Finite Time Blow-up for Heat Flows of Self-induced Harmonic Maps

Abstract

Let Mⁿ be an embedded closed submanifold of ℝ^k+1 or a smooth bounded domain in ℝⁿ, where n ≥ 3. We show that the local smooth solution to the heat flow of self-induced harmonic map will blow up at a finite time, provided that the initial map u₀ is in a suitable nontrivial homotopy class with energy small enough.