Let $(M,g)$ be a closed smooth Riemannian manifold, and let $(N,h)$ be a compact smooth Riemannian manifold whose sectional curvature satisfies $K_N \leq 0$. For every smooth map $u_0: M \to N$, there exists a smooth map
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\begin{align*}
u: M \times [0,\infty) &\to N
\end{align*}
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such that $u(\cdot,0)=u_0$ and $u$ solves the harmonic map heat flow equation
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\begin{align*}
\partial_t u = \tau_g(u)
\end{align*}
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on $M \times (0,\infty)$, where $\tau_g(u)$ denotes the tension field of $u$ with respect to $g$ and $h$.