Correctly solvable boundary value problems in an infinite layer for systems of linear partial differential equations

We study classes of correct solvability of boundary value problems for systems of linear equations with constant coefficients of the form $\frac{\partial\bar u(x,t)}{\partial t}=P\bigl(\frac\partial{\partial x}\bigr)\bar u(x,t)$ in the layer $R^m\times[0,T]$ with boundary conditions consisting in prescribing certain components of the vectors $\bar u(x,0)$ and $\bar u (x,T)$ for $x\in R^m$.