A modified Cauchy problem for an inhomogeneous equation of degenerate hyperbolic type of the second kind

In this study, a modified Cauchy problem was examined for an inhomogeneous equation of degenerate hyperbolic type of the second kind in a characteristic triangle. It is known that degenerate hyperbolic equations have a singularity, meaning that the well-posedness of the Cauchy problem with initial data on the line of parabolic degeneracy does not always hold for them. Therefore, in such cases, it is necessary to consider the problem with initial conditions in a modified form.
In present paper, modified Cauchy problems with initial conditions were formulated on the line of parabolic degeneracy for an inhomogeneous equation of degenerate hyperbolic type of the second kind. The considered problem is reduced to a modified Cauchy problem for a homogeneous equation and to a Cauchy problem for an inhomogeneous equation with zero initial conditions. The solutions of the modified Cauchy problem for a homogeneous equation are derived from the general solution of the considered equation. The explicit solutions of the modified Cauchy problem with homogeneous conditions for the inhomogeneous equation are found using the Riemann method.
It is proven that the discovered solutions indeed satisfy the equation and the initial conditions.