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In this paper, we elucidate the problem of gravitating Skyrmion governed byfield equations of the Einstein-Skyrme system with no potential term in theBondi coordinate. The spherical symmetry has to be assumed and both the metricfunctions and Skyrme ansatz depend on radial and retarded time coordinateswhich implies that the system is dynamic. We show that unique smooth solutionswith arbitrary initial data exist for a certain time interval with the extracondition that the time interval must be finite in order to have a non-zerotopological charge. Then, the strategy to show that global smooth solutionsexist is to restrict the initial data and proceed to find suitableconfigurations for extending the time interval to infinity. We also discuss thepossible configurations within The Einstein-Skyrme System which developsingularities in coordinate origin.

Global Existence and Singularity Formation of Classical Solutions in Einstein-Skyrme System