Existence and maximal regularity of solutions in $L_2(\mathbb{R}^2)$ for a hyperbolic type differential equation with quickly growing coefficients

In this paper the problem of the existence of solutions is studied for a hyperbolic type differential equation defined in an unbounded domain. The problem of the smoothness of solutions is also considered here. Such problems are of particular interest when the coefficients are unbounded. The novelty of the work is that the weighted coercive estimate is obtained for the solutions of a hyperbolic type differential equation with quickly growing coefficients.