A problem without initial conditions for linear degenerate hyperbolic equations of second order with infinite domain of dependence

The equation
$$ \psi^2(t,x)u_{tt}+\varphi(t,x)u_t-M\biggl(t,x,\frac{\partial}{\partial x}\biggr)u=f(t,x) $$
is considered on the strip $H=(0,T]\times\mathbf R_x^n$. Here $M$ is a linear elliptic operator of the second order, and $\psi$ and $\varphi$ are nonnegative on $H$ and have a zero at least of the first order on a hyperplane $t=0$. Hence for $t=0$ we cannot give the initial values. Precise restrictions on the growth of the desired function for $|x|\to\infty$ are found guaranteeing the existence and uniqueness of a generalized solution of the problem without initial conditions.
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