}u(x,y,t) dt=f_1(x,y)\), \(\int_{0}^{x}u(x,y,t) dt=f_2(x,y)\), \(0\leq x\leq a\), \(0\leq y\leq b\), \(\int_{0}^{\beta}u(x,t,z) dt=g(x,z)\), \(0\leq x,z\leq a\), \(0\leq \beta\leq b\), and conjunction conditions NEWLINE\[NEWLINE\begin{aligned} \lim_{z-x\to+0}u(x,y,z)&=\alpha_1(x,y)\lim_{z-x\to-0}u(x,y,x)+\delta(x,y),\\ \lim_{z-x\to+0}{\partial u\over\partial \bar n} &= \alpha_2(x,y)\lim_{z-x\to-0}{\partial u\over\partial \bar n}+\rho(x,y),\\ \lim_{z-x\to+0}{\partial^2 u\over\partial {\bar n}^2} &= \alpha_2(x,y)\lim_{z-x\to-0}{\partial^2 u\over\partial {\bar n}^2}+\rho(x,y).\end{aligned}NEWLINE\]NEWLINE Here \({\partial u\over\partial \bar n}={\partial u\over\partial x}-{\partial u\over\partial z}\), \({\partial^2 u\over\partial {\bar n}^2}= {\partial^2 u\over\partial x^2}+{\partial^2 u\over\partial z^2}-2{\partial^2 u\over\partial x\partial z}\).