Some blowup solutions about two systems derived from Landau–Lifshitz–Gilbert equation

Two different systems which can be regard as the extreme cases of Landau-Lifshitz–Gilbert equation are considered. When the gyromagnetic term of Landau–Lifshitz–Gilbert equation vanishes away, we consider some special solutions for a modified harmonic map heat flow which map from (2+1)(2+1)-dimensional space–time into the 2-sphere. The existence of regular initial data leading to blow up in finite time is established. When the Gilbert term is omitted, a blowup solution is obtained by constructing a blowup solution of Landau–Lifshitz equation.