Summary: We consider the problem \(u_ t (x,t) = \Delta u^ m (x,t)\) for \((x,t) \in D \times [0,+ \infty)\), \(u(x,0) = u_ 0 (x)\) for \(x \in D\), and \((\partial u^ m/ \partial n) (x,t) = h(x,t)\) for \((x,t) \in \partial D \times [0, + \infty)\). Here we assume \(D \subset \mathbb{R}^ N\), \(m>1\), \(u_ 0 \geq 0\), and \(h \geq 0\). It is well known that solutions to this problem have the property of finite speed propagation of the perturbations. By this we mean that if \(z\) is an interior point of \(D\) and exterior to the support of \(u_ 0\), then there exists a time \(T(z)>0\) so that \(u(z,t) = 0\) for \(t < T(z)\) and \(u(z,t) > 0\) for \(t > T(z)\). In this note we give, in an elementary way, an upper bound for \(T(z)\) for the case of bounded convex domains and in the case of a half space.