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An initial value approach to rotationally symmetric harmonic maps. (English) Zbl 1109.58021

Summary: We study the effect of the varying ${y}^{\text{'}}\left(0\right)$ on the existence and asymptotic behavior of solutions for the initial value problem

$\left\{\begin{array}{c}{y}^{\text{'}\text{'}}\left(r\right)+\left(n-1\right)\frac{{f}^{\text{'}}\left(r\right){y}^{\text{'}}\left(r\right)}{f\left(r\right)}-\left(n-1\right)\frac{g\left(y\left(r\right)\right){g}^{\text{'}}\left(y\left(r\right)\right)}{f{\left(r\right)}^{2}}=0,\hfill \\ y\left(0\right)=0,\hfill \end{array}\right\$

where $f$ and $g$ are some prescribed functions. Global solutions of this ODE on $\left[0,\infty \right)$ represent rotationally symmetric harmonic maps, with possibly infinite energies, between certain class of Riemannian manifolds. By studying this ODE, we show among other things that (i) all rotationally symmetric harmonic maps from ${ℝ}^{n}$ to the hyperbolic space ${ℍ}^{n}$ blow up in a finite interval; (ii) all such harmonic maps from ${ℍ}^{n}$ to ${ℝ}^{n}$ are bounded; and (iii) a trichotomy phenomenon occurs for such harmonic maps from ${ℍ}^{n}$ into itself, viz., they blow up in a finite interval, are the identity map, or are bounded according as the initial value ${y}^{\text{'}}\left(0\right)<1$, $=1$, or $>1$. Finally when $n=2$, the above equation can be solved exactly by quadrature method. Our results supplement those of A. Ratto and M. Rigoli [J. Differ. Equations 101, No. 1, 15–27 (1993; Zbl 0767.34029)] and A. Tachikawa [Tokyo J. Math. 11, No. 2, 311–316 (1988; Zbl 0686.58013)].

##### MSC:
 58E20 Harmonic maps between infinite-dimensional spaces 34A05 Methods of solution of ODE 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions