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Let $N$ be a compact Riemannian manifold. A quasi-harmonic sphere on $N$ is a harmonic map from $({ℝ}^m,e^{{\left|x\right|}^2/2(m-2)}/d{s}_0^2)$ to $N$ ($m\ge 3$) with finite energy ([LnW]). Here $ds{2}_0$ is the Euclidean metric in ${ℝ}^m$. Such maps arise from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target $N$. We also derive gradient estimates and Liouville theorems for positive...