  Under the validity of the positive mass theorem, the Yamabe flow on a smooth
compact Riemannian manifold of dimension $N \ge 3$ is known to exist for all
time $t$ and converges to a solution to the Yamabe problem as $t \to \infty$.
We prove that if a suitable perturbation, which may be smooth and arbitrarily
small, is imposed on the Yamabe flow on any given Riemannian manifold $M$ of
dimension $N \ge 5$, the resulting flow may blow up at multiple points on $M$
in the infinite time. Our proof is constructive, and indeed we construct such a
flow by using solutions of the Yamabe problem on the unit sphere $\mathbb{S}^N$
as blow-up profiles. We also examine the stability of the blow-up phenomena
under a negativity condition on the Ricci curvature at blow-up points.
