The paper deals with the initial value problem with zero Dirichlet boundary data for $$ u_t = u^p \Delta u \quad \mbox{in } \Omega \times (0,\infty) $$ with $p \ge 1$. The behavior of positive solutions near the boundary is discussed and significant differences from the case of the heat equation ($p=0$) and the porous medium equation ($p \in (0,1)$) are found. In particular, for $p \ge 1$ there is a large class of initial data for which the corresponding solution will never enter the cone $\{ v: \Omega \to \R \ | \ \exists \, c>0: \ v(x) \ge c \dist(x,\rO) \}$.\\ Finally, for $p>2$ a solution $u$ with $u(t) \in C_0^\infty(\Omega) \ \forall \, t \ge 0$ is constructed.