Global existence of Landau–Lifshitz–Gilbert equation and self-similar blowup of Harmonic map heat flow on [formula omitted

Abstract

Under the plane wave setting, the existence of small Cauchy data global solution (or local solution) of Landau–Lifshitz–Gilbert equation is proved. Some variable separation type solutions (include some small data global solution) and self-similar type solutions are constructed for the Harmonic map heat flow on S2. As far as we know, there is not any literature that presents the exact blowup solution of this equation. Some explicit solutions which include some finite time gradient-blowup solutions are provided. These blowup examples indicate a finite time blowup will happen in any spacial dimension.