In this paper, the blow-up of solutions of ordinary differential equations, which are deduced from the equation of equivariant harmonic maps, is studied. Its direct consequence is the non-existence or existence result of equivariant harmonic maps between warped product manifolds. As another application we prove the non-existence of a harmonic map from an Euclidean space to a Hadamard manifold with a certain nondegeneracy condition at infinity, provided sectional curvatures of the Hadamard manifold are bounded from above by a slowly decaying negative function of the distance from a fixed point.
"Blow-up solutions for ordinary differential equations associated to harmonic maps and their applications." J. Math. Soc. Japan 53 (2) 485 - 500, April, 2001. https://doi.org/10.2969/jmsj/05320485