We give a classification of rotationally symmetric $p$-harmonic maps between some model spaces such as $\mathbb{R}^n$ and $\mathbb{H}^n$ by their asymptotic behaviors. Among others, we show that, when $p>2$ and $n≥q 2$, all rotationally symmetric $p$-harmonic maps from $\mathbb{R}^n$ to $\mathbb{H}^n$ have to blow up at a finite point, while all rotationally symmetric $p$-harmonic maps from $\mathbb{H}^n$ to $\mathbb{H}^n$ observe the trichotomy property, i.e. the map $y$ is the identity map, is bounded or blows up according as its initial value $y'(0)$ is equal to, less than or greater than one. Our sharp estimates imply and improve a number of existence and non-existence results of certain $p$-harmonic maps on noncompact manifolds.
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