Estimates for a solution to one differential inequality

In a domain $D=\Omega\times(-T,T)$ we consider a differential inequality whose left-hand side contains a linear second-order hyperbolic operator with coefficients depending only on $x\in\mathbb{R}^n$, $n\geqslant2$, and the right-hand side contains the modulus of the gradient of the sought function. We supplement the inequality with the Cauchy data on the lateral part of the boundary of $D$ and consider the problem of estimating a solution to the differential inequality satisfying the Cauchy data. We establish the estimate under some assumptions that involves the upper bound of the sectional curvatures of the Riemannian space associated with the differential operator, the Riemannian diameter of $\Omega$, and the length of the interval $(-T,T)$. The result is generalized to the case of compact domains bounded from above and below by characteristic surfaces.