Characteristic problem for a fourth-order equation with a dominant derivative

In this article we consider the Goursat problem for an equation with a dominating fourth-order mixed derivative and prove its unique solvability. The equation under consideration can be interpreted as a generalized Boussinesq — Love equation, which arises when describing longitudinal waves in a rod, taking into account transverse deformations. To justify the solvability, we proposed a method that is based on the possibility of reducing the problem posed to two Goursat problems for second-order equations. One of the problems is the classical Goursat problem for the simplest hyperbolic equation, while the other equation is loaded, and the study of the Goursat problem for it is the main result of the work.