On correct solvability of a boundary value problem in an infinite slab for linear equations with constant coefficients

Conditions depending on the properties of the polynomials $P(s)$ and $Q(s)$ are found for the correct solvability of the boundary value problem
\begin{gather*} \frac{\partial^2u(x,t)}{\partial t^2}+P\left(\frac\partial{\partial x}\right)\frac{\partial u(x,t)}{\partial t}+Q\left(\frac\partial{\partial x}\right)u(x,t)=0,\\ u(x,0)=u_0(x),\qquad u(x,T)=u_T(x) \end{gather*}
($x\in R_m$, $t\in[0,T]$; $P(s)$ and $Q(s)$ are polynomials in $s_1,\dots,s_m$ with constant coefficients) in various classes of functions.