Landau-Lifshitz-Bloch equation on Riemannian manifold

In this article, we bring in Landau-Lifshitz-Bloch(LLB) equation on $m$-dimensional closed Riemannian manifold and prove that it admits a unique local solution. In addition, if $m⩾3$ and $L^{\mathrm{\infty }}-$norm of initial data is sufficiently small, the solution can be extended globally. Moreover, if $m=2$, we can prove that the unique solution is global without assuming small initial data.