On a Phragmen–Lindelöf type theorem in the strip

Let $u(x,t)$ be a solution of the equation $\frac{\partial^2u(x,t)}{\partial t^2}+Q\left(\frac{\partial}{\partial x}\right)u(x,t)=0$ in the strip $\Pi(T)=\left\{(x,t):x\in\mathbb R\land t\in [0,T]\right\}$, where $Q(s)$ is an arbitrary polynomial with respect to $s\in\mathbb C$ with constant complex coefficients. In the paper the dependence of the behavior of $u(x,t)$ on the functions
$$ u_1(x)=u(x,0), \quad u_2(x)=\frac{\partial u(x,T)}{\partial t} $$
or
$$ u_1(x)=\frac{\partial u(x,0)}{\partial t},\quad u_2(x)=u(x,T), $$
at infinity is studied.