The heat flow for sections of a Riemannian fiber bundle (Q2764825)

The author studies the vertical heat equation NEWLINE\[NEWLINE \tau^v(w_t)={\partial w_t\over \partial t}, \qquad w_0=u, NEWLINE\]NEWLINE for a map \(w:{\mathbb R}\times M\to N,\) where \((N,M,\pi)\) is a Riemannian fiber bundle and \(W_t:=w(t,\cdot).\) It is proved that if the fibers of \((N,M,\pi)\) have nonpositive curvature, the equation admits a global solution defined on \({\mathbb R}_+\times M\) which converges uniformly as \(t\to \infty\) to a harmonic section.