Increasing smoothness of solutions to a hyperbolic system on the plane with delay in the boundary conditions

Under consideration is a mixed problem in the half-strip $\Pi=\{(x,t)\colon0<x<1,\ t>0\}$ for a first order homogeneous linear hyperbolic system with delay in $t$ in the boundary conditions. We study the behavior of the Laplace transform of a solution to this problem for the large values of the complex parameter. The boundary conditions are found under which the smoothness of a solution to the corresponding mixed problem increases with $t$.