Finite Speed of Propagation for the Porous Media Equation

We consider the Cauchy--Dirichlet problem for the equation \[ u_{t} = \Delta u^{m}, \quad m>1, \quad {\rm on} \ D=\Re^{N}_{k} \times ( t>0), \] where $\Re^{N}_{k} = \Re^{N} \cap \left\{x_{1},...,x_{k}>0 \right\}, \ 1 \leq k \leq N, \ N \geq 1$. Sharp bounds of the interface (or free boundary) are obtained. We use a weighted energy method, which allows us to consider more general equations.