Global existence and blow-up solutions of the radial Schrödinger maps

Abstract

This paper studies the Cauchy problem of radial inhomogeneous Schrödinger maps (ISM) which arises from the integrable model of the inhomogeneous spherically symmetric Heisenberg ferromagnetic spin system. Through a complex transformation the radial ISM is equivalent to an integro-differential Schrödinger equation. A new weighted Sobolev space [Formula: see text] is introduced and the well-posedness of integro-differential Schrödinger equations, including the integral radial IMS, with small spherically symmetric initial data in one-dimensional energy space [Formula: see text] is established. Furthermore, for [Formula: see text], we prove the existence of blow-up solutions for the integral radial ISM.