Second-order hyperbolic equations with strong characteristic degeneracy at the initial hypersurface

Equations of the following form are considered:
\begin{equation} \psi^2(t,x)u_{tt}+\varphi(t,x)u_t-\sum_{i,j}\bigl(a^{ij}(t,x)u_{x_i}\bigr)_{x_j}+\sum_ib^i(t,x)u_{x_i}+c(t,x)u=f(t,x), \tag{1} \end{equation}
where
\begin{gather*} (t,x)\in H=(0,T)\times\mathbb R^n, \qquad \psi(t,x)\geqslant 0, \qquad \varphi(t,x)\geqslant0; \\ \sum_{i,j}a^{ij}(t,x)\xi_i\xi_j\geqslant0 \quad \forall\,(t,x)\in H, \quad \forall\,\xi=(\xi_1,\dots,\xi_n)\in\mathbb R^n. \end{gather*}

In place of the Cauchy problem for (1), a problem without initial data but with constraints on the admissible growth of the solution as $t\to0$ and as $|x|\to\infty$ is discussed. The unique solubility of (1) in certain Sobolev-type weighted spaces is proved. The smoothness properties of generalized solutions are studied.