Global existence and singularity formation of classical solutions in spherically symmetric Einstein–Skyrme system

In this paper, we elucidate the problem of gravitating skyrmion governed by the field equations of the Einstein–Skyrme system with no potential term in the Bondi coordinate. The spherical symmetry is assumed and both the metric functions and Skyrme ansatz depend on the radial and retarded time coordinates implying that the system is dynamic. We show that unique global smooth solutions of the Einstein–Skyrme system with topological charge [Formula: see text], that are assumed to be monotonically decreasing with respect to the radial coordinate and infinitely differentiable at the neighborhood of the origin, exist if we constrain the initial data and the coupling constant must fall in certain finite intervals that are deduced from the sufficient conditions of fixed-point theorem. We also discuss the possible configurations within the Einstein–Skyrme system which develop singularities in coordinate origin.