Let $\phi_0\colon S^n\to \mathbb{R}^{n+1}$ be the embedding of the standard Riemannian $n$-sphere in $\mathbb{R}^{n+1}$.
By a general result of Eells and Sampson, the solution $\phi_t$ of the harmonic map heat flow starting at $\phi_0$, exists for all $t\in [0,\infty)$ and converges to a constant map $\phi_{\infty}$ (since $\mathrm{Ric}_{S^n}>0$, $\mathbb{R}^{n+1}$ is flat, and a map to $\mathbb{R}^{k}$ is harmonic iff it is constant).
Questions: Is there an explicit formula for $\phi_t$? I am interested in how $\phi_t$ shrinks $\phi_t(S^n)$ to a point. Is it possible to rescale the flow so that the volume or diameter of $\phi_t(S^n)$ is constant?