Long time behaviour of leafwise heat flow for Riemannian foliations

For any Riemannian foliation F on a closed manifold M with an arbitrary bun dle-like metric, leafwise heat flow of differential forms is proved to pres erve smoothness on M at infinite time. This result and its proof have conse quences about the space of bundle-like metrics on M, about the dimension of the space of leafwise harmonic forms, and mainly about the second term of the differentiable spectral sequence of F.