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We consider the multidimensional dimensional inhomogeneous Landau–Lifshitz–Gilbert (ILLG) equation and its degenerate case, the Schrödinger map equation. We investigate the special solutions (under large initial values) and their energy property of the ILLG and Schrödinger map equations. Until now, we had not seen a paper presenting an explicit dynamic solution of the multidimensional ILLG. Using the stereographic method, an equivalent equation of ILLG is obtained. Based on this equivalent system, we obtain some exact solutions of the ILLG equation and present some implicit solutions of the Schrödinger map equation. Based on these solutions, by a careful estimation we give the decay rate of energy density.

Energy decay rate of multidimensional inhomogeneous Landau–Lifshitz–Gilbert equation and Schrödinger map equation on the sphere