Correct solvability of boundary value problem in cylindric domain (Q1897931)

In the infinite cylinder the following boundary value problem is considered \[ {\partial^2\over \partial t_1^2} u(x, t)+ {\partial^2\over \partial t^2_2} u(x, t)= P\Biggl({\partial\over \partial x}\Biggr) u(x, t),\;t^2_1+ t^2_2\leq r^2_0,\;x\in \mathbb{R}^n,\tag{1} \] \[ u(x,t)\Bigl|_{t^2_1+ t^2_2= r^2_0}= u_0(x, t).\tag{2} \] Under various assumptions on the polynomial \(P(- is)\), classes of functions are introduced in which (1), (2) is properly posed.