The exponentially harmonic heat flow on Riemannian manifolds and gradient estimates

Suppose that M is a complete Riemannian manifolds with nonnegative sectional curvature. We prove that for the exponentially harmonic heat flow (3) on bounded regular domain with the Dirichlet initial-boundary value data, there exists a unique global solution. We prove that for any bounded solution of the exponentially harmonic function heat flow on M, there is a gradient estimate. As a consequence of this estimate, we derive the Liouville type theorem for bounded ancient solutions to exponentially harmonic function heat flow on M. We also obtain Liouville type results for the exponentially harmonic functions with finite weighted $L^2$ norms.